P}^{\frac{n-2k}{2}+t}\times{\mathbb{R}}\\!{\rm
P}^{\frac{n-2k}{2}-t}\subset\bar{X}_{a\times\dot{{a}}}$. Take a small generic
(formal) alteration of the mapping $d_{X}$, restricted on this submanifold;
consider the polyhedron ${\bf N}(t)$ of iterated self-intersections of this
mapping, which are near ${\bf N}_{b\times\dot{{b}}}$. The image of the
fundamental class $[{\bf N}(t)]$ by the mapping (126) (after the 4-ordered
transfer) is the following element in $D^{loc}_{n-8k}({\bf
I}_{a\times\dot{{a}}};{\mathbb{Z}}/2[{\mathbb{Z}}])$, which is calculate as
$\varkappa_{a}^{\frac{n-8k}{2}+2t}\otimes\varkappa_{\dot{{a}}}^{\frac{n-8k}{2}-2t}$.

######  Lemma 23.

There exists a formal mapping, which satisfies Definition 22.

#### Proof of Lemma 23

A required deformation (formal) is defined by 3 steps (see Definition 22). On
the firs step intersection polyhedron of the bottom covering (121) is
deformed, on the second step self-intersection polyhedron of the top covering
(120) is deformed. Moreover, over a marked point on ${\mathbb{R}}\\!{\rm
P}^{2}$ the deformation may be a Cartesian product of the coordinate
deformations, described in Lemma 21. The deformation is vertical along
bisectors 2-planes in the normal bundle $D^{2}\rtimes_{\chi}D^{2}$ and is
equivariant with respect to the involution $\chi$. The extension over
${\mathbb{R}}\\!{\rm P}^{2}$ is possible, because of this property. By a
result of the first 2 steps of the deformation an iterated self-intersection
is splitted into two components.

The deformation of the third step is also vertical and is much smaller and a
decomposition of the iterated self-intersection polyhedron into two
subpolyhedra by the first two steps are keept. The goal of this step is to
eleminate control polyhedra $P$, $P^{\prime}$, where $P$ is an obstruction for
integer reduction (88) of the structure (90) for interior self-intersection,
the control polyhedron $P$ is an obstruction for exterior self-intersection.

The goal of this deformation is to make selection polyhedra $P_{1},P_{2}$
empty. This deformation (formal) keeps $\chi$-symmetry of the formal mapping.
The definition $P$, $P^{\prime}$ and an estimation of its dimension is
possible by a generalisation of Lemma 33 for ramified mapping with bi-cyclic
structure. The dimension $dim(P)$ is not greater then
$\frac{n-(2k-4)}{2}+d=\frac{n}{2}-(k-2)+d$, where $d=2$ is the denominator of
a defect of the standardization.

The dimension $\frac{n}{2}-(k-4)+d$ of double self-intersection inside $P$ by
a vertical generic deformation increases by $4k-8$ (because we get two pairs
of sheets, the codimension of a sheet equals to the dimension of the
deformation $2k-4$).

Because we get a restriction of dimension of internal self-intersections, the
controle polyhedron $P^{\prime\prime}$ for external self-intersections, which
is inside $P^{\prime}$, increses its codimension by $2k-4$, because a negative
contribution of its universal dimension. By a vertical deformation the
dimension of $P^{\prime\prime}\subset P^{\prime}$ decreases by $2k-4$. As the
result, the dimension of iterated self-intersection is
$\delta=\frac{n}{2}-9k+20+d$. In the case $n=254$ we have $\delta=127-9\cdot
15+20+d=12+d$. This calculation is not critical, because for Theorem 18 there
is an extra codimension $4$ of the manifold $N$ inside the source space of the
mapping $d^{(2+2)}_{X}$.

The analogous calculations one should apply for iterated self-intersection
with other order for two pairs of points in a $4$-uple. On the double self-
intersection of formal deformed mapping $d^{(2+2)}$ for all configurations of
points control polyhedra is empty.

Properties of the image of the fundamental class $[{\bf N}(t)]$ by the
structuring mapping follow from Lemma 20.

The component ${\bf N}_{\circ}$ of the iterated self-intersection polyhedron
of the resulting deformation mapping $\bar{d}^{(2+2)}\mapsto d^{(2+2)}$
contains only real residues. One formulated, see Definition 17, and one may
prove that this component is negligible. Lemma 23 is proved. ∎

### Proof of Theorem 18

The proof is analogous to the proof of Theorem 5. Additionally, the the
fundamental class $[]N]$ belongs to the subgroup (106) with coeffisients
modulo $2$, see Definition 15, because of the retraction condition. Integer
coefficients on this step is not possible, because $k=7$ is odd and the
manifold $N^{n-2k}$ is, generally speaking, non-orientable.

Differences with respect to Theorem 5 are the following. Now, a formal
deformation of a regular approximation is required, this formal deformation is
induced by the formal deformation, described in Lemma 23.

Consider a standardized ${\bf D}$-framed immersion $(g,\Psi,\eta_{N})$, where
$g$ is closed to the composition of the structuring mapping $N\to X_{{\bf
H}_{a\times\dot{{a}}}}\operatorname{\rtimes}_{\chi}{\mathbb{R}}\\!{\rm P}^{2}$
with the mapping (123) and with the embedding (171). Then a formal deformation
of the immersion $g$, which is induced by the formal deformation
$\bar{d}\mapsto d$ is considered. The bi-cyclic component of the iterated
self-intersection is defined as the component in a neighbourhood of a regular
domain of the component ${\bf N}_{b\times\dot{{b}}}$.

The structured group of the characteristic mapping has the required reduction,
as it is followed from Statement 2 in Definition 22. The characteristic number
in Statement 2 Definition 22 is restricted using the collection of
characteristic number for the polyhedra ${\bf N}(t)$ (see the last part of
Definition 22). ∎

##  7 ${\bf Q}\times{\mathbb{Z}}/4$–structure (quaternionic-cyclic structure)
on self-intersection manifold of a standardized ${\mathbb{Z}}/2^{[4]}$–framed
immersion

Let us recall the definition of the quaternionic subgroup ${\bf
Q}\subset{\mathbb{Z}}/2^{[3]}$, which contains the subgroup ${\bf
J}_{b}\subset{\bf Q}$.

Let us define the following subgroups:

$\displaystyle i_{{\bf J}_{a}\times{\dot{\bf J}}_{a},{\bf
Q}\times{\mathbb{Z}}/4}:{\bf J}_{a}\times{\dot{\bf J}}_{a}\subset{\bf
Q}\times{\mathbb{Z}}/4,$ (127)

$\displaystyle i_{{\bf Q}\times{\mathbb{Z}}/4}:{\bf
Q}\times{\mathbb{Z}}/4\subset{\mathbb{Z}}/2^{[5]},$ (128)

$\displaystyle i_{{\bf J}_{a}\times{\dot{\bf J}}_{a}\times{\mathbb{Z}}/2}:{\bf
J}_{a}\times{\dot{\bf J}}_{a}\times{\mathbb{Z}}/2\subset{\mathbb{Z}}/2^{[5]}.$
(129)

Define the subgroup $(\ref{QZ4})$. Define the epimorphism ${\bf
J}_{b}\times{\dot{\bf J}}_{b}\to{\mathbb{Z}}/4$ by the formula $(x\times
y)\mapsto xy$. The kernel of this epimorphism coincides with the antydiagonal
subgroup ${\dot{\bf I}}_{b}=antidiag({\bf J}_{b}\times{\dot{\bf
J}}_{b})\subset{\bf J}_{b}\times{\dot{\bf J}}_{b}$, and this epimorphism
admits a section, the kernel is a direct factor (the subgroup ${\bf
J}_{b}\subset{\bf J}_{b}\times{\dot{\bf J}}_{b}$). This kernel is mapped onto
the group ${\mathbb{Z}}/4$ by the formula: $(x\times x^{-1})\mapsto x$. The
subgroup $(\ref{QZ4})$ is well-defined.

Let us define subgroups $(\ref{iQaa})$, $(\ref{iaaa})$. Consider the basis
$({\mathfrak{h}}_{1,+},{\mathfrak{h}}_{2,+},{\mathfrak{h}}_{1,-},{\mathfrak{h}}_{2,-},{\bf\dot{h}}_{1,+},{\bf\dot{h}}_{2,+},{\bf\dot{h}}_{1,-},{\bf\dot{h}}_{2,-})$
of the space ${\mathbb{R}}^{8}$, which is defined by the formulas $(\ref{h})$,
$(\ref{hh})$.

Let us define an analogous basis of the space ${\mathbb{R}}^{16}$. This basis
contains $16$ vectors, the basis vectors are divided into two subset $16=8+8$.

$\displaystyle{\mathfrak{h}}_{1,\ast,\ast\ast},{\mathfrak{h}}_{2,\ast,\ast\ast};$
(130)

$\displaystyle{\bf\dot{h}}_{1,\ast,\ast\ast},{\bf\dot{h}}_{2,\ast,\ast\ast}.$
(131)

where the symbols $\ast$, $\ast\ast$ takes values $+,-$ independently.

Let us define the subgroup $(\ref{iQaa})$. The representation $i_{{\bf
Q}\times{\mathbb{Z}}/4}$ is given such that the generator ${\bf j}$ of the
quaternionic factor ${\bf Q}\subset{\bf Q}\times{\mathbb{Z}}/4$ acts in each
$4$-dimensional subspace from the following list:

$\displaystyle
diag(Lin({\mathfrak{h}}_{1,\ast,\ast\ast},{\mathfrak{h}}_{2,\ast,\ast\ast},{\mathfrak{h}}_{1,\ast,-\ast\ast},{\mathfrak{h}}_{2,\ast,-\ast\ast}),$
(132)
$Lin({\bf\dot{h}}_{1,\ast,\ast\ast},{\bf\dot{h}}_{2,\ast,\ast\ast},{\bf\dot{h}}_{1,\ast,-\ast\ast},{\bf\dot{h}}_{2,\ast,-\ast\ast})),$

$\displaystyle
diag(Lin({\mathfrak{h}}_{1,-\ast,\ast\ast},{\mathfrak{h}}_{2,-\ast,\ast\ast},{\mathfrak{h}}_{1,-\ast,-\ast\ast},{\mathfrak{h}}_{2,-\ast,-\ast\ast}),$
(133)
$Lin({\bf\dot{h}}_{1,-\ast,\ast\ast},{\bf\dot{h}}_{2,-\ast,\ast\ast},{\bf\dot{h}}_{1,-\ast,-\ast\ast},{\bf\dot{h}}_{2,-\ast,-\ast\ast})),$

$\displaystyle
antidiag(Lin({\mathfrak{h}}_{1,\ast,\ast\ast},{\mathfrak{h}}_{2,\ast,\ast\ast},{\mathfrak{h}}_{1,\ast,-\ast\ast},{\mathfrak{h}}_{2,\ast,-\ast\ast}),$
(134)
$Lin({\bf\dot{h}}_{1,\ast,\ast\ast},{\bf\dot{h}}_{2,\ast,\ast\ast},{\bf\dot{h}}_{1,\ast,-\ast\ast},{\bf\dot{h}}_{2,\ast,-\ast\ast})),$

$\displaystyle
antidiag(Lin({\mathfrak{h}}_{1,-\ast,\ast\ast},{\mathfrak{h}}_{2,-\ast,\ast\ast},{\mathfrak{h}}_{1,-\ast,-\ast\ast},{\mathfrak{h}}_{2,-\ast,-\ast\ast}),$
(135)
$Lin({\bf\dot{h}}_{1,-\ast,\ast\ast},{\bf\dot{h}}_{2,-\ast,\ast\ast},{\bf\dot{h}}_{1,-\ast,-\ast\ast},{\bf\dot{h}}_{2,-\ast,-\ast\ast}))$

by the standard matrix, which is defined in the standard basis of the
corresponding space. Each of $4$-space, described above, corresponds to one of
the pair of spaces ${\mathbb{R}}^{2}_{b,\ast}$, or, to one of the space
${\mathbb{R}}^{2}_{\dot{{b}},\ast}$, which is defined below the formulas
$(\ref{h})$, $(\ref{hh})$.

The generator ${\bf i}\in{\bf Q}$ acts in the direct sum of the two exemplars
of the corresponding space as the generator of the group ${\bf J}_{b}$ by the
corresponding matrix. The generator of the factor ${\mathbb{Z}}/4\subset{\bf
Q}\times{\mathbb{Z}}/4$ acts of the direct sum of the two exemplars of the
corresponding space as the generator of the group $antidiag({\bf
J}_{b}\times{\dot{\bf J}}_{b})\subset{\bf J}_{b}\times{\dot{\bf J}}_{b}$. The
representation $(\ref{iQaa})$ is well-defined.

Let us define the representation $(\ref{iaaa})$ as following. The factor ${\bf
J}_{b}\times{\dot{\bf J}}_{b}\subset{\bf J}_{b}\times{\dot{\bf
J}}_{b}\times{\mathbb{Z}}/2$ is represented in each $4$-dimensional subspace
$(\ref{11})$-$(\ref{14})$ by the formula $(\ref{iaa})$, which is applied
separately to standard basis of each spaces.

The factor ${\mathbb{Z}}/2\subset{\bf J}_{b}\times{\dot{\bf
J}}_{b}\times{\mathbb{Z}}/2$ is represented

–in 8-dimensional subspace, which is defined as the direct sum of the
subspaces $(\ref{11})$, $(\ref{13})$ by the identity transformation.

–in 8-dimensional subspace, which is defined as the direct sum of the
subspaces $(\ref{12})$, $(\ref{14})$ by the central symmetry.

The representation $(\ref{iaaa})$ is well-defined.

On the group ${\bf Q}\times{\mathbb{Z}}/4$ define the automorphism
$\chi^{[5]}$ of the order 4. This automorphism on the subgroup $(\ref{QZ4})$
is defined as the restriction of the automorphism $\chi^{[4]}$. The extension
of $\chi^{[4]}$ from the subgroup $\chi^{[5]}$ to the group is defined by the
simplest way: the automorphism $\chi^{[5]}$ keeps the generator${\bf j}$. It
is easy to see that the automorphism with such property exists and uniquely.

Consider the projection

$\displaystyle p_{{\bf Q}}:{\bf Q}\times{\mathbb{Z}}/4\to{\bf Q}$ (136)

on the first factor. The kernel of the homomorphism $p_{{\bf Q}}$ coincides
with the antidiagonal subgroup ${\dot{\bf I}}_{b}\subset{\bf
J}_{b}\times{\dot{\bf J}}_{b}\subset{\bf Q}\times{\mathbb{Z}}/4$. Evidently,
the following equality is satisfied:on the first factor. The kernel of the
homomorphism $p_{{\bf Q}}$ coincides with the antidiagonal subgroup ${\dot{\bf
I}}_{b}\subset{\bf J}_{b}\times{\dot{\bf J}}_{b}\subset{\bf
Q}\times{\mathbb{Z}}/4$.

Evidently, the following equality is satisfied:

$\displaystyle p_{{\bf Q}}\circ\chi^{[5]}=p_{{\bf Q}}.$ (137)

Analogously, on the group ${\bf J}_{b}\times{\dot{\bf
J}}_{b}\times{\mathbb{Z}}/2$ define the authomorphim $\chi^{[5]}$ of the order
2 (this new automorphism denote the same). Define the projection

$\displaystyle p_{{\mathbb{Z}}/4\times{\mathbb{Z}}/2}:{\bf
J}_{a}\times{\dot{\bf
J}}_{a}\times{\mathbb{Z}}/2\to{\mathbb{Z}}/4\times{\mathbb{Z}}/2,$ (138)

the kernel of this projection coincides with the diagonal subgroup

Obviously, the following formula is satisfied:the kernel of this projection
coincides with the diagonal subgroup ${\dot{\bf I}}_{b}\subset{\bf
J}_{b}\times{\dot{\bf J}}_{b}\times{\mathbb{Z}}/2$. Obviously, the following
formula is satisfied: $\chi^{[5]}\circ
p_{{\mathbb{Z}}/4\times{\mathbb{Z}}/2}=p_{{\mathbb{Z}}/4\times{\mathbb{Z}}/2}$.

This allos to define analogously with $(\ref{chi2Z})$, $(\ref{chi3Z})$,
$(\ref{chi4Z})$ the groups

$\displaystyle({\bf
Q}\times{\mathbb{Z}}/4)\operatorname{\rtimes}_{\chi^{[5]}}{\mathbb{Z}},$ (139)
$\displaystyle({\bf J}_{a}\times{\dot{\bf
J}}_{a}\times{\mathbb{Z}}/2)\operatorname{\rtimes}_{\chi^{[5]}}{\mathbb{Z}},$
(140)

as semi-direct products of the corresponding groups with automorphisms with
the group ${\mathbb{Z}}$.

Let us define the epimorphism:

$\displaystyle\omega^{[5]}:({\bf
Q}\times{\mathbb{Z}}/4)\operatorname{\rtimes}_{\chi^{[5]}}{\mathbb{Z}}\to{\bf
Q},$ (141)

the restriction of this epimorphism on the subgroup $(\ref{QZ4})$ coincides
with the epimorphism $(\ref{prQ})$. For this definition use the formula
$(\ref{chi5com})$ and define $z\in Ker(p_{{\bf Q}})$, where $z\in{\mathbb{Z}}$
is the generator.

Evidently, the epimorphism

$\displaystyle\omega^{[5]}:({\bf J}_{a}\times{\dot{\bf
J}}_{a}\times{\mathbb{Z}}/2)\operatorname{\rtimes}_{\chi^{[5]}}{\mathbb{Z}}\to{\mathbb{Z}}/4\times{\mathbb{Z}}/2,$
(142)

analogously is well-defined, denote this automorphism as the automorphism
$(\ref{pchi5Z})$.

In the group ${\mathbb{Z}}/2^{[5]}$ let us define the involution, which is
denoted by $\chi^{[5]}$ as on the resolution group. In the standard basis of
the spaces $(\ref{11})$-$(\ref{14})$ the authomorphism $\chi^{[5]}$ is defined
by the same formulas as $\chi^{[4]}$, the each considered space is a proper
space for $\chi^{[5]}$. This definition implies that $\chi^{[5]}$ is commuted
with the representations $(\ref{iQaa})$, $(\ref{iaaa})$.

Moreover, the following homomorphisms

$\displaystyle\Phi^{[5]}:({\bf
Q}\times{\mathbb{Z}}/4)\operatorname{\rtimes}_{\chi^{[5]}}{\mathbb{Z}}\to{\mathbb{Z}}/2^{[5]},$
(143) $\displaystyle\Phi^{[5]}:({\bf J}_{b}\times{\dot{\bf
J}}_{b}\times{\mathbb{Z}}/2)\operatorname{\rtimes}_{\chi^{[5]}}{\mathbb{Z}}\to{\mathbb{Z}}/2^{[5]},$
(144)

are analogously to $(\ref{Phi4})$ well-defined, they are included into the
following commutative diagrams $(\ref{a,Qaa})$, $(\ref{a,aaa})$.

$\displaystyle\begin{array}[]{ccc}\qquad({\bf J}_{a}\times{\dot{\bf
J}}_{a})\operatorname{\rtimes}_{\chi^{[4]}}{\mathbb{Z}}&\stackrel{{\scriptstyle\Phi^{[4]}\times\Phi^{[4]})}}{{\longrightarrow}}&\qquad{\mathbb{Z}}/2^{[4]}\times{\mathbb{Z}}/2^{[4]}\\\
i_{{\bf J}_{a}\times{\dot{\bf J}}_{a},{\bf
Q}\times{\mathbb{Z}}/4}\downarrow&&i_{[5]}\downarrow\\\ \qquad({\bf
Q}\times{\mathbb{Z}}/4)\operatorname{\rtimes}_{\chi^{[5]}}{\mathbb{Z}}&\stackrel{{\scriptstyle\Phi^{[5]}}}{{\longrightarrow}}&\qquad{\mathbb{Z}}/2^{[5]},\\\
\end{array}$ (148)

In this diagram the left vertical homomorphism

$i_{{\bf J}_{b}\times{\dot{\bf J}}_{b},{\bf Q}\times{\mathbb{Z}}/4}:({\bf
J}_{b}\times{\dot{\bf
J}}_{b})\operatorname{\rtimes}_{\chi^{[4]}}{\mathbb{Z}}\to({\bf
Q}\times{\mathbb{Z}}/4)\operatorname{\rtimes}_{\chi^{[5]}}{\mathbb{Z}}$

is induced by the homomorphism $(\ref{QZ4})$, the right vertical homomorphism

$i_{[5]}:{\mathbb{Z}}/2^{[4]}\times{\mathbb{Z}}/2^{[4]}\subset{\mathbb{Z}}/2^{[5]}.$

is the inclusion of the subgroup of the index 2.

The following diagram

$\displaystyle\begin{array}[]{ccc}\qquad({\bf J}_{a}\times{\dot{\bf
J}}_{a})\operatorname{\rtimes}_{\chi^{[4]}}{\mathbb{Z}}&\stackrel{{\scriptstyle\Phi^{[4]}\times\Phi^{[4]}}}{{\longrightarrow}}&\qquad{\mathbb{Z}}/2^{[4]}\times{\mathbb{Z}}/2^{[4]}\\\
i_{{\bf J}_{a}\times{\dot{\bf J}}_{a},{\bf J}_{a}\times{\dot{\bf
J}}_{a}\times{\mathbb{Z}}/2}\downarrow&&i_{[5]}\downarrow\\\ \qquad({\bf
J}_{a}\times{\dot{\bf
J}}_{a}\times{\mathbb{Z}}/2)\operatorname{\rtimes}_{\chi^{[5]}}{\mathbb{Z}}&\stackrel{{\scriptstyle\Phi^{[5]}}}{{\longrightarrow}}&\qquad{\mathbb{Z}}/2^{[5]},\\\
\end{array}$ (152)

the left vertical homomorphism

$i_{{\bf J}_{a}\times{\dot{\bf J}}_{a},{\bf J}_{a}\times{\dot{\bf
J}}_{a}\times{\mathbb{Z}}/2}:({\bf J}_{a}\times{\dot{\bf
J}}_{a})\operatorname{\rtimes}_{\chi^{[4]}}{\mathbb{Z}}\to({\bf
J}_{a}\times{\dot{\bf
J}}_{a}\times{\mathbb{Z}}/2)\operatorname{\rtimes}_{\chi^{[5]}}{\mathbb{Z}}$

is an inclusion.

######  Lemma 24.

The homomorphism (143) is extended to the homomorphism (81) from the subgroup
${\bf J}_{b}\times{\dot{\bf
J}}_{b}\rtimes_{\mu_{b\times\dot{{b}}}^{(4)}}{\mathbb{Z}}\times{\mathbb{Z}}\rtimes_{\chi^{[4]}}{\mathbb{Z}}$
of the index 2 on the group

$({\bf
Q}\times{\mathbb{Z}}/4)\rtimes_{\mu_{b\times\dot{{b}}}^{(5)}}{\mathbb{Z}}\times{\mathbb{Z}}\operatorname{\rtimes}_{\chi^{[5]}}{\mathbb{Z}}.$

### Proof of Lemma 24

Let us construct the extension

$({\bf Q}\times{\bf
Q})\rtimes_{\mu_{b\times\dot{{b}}}^{(5)}}{\mathbb{Z}}\times{\mathbb{Z}}\operatorname{\rtimes}_{\chi^{[5]}}{\mathbb{Z}},$

then let us pass to the required subgroup of the index $2$.

The automorphism $\mu_{b\times\dot{{b}}}^{(5)}$ is induced from the
automorphism $\mu_{{\bf Q}}:{\bf Q}\to{\bf Q}$ of the factors: $\mu_{{\bf
Q}}({\bf i})=-{\bf i}$, $\mu_{{\bf Q}}({\bf j})=-{\bf j}$, $\mu_{{\bf Q}}({\bf
k})={\bf k}$. ∎

Additionally, one has to consider, as this authomorphism is defined on
$3$-dimensional quaternionic lens $S^{3}/{\bf Q}$ (the quaternionic action on
$S^{3}$ is on the right).

The automorphism is given by the transformation of $S^{3}$, which is the right
multiplication of the quaternion unite. The automorphism $\mu_{{\bf Q}}$ is
commuted with the transformation of the quaternion units (the only non-obvious
calculation concerns the unite ) ${\bf k}$). Therefore on the quotient
$S^{3}/{\bf Q}$ the automorphism is well-defined.

The trivialisation of the tangent bundle on the quaternionic lens, which is
defined by the left action of the units $({\bf i},{\bf j},{\bf k})$, is
changed by corresponding automorphisms. Therefore the image of the quaternion
framing by the differential

$\displaystyle d(\mu_{{\bf Q}}):T(S^{3}/{\bf Q})\to T(S^{3}/{\bf Q})$ (153)

is fibred isotopic to the identity. This isotopy consists of a family of
rotations trough the angle $180^{0}$ in oriented planes, which are orthogonal
to the vectors ${\bf k}$.

The homomorphism (144) is extend to the homomorphism of the Laurent extension
from the subgroup ${\bf J}_{b}\times{\dot{\bf
J}}_{b}\rtimes_{\mu_{b\times\dot{{b}}}^{(4)}}{\mathbb{Z}}\times{\mathbb{Z}}\rtimes_{\chi^{[4]}}{\mathbb{Z}}$
of the index $2$ to the all group:

$\displaystyle({\bf J}_{b}\times{\dot{\bf
J}}_{b}\times{\mathbb{Z}}/2)\rtimes_{\mu_{b\times\dot{{b}}}^{(5)}}{\mathbb{Z}}\times{\mathbb{Z}}\operatorname{\rtimes}_{\chi^{[5]}}{\mathbb{Z}}\mapsto{\mathbb{Z}}/2^{[5]}.$
(154)

######  Definition 25.

Let us say that a ${\mathbb{Z}}/2^{[4]}$-framed immersion $(g,\Psi,\eta_{N})$
of the codimension $8k$, which is standardized in the sense of Definition 15,
is an immersion with ${\bf Q}\times{\mathbb{Z}}/4$-structure (with a
quaternion-cyclic structure), if $g$ is self-intersects along a
${\mathbb{Z}}/2^{[5]}$-framed immersion $(h,\Xi,\zeta_{L})$ with the self-
intersection manifold $L$ of the dimension $n-16k$ (for $k=7$, $n=126$ we have
$n-16k=14$).

Additionally, let us assume that $(g,\Psi,\eta_{N})$ is a pre-standardised
immersion, see Definition 15. Then the fundamental class $[\bar{L}]$ of the
canonical covering over the manifold $L$ of self-intersection,
$dim(L)=m_{\sigma}$, is mapped by the characteristic mapping into the element
of the dimension $m_{\sigma}$ in the subgroup (106).

######  Lemma 26.

Let $(g,\eta_{N},\Psi)$ a a pre-standardized ${\mathbb{Z}}/2^{[4]}$-framed
immersion in the codimension $8k$, see Definition 15. Then in a regular
homotopy class of this immersion there exists a standardised immersion with
quaternionic-cyclic structure, see Definition 25.

#### Proof of Lemma 26

Let us define the space

$\displaystyle ZZ_{{\bf J}_{a}\times{\dot{\bf
J}}_{a}}=S^{\frac{n+6}{2}+1}/{\bf i}\times S^{\frac{n+6}{2}+1}/{\bf i}$ (155)

Evidently, ${\operatorname{dim}}(ZZ_{{\bf J}_{a}\times{\dot{\bf
J}}_{a}})=n+8>n$.

Let us define a family $\\{Z_{0},\dots,Z_{j_{max}}\\}$ of standard
submanifolds in the manifold $ZZ_{{\bf J}_{a}\times{\dot{\bf J}}_{a}}$, where

$\displaystyle j_{max}=\frac{n+6}{8},$ (156)

by the following formula:

$\displaystyle Z_{0}=S^{\frac{n+12}{2}}\times S^{3}/{\bf i}\subset
S^{\frac{n+12}{2}}/{\bf i}\times S^{\frac{n+12}{2}}/{\bf i},$ (157)
$Z_{1}=S^{\frac{n+4}{2}}\times S^{7}/{\bf i}\subset S^{\frac{n+12}{2}}/{\bf
i}\times S^{\frac{n+12}{2}}/{\bf i},\dots$ $Z_{j}=S^{\frac{n+12-8j}{2}}/{\bf
i}\times S^{4j+3}/{\bf i}\subset S^{\frac{n+12}{2}}/{\bf i}\times
S^{\frac{n+12}{2}}/{\bf i},\dots$ $Z_{j_{max}}=S^{3}/{\bf i}\times
S^{\frac{n+12}{2}}/{\bf i}\subset S^{\frac{n+12}{2}}/{\bf i}\times
S^{\frac{n+6}{2}}/{\bf i}.$

In the formula $j_{max}$ is defined by the formula $(\ref{jmaxQ})$. A
subpolyhedron

$\displaystyle Z_{a\times\dot{{a}}}\subset ZZ_{{\bf J}_{a}\times{\dot{\bf
J}}_{a}}$ (158)

is defined as the union of submanifolds of the family $(\ref{Zj})$. Evidently,
${\operatorname{dim}}(Z_{{\bf J}_{a}\times{\dot{\bf J}}_{a}})=\frac{n+18}{2}$.

The standard involution, which is permuted the factors, is well-defined:

$\displaystyle\chi^{[4]}:ZZ_{{\bf J}_{a}\times{\dot{\bf J}}_{a}}\to ZZ_{{\bf
J}_{a}\times{\dot{\bf J}}_{a}},$ (159)

this involution is invariant on the subspace $Z_{a\times\dot{{a}}}$ (158).

Let us consider the standard cell-decomposition of the space $K({\bf
J}_{a}\times{\dot{\bf J}}_{a},1)=K({\bf J}_{a},1)\times K({\dot{\bf
J}}_{a},1)$, this cell-decomposition is defined as the direct product of the
cell-decomposition of the factors. The standard inclusion of the skeleton is
well-defined:

$\displaystyle Z_{a\times\dot{{a}}}\subset K({\bf J}_{a}\times{\dot{\bf
J}}_{a},1).$ (160)

The skeleton $(\ref{etaIaIaZ})$ is invariant with respect to the involution
$(\ref{ZZ3})$. Therefore the inclusion

$\displaystyle
Z_{a\times\dot{{a}}}\operatorname{\rtimes}_{\chi^{[4]}}S^{1}\subset
Z_{a\times\dot{{a}}}\operatorname{\rtimes}_{\chi^{[4]}}S^{1}\subset K(({\bf
J}_{a}\times{\dot{\bf
J}}_{a})\operatorname{\rtimes}_{\chi^{[4]}}{\mathbb{Z}},1)$ (161)

is well-defined.

Let us define a polyhedron $J_{Z}$, a subpolyhedron of the polyhedron $J_{X}$,
see the formula $(\ref{JX})$. Denote by $JJ_{{\bf J}_{a}}$ the join of
$j_{max}$ copies of the lence space $S^{3}/{\bf i}$. Denote by $JJ_{{\bf
J}_{a}}$ the join of $j_{max}$ copies of the lence space $S^{3}/{\bf i}$.

Denote by $JJ_{{\bf Q}}$ the join of $j_{max}$ copies of the quaternionic lens
space $S^{3}/{\bf Q}$, the second exemplar of this join. Recall that the
positive integer $j_{max}$ is defined by the formula $(\ref{jmaxQ})$.

Let us define the subpolyhedron $J_{\bar{Z},j}\subset JJ_{{\bf J}_{a}}\times
JJ_{{\dot{\bf J}}_{a}}$ by the formula

$J_{\bar{Z},j}=JJ_{{\bf J}_{a},j}\times JJ_{{\dot{\bf J}}_{a},j},1\leq i\leq
j_{max},$

where $JJ_{{\bf J}_{a},j}\subset JJ_{{\bf J}_{a}}$ is the subjoin with the
coordinates $1\leq i\leq j_{max}-j$, $JJ_{{\dot{\bf J}}_{a},j}\subset
JJ_{{\dot{\bf J}}_{a}}$ is the subjoin with the coordinates $1\leq i\leq j$.
Let us define the subpolyhedron $J_{Z,j}\subset JJ_{{\bf Q}}\times
JJ_{{\dot{\mathbf{Q}}}}$ by the formula:

$JJ_{Z,j}=JJ_{{\bf Q},j}\times JJ_{{\dot{\mathbf{Q}}},j},1\leq i\leq j_{max},$

where $JJ_{{\bf Q},j}\subset JJ_{{\bf Q}}$ is the subjoin with coordinates
$1\leq i\leq j_{max}-j$, $JJ_{{\dot{\mathbf{Q}}},j}\subset
JJ_{{\dot{\mathbf{Q}}}}$ is the subjoin with the coordinates $1\leq i\leq j$.

Define $J_{Z}$ by the formula:

$\displaystyle J_{Z}=\cup_{j=1}^{j_{max}}J_{Z,j}\subset JJ_{{\bf Q}}\times
JJ_{{\dot{\mathbf{Q}}}}.$ (162)

Define $J_{\bar{Z}}$ by the formula:

$\displaystyle J_{\bar{Z}}=\cup_{j=1}^{j_{max}}J_{\bar{Z},j}\subset JJ_{{\bf
J}_{a}}\times JJ_{{\dot{\bf J}}_{a}}.$ (163)

On the polyhedra $Z_{a\times\dot{{a}}}$, $J_{Z}$ a free involution, which is
denoted by $T_{{\bf Q}}$, and which is corresponded to the quadratic extension
$(\ref{QZ4})$ is well-defined. The polyhedron $(\ref{JZ})$ is invariant with
respect to the involution $T_{{\bf Q}}$, the inclusion of the factor-polyhedra
is defined by

$\displaystyle(J_{Z})/T_{{\bf Q}}=\cup_{j=1}^{j_{max}}J_{Z,j}/T_{{\bf
Q}}\subset JJ_{{\bf J}_{a}}/T_{{\bf Q}}\times JJ_{{\dot{\bf J}}_{a}}/T_{{\bf
Q}}.$ (164)

The standard $4$-sheeted covering (with ramifications)

$\displaystyle c_{Z}:Z_{a\times\dot{{a}}}\to J_{Z},$ (165)

is well-defined. This covering is factorized to the ramified covering
$\hat{c}_{Z}$. This covering is defined by the composition of the standard
$2$-sheeted covering

$\displaystyle Z_{a\times\dot{{a}}}/T_{{\bf Q}}\to(J_{Z})/T_{{\bf Q}}$ (166)

and the standard $2$-sheeted covering: $Z_{a\times\dot{{a}}}\to
Z_{a\times\dot{{a}}}/T_{{\bf Q}}$. The ramified covering $(\ref{nakr})$ is
equivariant with respect to the involution $(\ref{ZZ3})$, which acts in the
target and the source.

Let us consider the embedding $S^{3}/{\bf Q}\subset{\mathbb{R}}^{4}$, which is
constructed in [Hi]. The Cartesian product of joins of corresponding copies of
the embeddings gives the embedding

$\displaystyle J_{Z,j}\subset{\mathbb{R}}^{5j_{max}+3}.$ (167)

The following lemma is analogous to Lemma 21.

######  Lemma 27.

There exists an embedding

$\displaystyle\rm{i}_{J_{Z}}:J_{Z}\operatorname{\rtimes}_{\chi}S^{1}\subset{\bf
D}^{n-5}\times S^{1}\subset{\mathbb{R}}^{n}.$ (168)

The following extension of the polyhedron (161), which corresponds to the
Laurent extension of the structuring groups, as in Lemma 24, is well-defined.
Let us define a ramified covering:

$\displaystyle
c_{Z}:Z_{a\times\dot{{a}}}\rtimes_{\mu_{b\times\dot{{b}}}}S^{1}\times S^{1}\to
J_{Z}\rtimes_{\mu_{b\times\dot{{b}}}}S^{1}\times S^{1},$ (169)

which is an extension of the covering (165), as a parametrized covering over
$2$-dimensional torus. The factors of the torus acts on factors of the
coverings (165), correspondingly with the extension of the formula (153) on
the quaternion lens space.

The inclusion of the base of the covering (27) is extended to the following
immersion (which is denoted by the same):

$\displaystyle\rm{i}_{J_{Z}}:J_{Z}\operatorname{\rtimes}_{\mu_{b\times\dot{{b}}},\chi}S^{1}\looparrowright{\bf
D}^{n-5}\times S^{1}\times S^{1}\rtimes_{\chi}S^{1}\subset{\mathbb{R}}^{n}.$
(170)

An arbitrary immersion $g$ with a ${\bf Q}\times{\mathbb{Z}}/4$-structure is
defined as a small approximation of a mapping, which is the composition of the
mapping to the total space of the $2$-covering (169), the ramified covering
itself, and the immersion (170).

The decomposition to maximal strata of self-intersection points of the
ramified covering (169) determines a control of the image of the fundamental
class of self-intersection points of the immersion $g$ by the structuring
mapping. Cycles of the types 1,2 correspond to structuring groups of maximal
strata, cycles of type 3 can be exist, because (170) is an immersion and could
have additional closed component of a self-intersection. Lemma 26 is proved. ∎

In the following Lemma the main result is proved. This proof is analogous to
the proof in [A-P1].

######  Theorem 28.

Let $(g,\Psi,\eta_{N})$ ${\mathbb{Z}}/2^{[4]}$ be a standardised (see
Definition 15) immersion, with ${\bf Q}\times{\mathbb{Z}}/4$-structure (see
Definition 25). Let us assume that $(g,\Psi,\eta_{N})$ is pure (see Definition
16). Then $(g,\Psi,\eta_{N})$ is negligible.

#### Proof of Theorem 28

We use a non-standard modification of the Herbert formula with local
$\mu_{b\times\dot{{b}}},\chi$ coefficients valued in the group described in
Lemma 14, and for immersions with additional marked component, which gives an
extra contribution on the both sides of the Herbert formula. The Euler class
of the bundle (92) is an immersed submanifold in the codimension 2, possibly,
with double self-intersections. Denote this manifold by $N_{1}\looparrowright
N$.

Let us restrict the immersion $g$ on the submanifold $N_{1}\looparrowright N$
and let us consider the manifold $L_{1}\looparrowright L$ of self-intersection
points of the immersed manifold $g(N_{1})$.

The characteristic number, the image of the fundamental class $[\bar{L}_{1}]$
by the characteristic mapping $D_{m_{\sigma}}^{loc}({\bf J}_{b}\times{\dot{\bf
J}}_{b}\rtimes_{\mu_{b\times\dot{{b}}}}({\mathbb{Z}}\times{\mathbb{Z}});{\mathbb{Z}}[{\mathbb{Z}}/2])$,
is calculated using the Euler classes of the structured $4$-bundle over $L$
and the image $[\bar{L}]$, which is an element in $D_{m_{\sigma}}^{loc}({\bf
J}_{b}\times{\dot{\bf
J}}_{b}\rtimes_{\mu_{b\times\dot{{b}}}}({\mathbb{Z}}\times{\mathbb{Z}});{\mathbb{Z}}[{\mathbb{Z}}/2])$.
This calculation of also possible by the Herbert theorem, which is applied to
$[\bar{L}]$. The self-intersection points of the immersion
$N_{1}\looparrowright N$ give an extra contribution to self-intersection
points of $g|_{N_{1}}$. In the Herbert formula for $[\bar{L}_{1}]$ this extra
classes are cancelled in the both sides in the formula.

The characteristic classes takes values modulo 4. By arguments as in [A-P1],
this characteristic class in $D_{m_{\sigma}}^{loc}({\bf J}_{b}\times{\dot{\bf
J}}_{b}\rtimes_{\mu_{b\times\dot{{b}}}}({\mathbb{Z}}\times{\mathbb{Z}});{\mathbb{Z}}[{\mathbb{Z}}/2])$
is opposite to itself. Because $g$ is pure, this proves that the
characteristic number is even. Theorem 28 is proved. ∎

##  8 Classifying space for self-intersections of skew-framed immersions

#### Construction of axillary mappings $c:{\mathbb{R}}\\!{\rm
P}^{n-k}\to{\mathbb{R}}^{n}$, $c_{0}:{\mathbb{R}}\\!{\rm
P}^{n-k}\to{\mathbb{R}}^{n}$, which are used in Lemma 9, in Lemma 20

Let us denote by $J_{0}\subset{\mathbb{R}}^{n}$ the standard sphere of
dimension $(n-k)$ in Euclidean space. This sphere is $PL$ \- homeomorphic to
the join of $\frac{n-k+1}{2}=r_{0}$ copies of the circles $S^{1}$. Let us
denote by $i_{J_{0}}:J_{0}\subset{\mathbb{R}}^{n}$ the standard embedding.

Let us denote by

$\displaystyle i_{J}:J\subset{\mathbb{R}}^{n}$ (171)

the standard $(n-k)$-dimensional sphere of dimension $(n-k)$ in Euclidean
space. This sphere is $PL$ \- homeomorphic to the join of $\frac{n-k+1}{2}=r$
copies of the circles $S^{1}$.

Define the following mapping $p_{0}:S^{n-k}\to J_{0}$ as the join of $r$
copies of the standard 4-sheeted coverings $S^{1}\to S^{1}/{\bf i}$. The
standard antipodal action $-1\in{\bf I}_{d}\times S^{n-k}\to S^{n-k}$ (for the
group ${\bf I}_{d}$ and analogous denotations below see Section 2) commutes
with the mapping $p_{0}$. Therefore the ramified covering
$p_{0}:{\mathbb{R}}\\!{\rm P}^{n-k}\to J$ is well-defined. The mapping
$c_{0}:{\mathbb{R}}\\!{\rm P}^{n-k}\to{\mathbb{R}}^{n}$ is defined as the
composition $i_{J}\circ p_{0}$.

Replace the mapping $p_{0}$ defined above by the mapping $p$, the join of $r$
copies of $2$-sheeted coverings $S^{1}\to{\mathbb{R}}\\!{\rm P}^{1}$. The
ramified covering $p:{\mathbb{R}}\\!{\rm P}^{n-k}\to J$ is analogously well-
defined. The mapping $c:{\mathbb{R}}\\!{\rm P}^{n-k}\to{\mathbb{R}}^{n}$ is
the compopsition $i_{J}\circ p$.

### Deleted square

Let us consider the classifying space

$\displaystyle({\mathbb{R}}\\!{\rm P}^{n-k}\times{\mathbb{R}}\\!{\rm
P}^{n-k}\setminus\Delta_{{\mathbb{R}}\\!{\rm
P}^{n-k}})/T^{\prime}=\Gamma_{\circ},$ (172)

which is also called the "deleted product", or, the "deleted square" of the
space ${\mathbb{R}}\\!{\rm P}^{n-k}$. This space is defined by the factor on
the Cartesian product outside the diagonal $T^{\prime}:{\mathbb{R}}\\!{\rm
P}^{n-k}\times{\mathbb{R}}\\!{\rm P}^{n-k}\to{\mathbb{R}}\\!{\rm
P}^{n-k}\times{\mathbb{R}}\\!{\rm P}^{n-k}$, which permutes the coordinates.
The deleted product is an open manifold.

### The mapping $c$ as a formal mapping with a formal self-intersection

Denote by $T_{{\mathbb{R}}\\!{\rm P}^{n-k}\times{\mathbb{R}}\\!{\rm
P}^{n-k}}$, $T_{{\mathbb{R}}^{n}\times{\mathbb{R}}^{n}}$ the standard
involutions in the spaces ${\mathbb{R}}\\!{\rm
P}^{n-k}\times{\mathbb{R}}\\!{\rm P}^{n-k}$,
${\mathbb{R}}^{n}\times{\mathbb{R}}^{n}$, which permute the coordinates. Let

$\displaystyle d^{(2)}:{\mathbb{R}}\\!{\rm P}^{n-k}\times{\mathbb{R}}\\!{\rm
P}^{n-k}\to{\mathbb{R}}^{n}\times{\mathbb{R}}^{n}$ (173)

be an arbitrary $T_{{\mathbb{R}}\\!{\rm P}^{n-k}\times{\mathbb{R}}\\!{\rm
P}^{n-k}}$, $T_{{\mathbb{R}}^{n}\times{\mathbb{R}}^{n}}$–equivariant mapping,
which is transversal along the diagonal in the target. Denote the polyhedron
$(d^{(2)})^{-1}({\mathbb{R}}^{n}_{diag})/T_{{\mathbb{R}}\\!{\rm
P}^{n-k}\times{\mathbb{R}}\\!{\rm P}^{n-k}}$ by ${\bf N}={\bf N}(d^{(2)})$,
let us call this polyhedron a formal self-intersection polyhedron of the
mapping $d^{(2)}$. In the case when the formal mapping $d^{(2)}$ is a
holonomic mapping and is the formal extension of a mapping $d$ (the case
$d=c_{0}$ is required) the polyhedron ${\bf N}_{\circ}(d^{(2)})$ coincides
with the polyhedron of self-intersection points of $d$, which is defined by
the formula:

$\displaystyle{\bf N}_{\circ}(d)=Cl\\{([x,y])\in\Gamma_{\circ}:y\neq
x,d(y)=d(x)\\},$ (174)

where $\Gamma_{\circ}$ is the deleted square (172).

### The structuring mapping $\eta_{{\bf N}_{\circ}}:{\bf N}_{\circ}\to K({\bf
D},1)$

Let us define the mapping

$\displaystyle\eta_{\Gamma_{\circ}}:\Gamma_{\circ}\to K({\bf D},1),$ (175)

which is called the structuring mappiong.

Denote, that the inclusion $\bar{\Gamma}_{\circ}\subset{\mathbb{R}}\\!{\rm
P}^{n-k}\times{\mathbb{R}}\\!{\rm P}^{n-k}$ induces an isomorphism of the
fundamental groups, because the codimension of the diagonal
$\Delta_{{\mathbb{R}}\\!{\rm P}^{n-k}}\subset{\mathbb{R}}\\!{\rm
P}^{n-k}\times{\mathbb{R}}\\!{\rm P}^{n-k}$ equals $n-k$ and satisfies the
following inequality: $n-k\geq 3$. Therefore, the following equation is
satisfied:

$\displaystyle\pi_{1}(\bar{\Gamma}_{\circ})=H_{1}(\bar{\Gamma}_{\circ};{\mathbb{Z}}/2)={\mathbb{Z}}/2\times{\mathbb{Z}}/2.$
(176)

Let us consider the induced automorphism
$T^{\prime}_{\ast}:H_{1}(\bar{\Gamma}_{\circ};{\mathbb{Z}}/2)\to
H_{1}(\bar{\Gamma}_{\circ};{\mathbb{Z}}/2)$. This automorphism is not the
identity. Let us fix the standard isomorphism
$H_{1}(\bar{\Gamma}_{\circ};{\mathbb{Z}}/2)$ with ${\bf I}_{A\times\dot{A}}$,
this isomorphism maps the generator of the first factor (correspondingly, the
generator of the second factor) of the group
$H_{1}(\bar{\Gamma}_{\circ};{\mathbb{Z}}/2)$ (see (176)) to the element
$A=ab\in{\bf I}_{c}\subset{\bf D}$ (correspondingly, to the element
$\dot{A}=ba\in{\bf I}_{c}\subset{\bf D}$), which is represented in the
representation of ${\bf D}$ by the symmetry wioth respect to the second
(correspondingly, with respect to the first) coordinate axis.

It is easy to check that the authomorphism of external conjugation of the
subgroup ${\bf I}_{A\times\dot{A}}\subset{\bf D}$ on the element $b\in{\bf
D}\setminus{\bf I}_{A\times\dot{A}}$ (in this formula the element $b\in{\bf
I}_{b}$ is a generator), which is defined by the formula: $x\mapsto bxb^{-1}$,
is transformed by the considered isomorphism into the automorphism
$T^{\prime}_{\ast}$. The fundamental group $\pi_{1}(\Gamma_{\circ})$ is a
quadratic extension of the group $\pi_{1}(\bar{\Gamma})$ by the element $b$,
this extension is uniquely determined (up to an equivalence) by the
automorphism $T^{\prime}_{\ast}$. Therefore $\pi_{1}(\Gamma_{\circ})\simeq{\bf
D}$ and the mapping $\eta_{\Gamma_{\circ}}:\Gamma_{\circ}\to K({\bf D},1)$ is
well-defined.

The structuring mapping $\eta_{{\bf N}_{\circ}}:{\bf N}_{\circ}\to K({\bf
D},1)$ is defined by the restriction of $\eta_{\Gamma_{\circ}}$ on the open
subpolyhedron (174).

### A stratification of the sphere $J$

Let us ordered 1-dimensional lence spaces (circles), which are generators of
the join, by positive integers $1,\dots,r$, and denote by
$J(k_{1},\dots,k_{s})\subset J$ the subjoin, which is defined by the join of
the subcollection of circles $S^{1}/{\bf i}$ with the numbers $1\leq
k_{1}<\dots<k_{s}\leq r$, $0\geq s\geq r$. The stratification is induced by
the stratification of the standard $r$-dimensional simplex $\delta^{r}$ by the
projection $J\to\delta^{r}$. The inverse images of the vertexes of the simplex
are the lens spaces $J(j)\subset J$, $J(j)\approx S^{1}/{\bf i}$, $1\leq j\leq
r$, which are the factors of the join.

Let us define the space $J_{1}^{[s]}$ as a subspace in $J$ as the union of all
subspaces $J(k_{1},\dots,k_{s})\subset J$.

Therefore, the following stratification

$\displaystyle J^{(r)}\subset\dots\subset J^{(1)}\subset J^{(0)},$ (177)

of the space $J$ is well-defined. For a given stratum the number $r-s$ of
omitted coordinates is called the deep of the stratum.

Let us introduce the following denotations:

$\displaystyle J^{[i]}=J^{(i)}\setminus J^{(i+1)}.$ (178)

#### The stratification of ${\mathbb{R}}\\!{\rm P}^{n-k}$

Let us define the stratification of the projective space ${\mathbb{R}}\\!{\rm
P}^{n-k}$.Denote the maximal open cell $p^{-1}(J(k_{1},\dots,k_{s}))$ by
$U(k_{1},\dots,k_{s})\subset S^{n-k}/-1$. This cell is called an elementary
stratum of the deep $r-s$. A point on an elementary stratum
$U(k_{1},\dots,k_{s})\subset S^{n-k}/-1$ is defined by the collection of
coordinates $(\check{x}_{k_{1}},\dots,\check{x}_{k_{s}},l)$, where
$\check{x}_{k_{i}}$ is the coordinate on the circle, which is covered the
circle of the join with the number $k_{i}$, $l$ is a coordinate on
$(s-1)$-dimensional simplex of the join. Points on an elementary stratum
$U(k_{1},\dots,k_{s})$ belong to the union of subcomplexes of the join with
corresponding coordinates.

### A stratification of the polyhedron ${\bf N}_{\circ}$ (174)

The polyhedron ${\bf N}_{\circ}$, (174) is the disjoin union of elementary
strata, which are defined as inverce images of the strata $(\ref{strat[J]})$.
Let us denote tis strata by

$\displaystyle K^{[r-s]}(k_{1},\dots,k_{s}),\qquad 1\leq s\leq r.$ (179)

Let us consider exceptional antidiagonal strata on the anti-diagonal of the
deleted square $\Gamma_{\circ}$, this strata also are included into ${\bf
N}_{\circ}$. An open polyhedron, which is defined by the cutting of anti-
diagonal strata from ${\bf N}_{\circ}$ is denoted by $K_{\circ}\subset{\bf
N}_{\circ}$.

Let us describe an elementary stratum $K^{[r-s]}(k_{1},\dots,k_{s})$ by means
of coordinate system. For the simplicity of denotations, let us consider the
case $s=r$, this is the case of maximal elementary stratum. Assume a pair of
points $(x_{1}$, $x_{2})$ determines a point on $K^{[0]}(1,\dots,r)$, let us
fixes for the pair a pair of points $(\check{x}_{1},\check{x}_{2})$ on the
covering sphere $S^{n-k}$, which is mapped into the pair $(x_{1}$, $x_{2})$ by
the projection $S^{n-k}\to{\mathbb{R}}\\!{\rm P}^{n-k}$. With respect to
constructions above, denote by $(\check{x}_{1,i},\check{x}_{2,i})$,
$i=1,\dots,r$ the collection of spherical coordinates of each point. A
spherical coordinate determines a point on the circle with the same number
$i$, the circle is a covering over the corresponding coordinate lens
$J(i)\subset J$ in the join. Recall, that the pair of coordinates with a
common index determines a par in a fibre of the standard cyclic ${\bf
i}$-covering $S^{1}\to S^{1}/{\bf i}$.

Collections of coordinates $(\check{x}_{1,i},\check{x}_{2,i})$ are considered
up to antipodal transformations of the full collection into the antipodal
collection. Moreover, pairs of coordinates $(x_{1},x_{2})$ are non-ordered and
lifts of a point in $K$ to a pair of points $(\bar{x}_{1},\bar{x}_{2})$ on the
sphere $S^{n-k}$ are not uniquely defined. Therefore collection of coordinates
are defined up to $8$ different transformations. (Ap to transformation opf the
group ${\bf D}$ of the order $8$.)

Analogous statement is well-formulated for points in deeper elementary strata
$K^{[r-s]}(k_{1},\dots,k_{s})$, $1\leq s\leq r$.

### The involution $\tau_{c}$

An elementary stratum, except anti-diagonal stratum, is a 2-sheeted covering
with respect to a free involution, denoted by $\tau_{c}$ (see the formula
below) over the corresponding elementary stratum of a polyhedron
$\hat{K}_{\circ}=K_{\circ}/\tau_{c}$, a stratum of this polyhedron is denoted
by

$\displaystyle\hat{K}^{[r-s]}(k_{1},\dots,k_{s}),\qquad 1\leq s\leq r.$ (180)

Define the involution $\tau_{c}$ on the space $K_{\circ}$ by the formula:

$\tau_{c}:([x,y],\lambda)\mapsto([ix,iy],\lambda).$

This involution is free outside the antidiagonal (recall that by the
definition of $k_{\circ}$ the antidiagonal is removed) This is followed from
the coordinate description of elementary strata of $K_{\circ}$ (see below)),
coordinates are invariant by the involution $\tau_{c}$ and a factorization by
the involution is well-defined.

######  Lemma 29.

There exists a mapping $\eta_{c}$, which is included into the following
diagram, where vertical mappings are 2-sheeted coverings:

###### Proof.

In each elementary stratum, which consists of points $((x,y),\lambda)$ on the
sphere, let us define the action of the dihedral group ${\bf
D}=\\{a,b:[a,b]=b^{2};b^{4}=e\\}$, by the formulas:

$\displaystyle a$ $\displaystyle\colon((x,y),\lambda)\mapsto((y,x),\lambda)$
$\displaystyle b$ $\displaystyle\colon((x,y),\lambda)\mapsto((y,-x),\lambda).$

Define the group $C$ as the quadratic central extension of the group ${\bf D}$
by the element $c$, where $c^{2}=b^{2}$. Define the action of $C$, generated
by elements $a,b,c$, by the formulas:

$\displaystyle a$ $\displaystyle\colon((x,y),\lambda)\mapsto((y,x),\lambda)$
$\displaystyle b$ $\displaystyle\colon((x,y),\lambda)\mapsto((y,-x),\lambda)$
$\displaystyle c$
$\displaystyle\colon((x,y),\lambda)\mapsto((ix,iy),\lambda).$

This action is free and one may define $\hat{K}_{\circ}=K_{\circ}/\tau_{c}$.
The involution $\tau_{c}$, defined above, are given by the action on the
element $c\in C\setminus{\bf D}$. ∎

### Coordinates in $K_{\circ}$, $K_{\circ}/\tau_{c}$

Let $x\in K^{[r-s]}(k_{1},\dots,k_{s})$ be a point on an elementary stratum.
Let us consider collections of spherical coordinates $\check{x}_{1,i}$ ?
$\check{x}_{2,i}$, $k_{1}\leq i\leq k_{s}$, $1\leq i\leq r_{0}$ of the point
$x$. For each $i$ the following cases are possible: the pairs of $i$-th
coordinates coincides; antipodal; the second coordinate is the result of the
transformation of the first coordinate by the generator ${\bf i}$ of the
cyclic cover $S^{1}\mapsto S^{1}/{\bf i}$. Let us define for an ordered pair
of coordinates $\check{x}_{1,i}$, $\check{x}_{2,i}$, $1\leq i\leq r$ a residue
$v_{i}$ vith values $+1$, $-1$, $+{\bf i}$, $-{\bf i}$:
$v_{k_{i}}=\check{x}_{1,k_{i}}(\check{x}_{2,k_{i}})^{-1}$.

By transformation of the collection of coordinates $\check{x}_{2}$ into the
antipodal collection, the collection of residues are transformed by the
antipodal opposition:

$\\{(\check{x}_{1,k_{i}},\check{x}_{2,k_{i}})\\}\mapsto\\{(-\check{x}_{1,k_{i}},\check{x}_{2,k_{i}})\\},\quad\\{v_{k_{i}}\\}\mapsto\\{-v_{k_{i}}\\},$
$\\{(\check{x}_{1,k_{i}},\check{x}_{2,k_{i}})\\}\mapsto\\{(\check{x}_{1,k_{i}},-\check{x}_{2,k_{i}})\\},\quad\\{v_{k_{i}}\\}\mapsto\\{-v_{k_{i}}\\}.$

By transformation of the collections of coordinates by the renumeration:
$(\check{x}_{1},\check{x}_{2})\mapsto(\check{x}_{2},\check{x}_{1})$, the
collection of residues are transformed by the complex conjugated collection:
into the antipodal collection, the collection of residues are transformed by
the antipodal opposition:

$\\{(\check{x}_{1,k_{i}},\check{x}_{2,k_{i}})\\}\mapsto\\{(\check{x}_{2,k_{i}},\check{x}_{1,k_{i}})\\},\quad\\{v_{k_{i}}\\}\mapsto\\{\bar{v}_{k_{i}}\\},$

where $v\mapsto\bar{v}$ means the complex conjugation. Evidently, the
collection of residues remains fixed in the case a different point on the
elementary stratum is marked.

Elementary strata $K^{[r-s]}(k_{1},\dots,k_{s})$, accordingly with collections
of coordinates, are divided into 3 types: ${\bf I}_{b},{\bf
I}_{a\times\dot{{a}}},{\bf I}_{d}$. In the case residues takes values in
$\\{+{\bf i},-{\bf i}\\}$ (correspondingly, in $\\{+1,-1\\}$), one say that an
elementary stratum is of the type ${\bf I}_{b}$ (correspondingly, of the type
${\bf I}_{a\times\dot{{a}}}$). In the case there are residues of the complex
type $\\{+{\bf i},-{\bf i}\\}$, and the real type $\\{+1,-1\\}$
simultaneously, one shall say about an elementary stratum of the type ${\bf
I}_{d}$. It is easy to check that the restriction of the characteristic
mapping $\eta:K_{\circ}\to K({\bf D},1)$ ?on the elementary stratum of the
type ${\bf I}_{b},{\bf I}_{a\times\dot{{a}}},{\bf I}_{d}$ correspondingly, is
a composition of a mapping into the space $K({\bf I}_{b},1)$ (correspondingly,
into the space ${\bf I}_{a\times\dot{{a}}}$, or $K({\bf I}_{d},1)$) with the
mapping $i_{b}:K({\bf I}_{b},1)\to K({\bf D},1)$ (correspondingly with the
mapping $i_{a\times\dot{{a}}}:K({\bf I}_{a\times\dot{{a}}},1)\to K({\bf
D},1)$, or with $i_{d}:K({\bf I}_{d},1)\to K({\bf D},1)$). For strata of the
first two types a reduction of the structuring mapping (up to homotopy) is not
well-defined; this reduction is well-defined only up to the conjugation of the
subgroup ${\bf I}_{b}$ (correspondingly, in the subgroup ${\bf
I}_{a\times\dot{{a}}}$).

The polyhedron ${\bf N}_{\circ}$ is defined by addition elementary anti-
diagonal strata to the polyhedron $K_{\circ}$. On each elementary anti-
diagonal stratum $K(k_{1},\dots,k_{s})$ a residue of each coordinate equals
$+{\bf i}$. This anti-diagonal stratum (formally) is a stratum of the type
${\bf I}_{b}$.

Define the following open polyhedra:

$\displaystyle K_{b\circ}\subset{\bf N}_{\circ}$ (181) $\displaystyle
K_{a\times\dot{{a}}\circ}\subset K_{\circ}\subset{\bf N}_{\circ}$ (182)
$\displaystyle K_{{\rm d}\circ}\subset K_{\circ}\subset{\bf N}_{\circ}$ (183)

as polyhedra, which are obtain as union of all elementary strata of the
corresponding type.

All the statements above are formulated for elementary strata of the
polyhedron $\hat{K}_{\circ}=K_{\circ}/\tau_{c}$. Elementary strata of this
polyhedra (180) are divided into the following types: ${\bf I}_{b,c},{\bf
I}_{a\times\dot{{a}},c},{\bf I}_{d,\rm{c}}$, respectively to the type of the
subgroup in $C$. Surprisingly, strata of the ype ${\bf I}_{b,c}$, ${\bf
I}_{a\times\dot{{a}},c}$ are homeomorphic. A homeomorphism in prescribed
coordinate systems are given by the formula:

$\displaystyle([x,y],\lambda)\mapsto([ix,y],\lambda).$ (184)

Homeomorphism (184) induces an isomorphism of the corresponding structuring
subgroups ${\bf I}_{b,c}\mapsto{\bf I}_{a\times\dot{{a}},\rm{c}}$, which is
given on the generators by the formula: $b\mapsto ac$, $c\mapsto c$.

Analogously to (181),(182),(183) let us define the following open
subpolyhedra:

$\displaystyle\hat{K}_{\b{,}\rm{c}\circ}\subset\hat{K}_{\circ},$ (185)
$\displaystyle\hat{K}_{\rm{a}\times\dot{{a}},\rm{c}\circ}\subset\hat{K}_{\circ},$
(186) $\displaystyle\hat{K}_{{\rm d},\rm{c}\circ}\subset\hat{K}_{\circ}.$
(187)

The polyhedra $(\ref{Ibc})$, $(\ref{Iac})$ are homeomorphic, where the
homeomorphism is defined by the formula $(\ref{hom})$ for each elementary
strata of the considered polyhedra.

### Description of the structuring mapping using coordinate systems

Let us describe the structuring mapping

$\displaystyle\hat{\eta}_{\circ}:\hat{K}_{\circ}\to K(C,1)$ (188)

by coordinates.

Let $x=[(x_{1},x_{2})]$ be a marked point on $\hat{K}_{\circ}$, which is
inside a maximal stratum. Let us consider a closed path
$l:S^{1}\to\hat{K}_{\circ}$, which is started and ended in the marked point,
the path intersects strata of the deep $1$ in a finite number of points.

Let $(\check{x}_{1},\check{x}_{2})$ be spherical inverse images of $x$. Let us
define another pair $(\check{x}^{\prime}_{1},\check{x}^{\prime}_{2})$ of
spherical inverse images of the point $x$, which is called a result of a
naturally transformation of the coordinates $(\check{x}_{1},\check{x}_{2})$
along the path $l$.

In regular points of the path $l$ a family of spherical inverse-images in one-
parameter family are continuously changed; this uniquely defined
$(\check{x}^{\prime}_{1},\check{x}^{\prime}_{2})$ by
$(\check{x}_{1},\check{x}_{2})$. In an intersection point of $l$ with a strata
of the deep $1$ a corresponding pair of spherical coordinates jump. Because in
this critical point all regular coordinates remain fixed, the extension of the
singular coordinate is uniquely defined. For a given point $x$ on an
elementary maximal stratum a single pair of spherical coordinates determines
uniquely all the last coordinates.

The transformation
$(\check{x}_{1},\check{x}_{2})\mapsto(\check{x}^{\prime}_{1},\check{x}^{\prime}_{2})$
determines an element in the group $C$. This element does not depend of the
path $l$ up to homotopy in $\pi_{1}(\hat{K}_{0\circ},x)$. The homomorphism
$\pi_{1}(\hat{K}_{\circ},x)\to C$ is well-defined, moreover, the induced
mapping

$\displaystyle\hat{\eta}:\hat{K}_{\circ}\to K(C,1)$ (189)

coincides with the structuring mapping, was defined above. Obviously,
restrictions of the structuring mapping $\hat{\eta}$ on connected components
of a single stratum $\hat{K}_{\circ}(1,\dots,r)$ of the type ${\bf I}_{b,c}$
(correspondingly, of the type ${\bf I}_{\rm{a}\times\dot{{a}},\rm{c}}$) is
homotopic into the subspace $K({\bf I}_{b,c},1)\subset K(C,1)$
(correspondingly, into the subspace $K({\bf
I}_{\rm{a}\times\dot{{a}},\rm{c}},1)\subset K(C,1)$). Obviously, on the hole
polyhedron $\hat{K}_{b,c;\circ}$ the structuring mapping takes values in
$K(C,1)=BC$.

### Description of canonical covering over an elementary stratum by
coordinates

Let us consider an elementary stratum in the polyhedron $\hat{K}_{\circ}$
$\hat{K}^{[r-s]}(k_{1},\dots,k_{s})\subset\hat{K}^{(r-s)}_{\circ}$ of the deep
$(r-s)$. Denote by

$\displaystyle\hat{\pi}:\hat{K}^{[r-s]}(k_{1},\dots,k_{s})\to
K({\mathbb{Z}}/2,1)$ (190)

the classifying mapping, which corresponds to permutation of coordinates along
paths. This mapping is called the classifying mapping of the canonical
$2$-sheeted covering.

The mapping $\hat{\pi}$ coincides with the composition

$\hat{K}^{[r-s]}(k_{1},\dots,k_{s})\stackrel{{\scriptstyle\eta}}{{\longrightarrow}}K(C,1)\stackrel{{\scriptstyle
p}}{{\longrightarrow}}K({\mathbb{Z}}/2,1),$

where $K(C,1)\stackrel{{\scriptstyle
p}}{{\longrightarrow}}K({\mathbb{Z}}/2,1)$ is defined as the mapping, which is
induced by the epimorphism $C\to{\bf D}\to{\mathbb{Z}}/2$ with the kernel
${\bf I}_{A\times\dot{A},\rm{c}}\subset{\bf D}$. The canonical $2$-sheeted
covering, associated with the mapping $\hat{\pi}$, is denoted by

$\displaystyle\bar{K}^{[r-s]}(k_{1},\dots,k_{s})\to\hat{K}^{[r-s]}(k_{1},\dots,k_{s}).$
(191)

Additionally, with the mapping $(\ref{piii})$ the following equivariant
mapping

$\displaystyle\bar{\pi}:\bar{K}^{[r-s]}(k_{1},\dots,k_{s})\to S^{\infty}$
(192)

will be considered, where the involution in the image is the standard
antipodal involution, and the mapping itself is the double covering over the
mapping $(\ref{piii})$.

######  Lemma 30.

A restriction of the equivariant mapping $(\ref{eqvpi})$ on the canonical
covering over an elementary stratum of one of the following types ${\bf
I}_{\rm{a}\times\dot{{a}},\rm{c}}$, ${\bf I}_{\b{,}\rm{c}}$ is homotopic to
the mapping

$\displaystyle\bar{\pi}:\bar{K}^{[r-s]}(k_{1},\dots,k_{s})\to S^{1}\subset
S^{\infty},$ (193)

where $S^{1}\subset S^{\infty}$ is the inclusion of the double covering over
1-skeleton of the classifying space.

The involution (184) transforms canonical coverings over strata of different
types one into another.

#### Proof of Lemma 30

Let us prove the lemma using explicit formulas for the mapping $(\ref{pi1a})$.
An arbitrary point $[(x_{1},x_{2})]\in\hat{K}^{[r-s,i]}(k_{1},\dots,k_{s})$
are determined by equivalent class of the collection of angles and the
momentum. The structuring mapping $\hat{\eta}_{\circ}$ is determined by
transformations of coordinates. Let us define the mapping $(\ref{pi1a})$ by
the transformation of a marked pair of the angles. The homeomorphism between
elementary strata of types ${\bf I}_{\rm{a}\times\dot{{a}},\rm{c}}$, ${\bf
I}_{\b{,}\rm{c}}$ (see (184)) is corresponded to the isomorphism of the
structuring subgroups in $C$, the subgroups are denoted using its generators
by: $\\{b,c\\}$ $\\{a,\dot{{a}},c\\}$; this isomorphism transforms $b\mapsto
ac$, $\rm{c}\mapsto\rm{c}$.

To prove, it is sufficiently to consider the only case, let us consider the
case of a stratum of the type ${\bf I}_{a\times\dot{{a}},c}$. Below a
prescribed coordinate system of angles on an elementary stratum is defined;
also a marked pair of angles is defined.

Let a point
$[(\check{x}_{1},\check{x}_{2})]\in\hat{K}^{[r-s]}(k_{1},\dots,k_{s})$ be in a
stratum of the type ${\bf I}_{\rm{a}\times\dot{{a}},\rm{c}}$. Because the
residue of the prescribed coordinate system is uniquely defined, a non-ordered
pair of coordinates with the residue $+1$ denote by
$[(\check{x}_{1,-},\check{x}_{2,-})]$, a non-ordered pair of coordinates with
the residue $-1$ denote by $[(\check{x}_{1,+},\check{x}_{2,+})]$.

Each angle coordinate $\check{x}_{1,-}$, $\check{x}_{2,-}$, $\check{x}_{1,+}$,
$\check{x}_{2,+}$ determines a point on $S^{1}$. Obviously,
$\check{x}_{1,+}=\check{x}_{2,+}$, $\check{x}_{1,-}=-\check{x}_{2,-}$.
Therefore the mapping
$(\hat{x}_{1},\hat{x}_{2})\mapsto(\check{x}_{1,-}^{-1}\check{x}_{1,+},\check{x}_{2,-}^{-1}\check{x}_{2,+})$
transforms points of ordered positive negative into the opposite points on
$S^{1}$. By a translation of the pair of angle coordinates, which keeps an
order of points in the pair, the equivariant mapping remains unchanged. By a
translation of the pair, which inverses an order of points in the pair, the
equivariant mapping is translated into the antipodal. The described mapping is
the required equivariant mapping $(\ref{pi1a})$ for a strata of the type ${\bf
I}_{\rm{a}\times\dot{{a}},\rm{c}}$.

Let us remark that the constructed equivariant mapping $(\ref{pi1a})$ on an
elementary stratum of the type ${\bf I}_{\rm{a}\times\dot{{a}},\rm{c}}$ is not
homotopic to the identity.

The second statement is satisfied, because the structuring group ${\bf
I}_{a,c}$, ${\bf I}_{\b{,}\rm{c}}$ are transformed such that the subgroup
$(a,\dot{{a}},c)$ of the covering over a strata of the type ${\bf I}_{a,c}$
transforms into the subgroup $(b^{2},c)$ of the covering over a stratum of the
type ${\bf I}_{b,c}$. Lemma 30 is proved.

#### A prescribed coordinate system and marked pairs of angles on an
elementary stratum of the type ${\bf I}_{a\times\dot{{a}},c}$ in the
polyhedron $\hat{K}_{\circ}$

Let us recall that the space $K_{a\times\dot{{a}},\rm{c}}$ is decomposed into
a disjoint union of elementary strata $\hat{K}(k_{1},\dots,k_{s}))$, $0\leq
s\leq r$, residues of angles coordinates take values $\\{+1,-1\\}$. On each
elementary stratum $\alpha$ of the type ${\bf I}_{a\times\dot{{a}},\rm{c}}$
let us define a coordinate system, which is called a prescribed coordinate
system $\Omega(\alpha)$.

Let us call a coordinate system a prescribed system, if a number of
coordinates with the residues $+1$ is greater then the number of coordinates
with the residues $-1$; in the case the residues are divided in the middle,
let us define the prescribed coordinate system such that the residues with the
smallest index equals to $+1$.

The angle coordinate with the residue $-1$ in the prescribed coordinate system
with the smallest index is called a marked coordinate on
$K^{[r-s,i]}(k_{1},\dots,k_{s})$.

#### Admissible boundary strata of elementary strata

For an arbitrary elementary stratum
$\beta\subset\hat{K}^{[r-s,i]}(k_{1},\dots,k_{s})$ of the type ${\bf
I}_{\rm{a}\times\dot{{a}},\rm{c}}$ (correspondingly, of the type ${\bf
I}_{\rm{a}\times\dot{{a}},\rm{c}}$) of $\hat{K}_{\circ}$ let us consider a
smallest stratum in its boundary $\alpha$ of the same type, $\beta\neq\alpha$,
this means that $\alpha$ in inside the closure of $\beta$: $\alpha\subset
Cl(\beta)\subset Cl(\hat{K}^{[r-s,i]}(k_{1},\dots,k_{s}))$.

Prescribed coordinate system $\Omega(\beta)$ on the stratum
$\beta\subset\hat{K}(k_{1},\dots,k_{s})$ is restricted to the elementary
stratum $\alpha\subset Cl(\beta)$ inside its boundary.We shall write:
$\alpha\prec\beta$. In the case when the restriction of a prescribed
coordinate system $\Omega(\beta)|_{\alpha}$ coincide with the prescribed
coordinate system $\Omega(\alpha)$ on the smallest stratum, we shall call that
the pair of strata $(\alpha,\beta)$ is admissible. In the opposite case, the
pair of strata $(\alpha,\beta)$ is not admissible.

In the case a pair $(\alpha,\beta)$ is not admissible, the transformation
$\Omega(\beta)|_{\alpha}$ into $\Omega(\alpha)$ are described below for all
types of strata.

For pair of strata of the type ${\bf I}_{\b{,}\rm{c}}$ a non-admissible of the
pair means that the system of coordinates $\Omega(\beta)|_{\alpha}$ are
related with $\Omega(\alpha)$ by one of the following transformations:

$\displaystyle(\check{x_{1}},\check{x_{2}})\mapsto(\check{x_{2}},\check{x_{1}}),$
(194)
$\displaystyle(\check{x_{1}},\check{x_{2}})\mapsto(-\check{x_{2}},-\check{x_{1}}),$
(195)
$\displaystyle(\check{x_{1}},\check{x_{2}})\mapsto(-\check{x_{1}},\check{x_{2}}),$
(196)
$\displaystyle(\check{x_{1}},\check{x_{2}})\mapsto(\check{x_{1}},-\check{x_{2}}).$
(197)

## The polyhedron $\hat{K}_{\circ}$, the lift of its structuring mapping

Let us consider the polyhedron $\hat{K}_{\circ}$, whit coordinate system of
the type ${\bf I}_{b,c}$, for short re-denote this polyhedron by $\hat{K}$.

Consider in ${\bf D}\subset C$ subgroups, which are generated by the
prescribed elements:

In this diagram all mappings are induced by inclusions of subgroups of the
index $2$. The classifying mapping $K_{\circ}\to B{\bf D}$ be extended to the
classifying mapping $\hat{K}_{\circ}\to BC$. Because ${\bf D}\subset C$ is the
central extension, the subgroups $\\{b^{2}\\},\\{b\\},\\{A,\dot{A}\\}$ in
$\\{D\\}$ determines the corresponding subgroups in $C$ and the diagram of the
2-sheeted coverings is well-defined:

In this section points of the space $\hat{K}_{b^{2}}=K_{b^{2},c}$, denoted by
$[(x,y),\lambda]$, and the momentum coordinate $\lambda$ should be omitted. In
the space $\hat{K}_{b^{2}}$ a point $(x,y)$ coincides with points $({\bf
i}x,{\bf i}y)$,$(-x,-y)$,$(-{\bf i}x,-{\bf i}y)$.

### Construction of equivariant morphism of a bundle over $\hat{K}_{b^{2}}$
with fibres non-ordered pairs of oriented real line into a bundle with fibres
${\mathbb{C}}$

Let us divide the space $\hat{K}_{b^{2}}$ into two spaces $\hat{K}_{b^{2},+}$,
$\hat{K}_{b^{2},-}$ with the common boundary. This common boundary let us
denote by $\Sigma\subset\hat{K}_{b^{2}}$. Define the subspaces in all
elementary strata separately. Let us consider the momenta coordinate
$\lambda$, which takes value in the simplex $\Delta^{r-j}$, where $j$ is the
deep of the strata. Let us consider two faces of simplex $\Delta^{r-j}$, which
are defined by momenta, corresponded to angles with positive and negative
imaginary residues. The simplex itself is the join of this two faces, let us
assume that a parameter of the join $t\in[0,1]$ takes the value $0$ on the
negative face and the value $1$ to the positive face.

Define $\Delta^{r-j}=\Delta_{+}\cup\Delta_{-}$, where the first subspace
$\Delta_{+}$ is defined as points of the join, which are attached to positive
faces, with $t$ coordinate on the segment $[1/2,1]$. The second subspace
$\Delta_{-}$ is defined as points, which are attached to negative face, with
coordinate on the segment $[0,1/2]$.

This decomposition of the simplex of momenta for each elementary stratum
defines the decomposition of the hole space $\hat{K}_{b^{2}}$ into two
subspaces, which is required. The involution $\tau_{a}$ is mapped the positive
subspace to the negative. On a maximal stratum this involution $\tau_{a}$
permutes a point, for which the number of positive residues is greater then
half number $r$ of residues, with the corresponding point, for which the
number of positive residues is less then $r$.

This means that in $x$ is strictly inside $\hat{K}_{b^{2},+}$, then
$\tau_{a}(x)$ is strictly inside $\hat{K}_{b^{2},-}$.

Over the space $\hat{K}_{b^{2}}=\hat{K}_{b^{2},+}\cup\hat{K}_{b^{2},-}$ let us
define a pair of bundles, equipped with actions of the involutions
$\tau_{a},\tau_{b},\tau_{i}$. As equivariant bundles, the bundles are non-
trivial, but without the involutions the considered bundles are trivial.

A fiber of the bundle $E_{\mathbb{R}}$ is a disjoin union of the oriented real
lines ${\mathbb{R}}\sqcup{\mathbb{R}}$, i.e.
$E_{\mathbb{R}}=\hat{K}_{b^{2}}\times({\mathbb{R}}\sqcup{\mathbb{R}})$. The
base vectors in a fibre ${\mathbb{R}}\sqcup{\mathbb{R}}$ over
$\hat{K}_{b^{2},+}$ will be denoted by $\pm t_{+}$, $\pm t_{-}$. The
coordinates $t_{+}$, $t_{-}$ are fixed such that over $\hat{K}_{b^{2},+}$ the
base vector in the first fibre ${\mathbb{R}}\sqcup{\mathbb{R}}$ is
parametrized by the coordinate $t_{+}$, the second fibre by the coordinate
$t_{-}$. Over $\hat{K}_{b^{2},-}$ numbers of fibres are fixed in the opposite
way: the second fibre is parametrized by the coordinate $t_{+}$, the first
fibre by the coordinate $t_{-}$. The coordinate $t_{+}$ over
$\hat{K}_{b^{2},+}$ and the coordinate $t_{-}$ over $\hat{K}_{b^{2},-}$
corresponds each other by the identification of fibres, i.e. a positive ray in
the fibre $t_{+}$ over $\hat{K}_{b^{2},+}$ is identified with the positive ray
in the fibre $t_{-}$ over $K_{b^{2},-}$ along the hypersurface $\Sigma$.
Analogously rule of the identification of the fibres is used by the
identification of $t_{-}$ over $\hat{K}_{b^{2},+}$ with $t_{+}$ over
$\hat{K}_{b^{2},-}$ along $\Sigma$.

The trivialization of the bundles with fibres ${\mathbb{C}}$ over
$\hat{K}_{b^{2}}$ is given by the variable $z$ and this rule is used for
identification over $\hat{K}_{b^{2},+}$, and over $\hat{K}_{b^{2},-}$.

Let us define the following triple of involutions on the bundle
$E_{\mathbb{R}}$ by the following formulas:

$\displaystyle\tau_{a}$ $\displaystyle:((x,y),t_{\pm})\mapsto((y,x),t_{\mp})$
(198) $\displaystyle\tau_{b}$
$\displaystyle:((x,y),t_{\pm})\mapsto((-y,x),t_{\pm})$ (199)
$\displaystyle\tau_{i}$ $\displaystyle:((x,y),t_{\pm})\mapsto((x,y),-t_{\mp})$
(200)

The involution $\tau_{a}$ transforms the map $\hat{K}_{b^{2},+}$ with
coordinates $(t_{\pm},z)$ into the map $\hat{K}_{b^{2},-}$ with coordinates
$(t_{\mp},z)$. Involutions $\tau_{b}$, $\tau_{i}$ are transformed the map into
itself.

The fiber of the bundle $E_{\mathbb{C}}$ is the complex line:
$\hat{K}_{b_{2}}\times{\mathbb{C}}\to\hat{K}_{b_{2}}$. Let us define the
action of the triple of involutions by the formulas:

$\displaystyle\tau_{a}$ $\displaystyle:((x,y),z)\mapsto((y,x),\overline{z})$
(201) $\displaystyle\tau_{b}$ $\displaystyle:((x,y),z)\mapsto((-y,x),z)$ (202)
$\displaystyle\tau_{i}$ $\displaystyle:((x,y),z)\mapsto((x,y),\overline{z})$
(203)

In the all cases involutions are commuted.

The main result of the section is the following.

######  Theorem 31.

There exist $(\tau_{a},\tau_{b},\tau_{i})$-equivariant mapping $F$ of the
bundle $E_{\mathbb{R}}$ into the bundle $E_{\mathbb{C}}$, which is a fibrewise
monomorphism.

###### Proof.

The mapping $F$ is constructed by the induction. Let us start the mapping $F$
over centres of maximal simplexes, then over the strata of the deep $1$.

For each maximal simplex in $\hat{K}_{b^{2}}$ with a centre on $\Sigma$
corresponds exactly two centres, the first centre is in $\hat{K}_{b^{2},+}$,
the second is in $\hat{K}_{b^{2},-}$. The constructed mapping will be denote
by $F^{(0)}$. Then we extend this mapping over the simplexes, which are trough
a common face of the codimension $1$, this simplexes join corresponding
centres of $\hat{K}_{b^{2},+}$ and $\hat{K}_{b^{2},-}$, or, simplexes with the
single degenerated angle coordinate. This extension is denoted by $F^{(1)}$.
Is required to check that the obstruction for equivariant extension of
$F^{(1)}$ to the hole complex $\hat{K}_{b^{2}}$ are trivial. This calculation
is in the next subsection.

THe base of the induction is the construction $F^{(0)}$, the restriction of
$F$ on the set of points
$((x,y),\lambda)\in\hat{K}_{b^{2},+}\cup\hat{K}_{b^{2},-}$, for which for all
$j$ $\lambda_{j}\neq 0$. In the case the collection of residues
$\\{\delta_{j}\\}$ contains residues for the full collection. $j$. Denote
$\theta=\prod\limits_{j=1}^{r}\delta_{j}$. The number $r$ is odd, therefore
$\theta=\pm i$, in particular, $\overline{\theta}=-\theta$.

Let us define over $K_{b^{2},+}$ the morphism $F^{(0)}$ by the formula:

$F^{(0)}:((x,y),t_{+})\mapsto((x,y),-\theta t).$ (204)
$F^{(0)}:((x,y),t_{-})\mapsto((x,y),\bar{\theta}t).$ (205)

Let us define over $\hat{K}_{b^{2},-}$ this morphism as follows.

$F^{(0)}:((x,y),t_{+})\mapsto((x,y),\bar{\theta}t).$ (206)
$F^{(0)}:((x,y),t_{-})\mapsto((x,y),-\theta t).$ (207)

Let us check that the mapping defined above is
$(\tau_{a},\tau_{b},\tau_{i})$-equivariant. The following formulas proves
correction of the formula $(\ref{1.1+})$ for the only fibre $t_{+}$ over
$\hat{K}_{+}$.

We get:

$\displaystyle
F^{(0)}\tau_{i}((x,y),t_{+})=F^{(0)}((x,y),-t_{-})=((x,y),-\overline{\theta}t)$
$\displaystyle\tau_{i}F^{(0)}((x,y),t_{+})=\tau_{i}((x,y),-\theta
t)=((x,y),-\overline{\theta}t)$

$\displaystyle
F^{(0)}\tau_{a}((x,y),t_{+})=F^{(0)}((y,x),t_{-})=((y,x),-\bar{\theta}t)$
$\displaystyle\tau_{a}F^{(0)}((x,y),t_{+})=\tau_{a}((x,y),-\theta
t)=((y,x),-\bar{\theta}t)$

$\displaystyle
F^{(0)}\tau_{b}((x,y),t_{+})=F^{(0)}((-y,x),t_{+})=((-y,x),-\theta t)$
$\displaystyle\tau_{b}F^{(0)}((x,y),t_{+})=\tau_{b}((x,y),-\theta
t)=((-y,x),-\theta t)$

Now, extend the mapping $F^{(0)}$ on $E_{\mathbb{R}}$ over segments in
$\hat{K}_{b^{2}}$, which joins centres of maximal simplexes, the both vertexes
of the simplexes are over $\hat{K}_{b^{2},+}$, or over $\hat{K}_{b^{2},-}$.

Let us consider simplexes in $\hat{K}_{b^{2},+}$, which have common face of
the codimension $1$. This means that, in particular, that in centers of
simplexes collections of residues a single angle is degenerated. Let us define
by $\hat{\theta}^{\prime}$ the sum of all residues over the first maximal
simplex, and by $\hat{\theta}^{\prime\prime}$ the sum of the second maximal
simplex. Then $\theta^{\prime}=\exp(\pi\hat{\theta}^{\prime}/2)$ and
$\theta^{\prime\prime}=\exp(\pi\hat{\theta}^{\prime\prime}/2)$.

Let us consider a segment, which is inside $\hat{K}_{b^{2},+}$ (the case
$C+$). On the segment, which is join centers
$((x^{\prime},y^{\prime}),(y^{\prime},x^{\prime}))$ ?
$((x^{\prime\prime},y^{\prime\prime}),(y^{\prime\prime},x^{\prime\prime}))$ of
the first and the second maximal simplexes correspondingly, define the mapping
$F^{(1)}$ by the formula:

$\displaystyle F^{(1)}:$
$\displaystyle((1-s)(x^{\prime},y^{\prime})+s(x^{\prime\prime},y^{\prime\prime}),t_{+})\mapsto$
$\displaystyle\mapsto((1-s)(x^{\prime},y^{\prime})+s(x^{\prime\prime},y^{\prime\prime}),-\exp(((1-s)\hat{\theta}^{\prime}+s\hat{\theta}^{\prime\prime})\pi/2)t)$
(208) $\displaystyle F^{(1)}:$
$\displaystyle((1-s)(x^{\prime},y^{\prime})+s(x^{\prime\prime},y^{\prime\prime}),t_{-})\mapsto$
$\displaystyle\mapsto((1-s)(x^{\prime},y^{\prime})+s(x^{\prime\prime},y^{\prime\prime}),\overline{\exp(((1-s)\hat{\theta}^{\prime}+s\hat{\theta}^{\prime\prime})\pi/2)}t)$
(209)

Consider the case, when the segment is inside $\hat{K}_{b^{2},-}$ (case $C-$).

$\displaystyle F^{(1)}:$
$\displaystyle((1-s)(x^{\prime},y^{\prime})+s(x^{\prime\prime},y^{\prime\prime}),t_{+})\mapsto$
$\displaystyle\mapsto((1-s)(x^{\prime},y^{\prime})+s(x^{\prime\prime},y^{\prime\prime}),\overline{\exp(((1-s)\hat{\theta}^{\prime}+s\hat{\theta}^{\prime\prime})\pi/2)}t)$
(210) $\displaystyle F^{(1)}:$
$\displaystyle((1-s)(x^{\prime},y^{\prime})+s(x^{\prime\prime},y^{\prime\prime}),t_{-})\mapsto$
$\displaystyle\mapsto((1-s)(x^{\prime},y^{\prime})+s(x^{\prime\prime},y^{\prime\prime}),-\exp(((1-s)\hat{\theta}^{\prime}+s\hat{\theta}^{\prime\prime})\pi/2)t)$
(211)

Check that this formula determines $(\tau_{a},\tau_{b},\tau_{i})$-equivariant
mapping. For short, let us consider the case $C+$. The case $C-$ is completely
analogous.

One has:

$\displaystyle F^{(1)}\tau_{a}:$
$\displaystyle((1-s)(x^{\prime},y^{\prime})+s(x^{\prime\prime},y^{\prime\prime}),t_{+})=F^{(1)}((1-s)(y^{\prime},x^{\prime})+s(y^{\prime\prime},x^{\prime\prime}),t_{-})\mapsto$
$\displaystyle\mapsto((1-s)(y^{\prime},x^{\prime})+s(y^{\prime\prime},x^{\prime\prime}),-\exp(((1-s)\bar{\hat{\theta^{\prime}}}+s\bar{\hat{\theta^{\prime\prime}}})\pi/2)t).$

Additionally:

$\displaystyle\tau_{a}F^{(1)}:$
$\displaystyle((1-s)(x^{\prime},y^{\prime})+s(x^{\prime\prime},y^{\prime\prime}),t_{+})\mapsto$
$\displaystyle\mapsto\tau_{a}((1-s)(x^{\prime},y^{\prime})+s(x^{\prime\prime},y^{\prime\prime}),-\exp(((1-s)\hat{\theta}^{\prime}+s\hat{\theta}^{\prime\prime})\pi/2)t)=$
$\displaystyle=((1-s)(y^{\prime},x^{\prime})+s(y^{\prime\prime},x^{\prime\prime}),-\overline{\exp(((1-s)\hat{\theta}^{\prime}+s\hat{\theta}^{\prime\prime})}\pi/2)t).$

The commutativity property requires to compare formulas $(\ref{2.1bis})$ and
$(\ref{2.1+})$, because after the back transformation from the map
$\hat{K}_{b^{2},-}$ to the map $\hat{K}_{b^{2},+}$ the morphism $\tau_{a}$
changes the number of the line: $t_{+}\mapsto t_{-}$.Because
$\overline{\exp(z)}=\exp(\overline{z})$, the $\tau_{a}$-equivariant property
is proved.

Then we assume the case $C+$ is considered:

$\displaystyle F^{(1)}\tau_{i}:$
$\displaystyle((1-s)(x^{\prime},y^{\prime})+s(x^{\prime\prime},y^{\prime\prime}),t_{+})=F^{(1)}((1-s)(x^{\prime},y^{\prime})+s(x^{\prime\prime},y^{\prime\prime}),-t_{-})\mapsto$
$\displaystyle\mapsto((1-s)(x^{\prime},y^{\prime})+s(x^{\prime\prime},y^{\prime\prime}),-\overline{\exp(((1-s)\hat{\theta}^{\prime}+s\hat{\theta}^{\prime\prime})\pi/2)}t).$

Additionally,

$\displaystyle\tau_{i}F^{(1)}:$
$\displaystyle((1-s)(x^{\prime},y^{\prime})+s(x^{\prime\prime},y^{\prime\prime}),t_{+})\mapsto$
$\displaystyle\mapsto\tau_{i}((1-s)(x^{\prime},y^{\prime})+s(x^{\prime\prime},y^{\prime\prime}),-\exp(((1-s)\hat{\theta}^{\prime}+s\hat{\theta}^{\prime\prime})\pi/2)t)=$
$\displaystyle=((1-s)(x^{\prime},y^{\prime})+s(x^{\prime\prime},y^{\prime\prime}),-\overline{\exp(((1-s)\hat{\theta}^{\prime}+s\hat{\theta}^{\prime\prime})\pi/2)t})=$

We have used the formulas $(\ref{2.1+})$, $(\ref{2.1-})$. The equivariant
property of the involution $\tau_{i}$ is proved.

The equivariant property for $\tau_{b}$ is evident. ∎

## Construction of an antipodal morphism of a covering space $\hat{K}_{b}$
into the circle over $\hat{K}$

Let us consider a $(\tau_{a},\tau_{b},\tau_{i})$-equivariant morphism $F$ of
the bundle $E_{\mathbb{R}}$ into the bundle $E_{\mathbb{C}}$, constructed in
Theorem 31. Let us define the extension of this morphism by an extra
coordinate $w$. Then, after a suitable transformation we get:
$(\tau_{a},\tau_{b},\tau_{i})\mapsto(\tilde{\tau}_{a},\tilde{\tau}_{b},\tilde{\tau}_{i})$.

Define $(\tilde{\tau}_{a},\tilde{\tau}_{b},\tilde{\tau}_{i})$-equivariant
$S^{0}$-covering $\hat{K}_{b^{2}}$ (a 2-sheeted covering), which is considered
as the bundle $E_{S}$ over $\hat{K}_{b^{2}}$ with the fiber $S^{0}=\\{\pm
1\\}$. Without equivariant structure we get: $E_{S}=\hat{K}_{b^{2}}\times
S^{0}$. The involutions are defined by the following formulas:

$\displaystyle\tilde{\tau}_{a}$ $\displaystyle:((x,y),w)\mapsto((-x,y),w)$
(212) $\displaystyle\tilde{\tau}_{b}$
$\displaystyle:((x,y),w)\mapsto((-y,x),w)$ (213)
$\displaystyle\tilde{\tau}_{i}$ $\displaystyle:((x,y),w)\mapsto((x,y),-w).$
(214)

Let us consider extensions $\tilde{\tau}_{b}$, $\tilde{\tau}_{i}$ of the
involutions $\tau_{b}$, $\tau_{i}$ in the extended bundle
$E_{\mathbb{R}}\otimes E_{S}$. Define an extension $\tilde{\tau}_{a}$ of the
involution $\tau_{i}\circ\tau_{b}\circ\tau_{a}$, i.e. define the involution
$\tilde{\tau}_{a}$ by the following formulas:

$\tilde{\tau}_{a}:((x,y),t_{\pm})\mapsto((-x,y),-t_{\pm})\\\ $
$\tilde{\tau}_{a}:((x,y),z)\mapsto((-x,y),z)$

and by the formula (212). If $(x,y)\in\hat{K}_{b^{2},+}$, then
$(-x,y)\in\hat{K}_{b^{2},-}$.

The involutions $(\tilde{\tau}_{b},\tilde{\tau}_{a},\tilde{\tau}_{i})$ act in
the bundle $E_{{\mathbb{C}}}$. Over the coordinate ${\mathbb{C}}$ the
involutions $\tilde{\tau}_{b}$, $\tilde{\tau}_{a}$ are trivial, the involution
$\tilde{\tau}_{i}$ acts by the conjugation in the fibres.

Recall that the involution $\tau_{a}$ transforms fibres of the bundle
$E_{\mathbb{R}}$ by preserving fibred transformation. The involution
$\tilde{\tau}_{a}$ changes orientations of the fibres of the bundle
$E_{\mathbb{R}}$. The
$(\tilde{\tau}_{a},\tilde{\tau}_{b},\tilde{\tau}_{i})$-equivariant morphism,
defined denote by $\tilde{F}$.

Let us define another one involution $\hat{\tau}_{i}$ in the bundles
$E_{{\mathbb{R}}}$, $E_{{\mathbb{C}}}$ by the formula:
$\hat{\tau}_{i}=(-1)\tilde{\tau}_{i}$. In the bundle $E_{S}$ the involution
$\hat{\tau}_{i}$ acts trivial. Let us consider the formulas:

$\hat{\tau}_{i}:((x,y),t_{\pm})\mapsto((x,y),t_{\mp}),$
$\hat{\tau}_{i}:((x,y),z)\mapsto((x,y),-\bar{z}),$
$\hat{\tau}_{i}:((x,y),w))\mapsto((x,y),w).$

It is easy to see that the morphism $\tilde{F}$ is defined as
$\hat{\tau}_{i}$-equivariant morphism $E_{{\mathbb{R}}}\to E_{{\mathbb{C}}}$,
which we denote the same. The morphisms
$\tilde{\tau}_{a},\tilde{\tau}_{i},\tilde{\tau}_{b},\hat{\tau}_{i}$ are
pairwise commuted. Let us consider
$(\tilde{\tau}_{a},\tilde{\tau}_{b},\tilde{\tau}_{i},\hat{\tau}_{i})$-equivariant
tensor product $E_{{\mathbb{R}}}\otimes E_{S}$, $E_{{\mathbb{C}}}\otimes
E_{S}$.

######  Lemma 32.

The morphism $F$ extends to the
$(\tilde{\tau}_{a},\tilde{\tau}_{b},\tilde{\tau}_{i},\hat{\tau}_{i})$\-
equivariant morphism

$\tilde{F}:E_{\mathbb{R}}\otimes E_{S}\mapsto E_{\mathbb{C}}\otimes E_{S}.$

### Proof of Lemma 32

This is a well-known fact from the algebra. ∎

### The convolution of the tensor product; the rule of marked vector in
equivariant orbite in the source

1\. Take the convolution of the tensor product over all involutions
$(\tilde{\tau}_{a},\tilde{\tau}_{b},\tilde{\tau}_{i},\hat{\tau}_{i})$; let us
calculate bundles in the image and in the preimage over $\hat{K}$.

2\. Construct a morphism $F^{\downarrow}$ of the factor-bundles over
$\hat{K}$, take a marked vector in an orbit in the preimage and calculate the
image of this vector by $F$. The rule of marked vector is formulated
explicitly. The formula of the morphism $F$ valued on the marked vectors
determines a morphism of the covering $K_{b,c}\to\hat{K}$ into the bundle
$S^{1}\times\hat{K}$, which is a discontinuous along a hypersurface (below
this hypersurface $\Delta$ is described). The jump of the morphism is
investigated and the factormorphism $F^{\downarrow}$ is modified into a well-
defined continuous morphism $(\ref{piF})$ of the covering
$K_{b,c}=\hat{K}_{b}\to\hat{K}$ into the trivial bundle
$S^{1}\times\hat{K}\to\hat{K}$.

#### Calculation of a factormorphism $F^{\downarrow}$

Calculation of the convolutions for the triple of involutions
$(\tilde{\tau}_{a},\tilde{\tau}_{b},\tilde{\tau}_{i})$ was, in fact, done. In
the pre-image we get the bundle with the wedge of the two real linear lines as
fibres. Between the linear lines an isomorphism by the transformation
$t_{+}\leftrightarrow t_{-}$ is defined; this transformation corresponds to
the transformation of the involution $\hat{\tau}_{i}$. The line bundle (after
the identification) has the characteristic class in the subgroup $(b,c)\subset
C$. Recall that along the hypersurface $\Sigma$, which is divided
$\hat{K}_{b^{2},+}$ and $\hat{K}_{b^{2},-}$, the fibres of the bundle
$E_{\mathbb{R}}\otimes E_{S}$ are identified by $\hat{\tau}_{i}$, by the
remuneration of the real lines. In the pre-image by the convolution over the
total collection of the involutions we get a line bundle with characteristic
subgroup $(b,c)\subset C$, the associated spherical bundle is the covering
$\hat{K}_{b}\to\hat{K}$, or, with equivalent denotations, $K_{b,c}\to\hat{K}$.

The involution $\hat{\tau}_{i}$ translates line fibres in the preimage (this
translation changes the coordinate $w$). In the target the result of the
convolution over the involutions
$(\tilde{\tau}_{a},\tilde{\tau}_{b},\tilde{\tau}_{i})$ is the trivial line
bundle over $\hat{K}$ with the projective fibre $P({\mathbb{C}})$.

The involution $\tilde{\tau}_{i}$ in the target acts by conjugations with
respect to the imaginary (vertical) line. Therefore the factormorphism, which
was described below, is a morphism of the line bundle, associated with the
covering $K_{b,c}\to\hat{K}$ into the projectivization of the trivial
${\mathbb{C}}$-bundle over $\hat{K}$.

#### Marked vector rule

This rule is formulated for fibres with the coordinate $w=+1$. Then the
equivariant property is proved with respect to the action of the involutions
$(\tilde{\tau}_{a},\tilde{\tau}_{b},\hat{\tau}_{i})$, which keeps the
coordinate $w$, and the rule is reformulated for the action $\tilde{\tau}_{i}$
to keep the equivariant property.

Let us take the base vector (the ray) on the line $t_{+}$, over
$\hat{K}_{b^{2},+}$, or over $\hat{K}_{b^{2},-}$. The sign of the vector is
defined such that the image by $F$ of this vector is on the hyper-halph-plane
(the imaginary coordinate is positive). In the case the vector by $F$ in the
target is in the right half plane (the real coordinate is positive), this
vector is defined the value of the convolution $F^{\downarrow}$. In the case
the target marked vector is in the left half-plane (the real coordinate of the
marked vector in the target is positive) the generator $\hat{\tau}_{i}$ is
applied in the source and the target. In the source the vector is transformed
to the baser vector on the line $t_{-}$ (with keeping the sign), with the
image by $F$ is in the right halph-plane (and in the upper halph-plane,
because $\hat{\tau}_{a}$ keeps the vertical coordinate in the target).

The marked vector rool is formulated. This defines a morphism $F^{\downarrow}$
over the base $\hat{K}_{b^{2}}$ outside the hypersurface $\Delta$, where the
line fibres are horizontal in the target. This morphism is well-defined as a
morphism of the factors over $\hat{K}$.

Along a hypersurface with horizontal fibres by $F$ the factormorphism
$F^{\downarrow}$ is not continuous. This hypersurface of jumps is denoted by
$\Delta\subset\hat{K}_{b^{2}}$. The hypersurface $\Delta$ on $1$-skeleton
$\hat{K}^{1}_{b^{2}}\subset\hat{K}_{b^{2}}$ is in a neighbourhood of strata of
the deep $1$, as it is follows by the formula of $F$.

Evidently, trough the hypersurface $\Delta$ the marked vector rool changes the
number of the line $t_{\pm}$ in the preimage, and simultaneously, a direction
of the base vector on the line $t_{-}$ (say, $t_{+}\mapsto-t_{-}$); the image
of a new base vector is in the right halph-plane. The marked vector rule has a
jump in the source, because on the factor the fibers $t_{+}$ is transformed
into $t_{-}$ with keeping the base vector. In the target the marked vectors
are mapped by $F$ continuously.

#### Correction of the factormorphism $F^{\downarrow}:E_{{\mathbb{R}}}\to
E_{{\mathbb{C}}}$ over $\hat{K}$ into a continuous factormorphism; fast angle
rule

The factormorphism $F^{\downarrow}$ of the corresponding bundles is well-
defined over $1$-dimensional skeleton of the polyhedron $K_{b^{2}}$, and this
factormorphism is discontinuous over the hypersurface $\Delta$; the goal of
the section is to transform this factormorphism into a continuous
factormorphism.

The construction of elimination of the gaps is formulated over
$\hat{K}_{b^{2},+}$ and $\hat{K}_{b^{2},-}$ separately, let us call the
construction fast angle rule. The constructions are agree to each other along
$\Sigma$.

Recall that $\hat{K}_{b^{2},+}\cup_{\Sigma}\hat{K}_{b^{2},-}$ are two
polyhedra with the common boundary $\Sigma$:
$T_{a}(\hat{K}_{b^{2},+})=\hat{K}_{b^{2},-}$.

The fast angle rule is formulated for vectors with $w=+1$, for vectors with
$w=-1$ the construction is translated by $\tilde{\tau}_{i}$ with keeping
equivariant properties.

Let $\Delta_{+}\subset\hat{K}^{1}_{b^{2},+}$ be a hyperspace of jumps in the
positive polyhedron, over the surface images of fibres in the target are
horizontal. The fast rotation rule is the following: rotate the vector
$t=+1,\varepsilon=+1$ along a short path of the trajectory by $F$ trough
$\Delta$ at the angle $\pi$ in the projective fibre $P({\mathbb{C}})$ in the
opposite direction with respect to the rotation of the trajectory of the
vector. This rule is formulated for an arbitrary vector $t=+1,\varepsilon=+1$
with an arbitrary (horizontal) image.

The same (an equivalent) fast rotation rule along $\Delta_{+}$ over
$\hat{K}_{b^{2},+}$ is the following: rotate the vector $t=-1,\varepsilon=+1$
along a short path of the trajectory by $F$ trough $\Delta$ at the angle
$+\pi$ in the projective fiber $P({\mathbb{C}})$ in the opposite direction of
the rotation trajectory.

The first and the second formula for fast angle rule for vectors on the line
$\varepsilon=+1$ is reformulated for vectors on the line $\varepsilon=-1$,
into the opposite angle of the fast rotation, because rotations of the lines
$\varepsilon=+1$ are $\varepsilon=-1$ towards each other.

Let $\Delta_{-}\subset\hat{K}^{1}_{b^{2},-}$ be a hypersutrface of jumps in
the negative polyhedron, over the surface images of fibres in the target are
horizontal. The fast rotation rule is the following: rotate the vector
$t=+1,\varepsilon=-1$ along a short path of the trajectory by $F$ trough
$\Delta_{-}$ at the angle $+\pi$ in the projective fibre $P({\mathbb{C}})$
along the rotation of the trajectory.

For transformation $t_{+}\mapsto-t_{+}$ the fast angle rule is kept, for
transformation $t_{+}\mapsto t_{-}$ the rule is changed into the opposite.

As a conclusion, the fast rotation rule trough $\Delta$ over
$\hat{K}_{b^{2},+}$ and over $\hat{K}_{b^{2},+}$ are opposite with respect to
the coordinate $\varepsilon$ and are agree with respect to the coordinate $t$.
This implies that along $\Sigma$ the coordinate $\varepsilon$ is changed into
the opposite, the coordinate $t$ is kept and the two fast angle rules
coincide.

To say that fast angle rule is correct, it is required to prove that this rule
is commuted with the generator action $\tilde{\tau}_{a}$ (in this case it
should compare the rules for a point in $\hat{K}_{b^{2},+}$ and for the
corresponding point in $\hat{K}_{b^{2},-}$) and with the generator
$\hat{\tau}_{i}$ (in this case it should compare the rules over
$\hat{K}_{b^{2},+}$ and over $\hat{K}_{b^{2},-}$ independently).

Let us check it for the generator $\tilde{\tau}_{a}$. We get:
$\tilde{\tau}_{a}:t\mapsto-t;\varepsilon\mapsto\varepsilon$ (geometrically, by
the action of $\tilde{\tau}_{a}$ the line fibre is changed, but the coordinate
$\varepsilon$ is preserved, because a side of the $\Sigma$ is changed). For
example, assume that the vector $t=+1,\varepsilon=+1$ is over
$\hat{K}_{b^{2},+}$ and a trajectory of this vector through $\Delta$ is
horizontal right-down. Then the fast angle is counter-clockwise trajectory.
The image of the vector by the generator $\tilde{\tau}_{a}$ is over
$\hat{K}_{b^{2},-}$ and has the coordinates $t=-1,\varepsilon=+1$, this vector
trout $\Delta$ is horizontal left-up, along the trajectory of rotation. It is
proved that the fast angle rule is commuted with the generator
$\tilde{\tau}_{a}$.

Let us prove the same statement for the generator $\hat{\tau}_{i}$. We get:
$\hat{\tau}_{i}:t\mapsto t;\varepsilon\mapsto-\varepsilon$. Because with
respect to the generator $\hat{\tau}_{i}$ a point keeps its position over the
base, the side of the hypersurface $\Sigma$ is preserved. Let us assume that
the point belongs to $\hat{K}_{b^{2},+}$ for simplification of denotations,
the last case is analogous. Assume, the vector $t=+1,\varepsilon=+1$ trough
$\Delta_{+}$ is right-down. The fast angle is counter-clockwise, against the
trajectory of the vector. The image of the vector by $\hat{\tau}_{i}$ is left-
down, the fast angle is clockwise, contr the trajectory. Because the generator
$\hat{\tau}_{i}$ changes the orientation in the target, the fast angles are
agree. This is proved that the generator $\hat{\tau}_{i}$ keeps the fast angle
rule.

We define a continuous convolution of the equivariant morphism $F$ over the
skeleton $\hat{K}^{1}\subset\hat{K}$. This convolution is a morphism of a real
(non-trivial) line bundle into the complex trivial line bundle, see the
formula (215) below.

### An extension of the morphism (215) over $\hat{K}^{1}$ to a morphism over
$\hat{K}^{2}$ and over $\hat{K}$

Let us extend the constructed morphism onto a morphism of the prescribed
bundles $2$-skeleton. Because over $\hat{K}_{b}$ the boundary of of a strata
of the deep $2$ is inside one map, the problem of the extension is local.

In a prescribe map the formula of $F$ has the trivial monodromy over a path in
the boundary of a $2$-cell. An algebraic numbers of fast angles is trivial,
because the rotations are defined along close path, which is decomposed into
$4$ segments. Along this path the full rotation angle of a base vector by $F$
equals to zero. The fast angle rule, formulated for this base vector gives the
trivial total fast angle. The constructed morphism (restricted over the
2-skeleton of the polyhedron) denote by

$\displaystyle\bar{\pi}:\bar{K}_{b,c}\to S^{1},$ (215)

where $\bar{K}_{b,c}$ is the covering over $\hat{K}$ with the structuring
group $C$, by the subgroup $(b,c)\subset C$. On each elementary strata the
morphism (215) is trivial.

The extension of the morphism (215) from $\hat{K}^{2}$ to a morphism over the
polyhedron $\hat{K}$ is obvious, because obstructions belong to
$\pi_{i}(S^{1})$, $i\geq 2$.

### Properties of the morphism (215)

Recall, $dim(\hat{K}_{\circ})=n-k,\quad n-k\equiv 1\pmod{2}$. Let us consider
a thin regular neighbourhood of the boundary
$\hat{U}_{\circ}\subset\hat{K}_{\circ}$ of the open polyhedron.

Consider the mapping $\hat{U}_{\circ}\to S^{1}$, which is the restriction of
the mapping $\hat{\pi}:\hat{K}_{\circ}\to S^{1}$, the quotient of the
equivariant mapping (215). Because the structuring group $\hat{U}_{\circ}$ is
reduced to the subgroup $(b,c)\subset C$, the mapping $\hat{U}_{\circ}\to
S^{1}$ contains values in the even index 2 subgroup
$2[\pi_{1}(S^{1})]\subset\pi_{1}(S^{1})$.

######  Lemma 33.

There exists a polyhedron $\hat{U}_{reg,\circ}\subset\hat{U}_{\circ}$, such
that the restriction of the mapping $\hat{\pi}$ to the complement
$\hat{U}_{\circ}\setminus\hat{U}_{reg,\circ}$ is homotopy to the mapping into
a point, and the polyhedron itself admits a mapping

$\hat{U}_{reg,\circ}\stackrel{{\scriptstyle\eta_{reg}}}{{\longrightarrow}}P\stackrel{{\scriptstyle\zeta_{reg}}}{{\longrightarrow}}K((b,c),1),$

where $P$ is a polyhedron of the dimension $\frac{n-k+1}{2}+d$, $d=2$ (i.e. of
the dimension a little greater then the half of the dimension
$dim(\hat{K}_{\circ})$), $\eta_{reg}$,$\zeta_{reg}$ be mappings, which are
called the control mapping and the structuring mapping for $P$.

#### Proof of Lemma 33

Consider in the polyhedron $\hat{U}_{reg,\circ}\subset\hat{K}_{\circ}$
elementary strata with prescribes coordinate system. A "nice" coordinate
system, this is a coordinate system, which is attached to the anti-diagonal
boundary, in which coordinates with residues $-{\bf i}$ are singular and with
residue $+{\bf i}$ are regular along the anti-diagonal. The prescribed
coordinate system on an elementary stratum could be different with respect to
"nice" coordinate system.

Let us consider all strata for which the number of "nice" angle coordinates
with the residue $+{\bf i}$ less or equals to the number of "nice" angles with
the residue $-{\bf i}$. This condition for a maximal stratum with odd number
of angle coordinates implies that the prescribed coordinate system coincides
with the "nice" coordinate system. Such strata let us call singular.

Let us consider the subpolyhedron $\Pi\subset\hat{U}_{\circ}$ consists of all
singular strata. Let us define a subpolyhedron $P\subset\Pi$ and the mapping
$\eta:\Pi\to P$, which is called the selection.

Let us define $P$. Collect all strata in $\Pi$ equipped by the nice coordinate
system, such that the the number of angle coordinates less, or equals, then
$s=[\frac{n-k+3}{4}]$ (approximately, the half of the total number of the
angle coordinate $\frac{r+1}{2}+1=[\frac{n-k+3}{2}]$ in $K$).

In each elementary singular stratum of the polyhedron $\Pi$ let us take
coordinates with the residue $+{\bf i}$ with smallest numbers, such that the
total number of the selected coordinates with the residue $+{\bf i}$ and all
the coordinates with the residue $-{\bf i}$ equals to $s=[\frac{n-k+3}{4}]$.
Let us call this operation the selection of strata. In the case the total
number of the coordinates in a strata is not greater then $s$, the selection
of this stratum define as the identity. The collection of the selections of
each elementary strata of $\Pi$ determines the selection mapping $\eta:\Pi\to
P$.

The structuring mapping, restricted on $\Pi$ is induced by the structuring
mapping on $P$. To prove this a $C$-equivariant reformulation of the mapping
$\eta$ is well-defined on subpolyhera $\bar{P}$, $\bar{\Pi}$ in $C$-coverings
$\bar{K_{\circ}}$ over $\hat{K}_{\circ}$. The covering equivariant mapping
$\bar{\eta}$ over $\eta$ is well-defined and the action of the group $C$ in
the source is induced from the action in the target. This proves that the
structuring mapping on $\Pi$ is induced by the structuring mapping on $P$.

Obviously, $\hat{K}_{\circ}\setminus\Pi$ is a polyhedron, of elementary strata
for which the number of nice coordinates with the residue $+{\bf i}$ is
greater then the number of nice coordinate with the residue $-{\bf i}$. In
particular, the prescribed coordinate system on a strata of
$\hat{K}_{\circ}\setminus\Pi$ coincides with nice coordinate system of the
strata. In particular, the mapping $\hat{\pi}$, restricted on
$\hat{K}_{\circ}\setminus\Pi$ homotopic to the identity. ∎

### Corollaries: Lemma 9 and Lemma 20

### Proof of Lemma 9

Let us consider the mapping (15), which is defined by a generic lift $c\mapsto
d$. The polyhedron ${\bf N}_{\circ}$ of the formal self-intersection of the
mapping $d$ is a subpolyhedron: ${\bf N}_{\circ}\subset K_{\circ}$, this
polyhedron is equipped with the structuring mapping (189). The polyhedron
$K_{\circ}$ (recal, we have a more general parameter $k^{\prime}$, instead of
the parameter $k$, in the construction of $c$) coincides with the $4$-sheeted
covering over the polyhedron $\hat{K}_{\circ}$ with respect to the subgroup
$(A,\dot{A})\subset C$. Additionally, the transfer of the morphism (215) by
this covering determines a lift (13) of the structuring mapping.

Let us prove that Condition 2 and the formula (17) are corollaries of Lemma
33. In Lemma 33, for the mapping $c$ the dimension of the polyhedron $P$
equals to $\frac{n-k^{\prime}-1}{2}$ up to a small constant $d$. After we get
a vertical lift of the mapping $c\mapsto d$ into a generic mapping, the
polyhedron $K_{\circ}$ is transformed into a subpolyhedron ${\bf N}(d)\subset
K_{\circ}$ of self-intersection of the mapping $d$. The dimension of the
polyhedron ${\bf N}(d)$ less then the dimension of $K_{\circ}$ by the
codimension $k^{\prime}$ of the lift.

By the same argument $P(d)\subset P(c)$ as a subpolyhedron of the codimension
$k^{\prime}$. The structuring mapping on $P(c)$ is induced from the
structuring mapping on $P(d)$, This proves Condition 2. Lemma 9 is proved.

∎

### Proof of Lemma 20

Let us consider the mapping $\bar{d}$, which is described in the lemma. Let us
consider the polyhedron $K_{\circ}$ of self-intersection points of this
mapping. Consider the polyhedron ${\bf N}_{\circ}$ of the formal self-
intersection of the mapping $\bar{d}$, which is a subpolyhedron in
$K_{\circ}$, because therequired deformation $\bar{d}^{(2)}$ is vertical.
Accordingly to (181), (182), (181) the decomposition on the subpolyhedron,
which corresponds to the reduction of the structuring group of elementary
strata to the subgroups in ${\bf D}$ is well-defined. The type of strata
corresponds to components (110),(111). For $k^{\prime}\geq 2$ there exists of
a formal small vertical deformation $\bar{d}^{(2)}\mapsto d^{(2)}$ of formal
extensions, such that the polyhedron self-intersection of the deformed formal
mapping $d^{(2)}$ is splitted into two disjoin components. This is a corollary
of the following lemma.

######  Lemma 34.

The classifaing mapping

$\displaystyle\pi_{d}=p_{A,\dot{A}}\circ\eta_{d}:K_{d}\to K({\bf D},1)\to
K({\mathbb{Z}}/2,1)$ (216)

for the cannonical covering over the polyhedron $K_{d}$, where $\eta_{d}$ is
the structuring mapping, $p_{A,\dot{A}}$ is the projection onto the residue
classes of the subgroup $(A,\dot{A})\subset{\bf D}$, is lifted to a mapping

$\bar{\pi}_{d}:K_{d}\to S^{1},$

and, additionaly, this mapping $\bar{\pi}$ on each elementary stratum of the
polyhedron, defined by formula (192), is the trivial mapping.

#### Proof of Lemma 34

Let us assume $N>>n$ ($N=n+4$ is suffisient) and consider a "stabilisation" of
the mapping $c_{0}$, by the substitution $n\mapsto N$ of its definition. Then
the following inclusion of polyhedra is well-defined:

$\displaystyle K_{d}(n)\subset K_{d}(N).$ (217)

Consider a subpolyhedron in $K_{d}(N)$, which is denoted by the same, which
consists of all strata with the full collection of residues the coordinates
$\\{+1,-1,+{\bf i},-{\bf i}\\}$. Evidently, that the inclusion (217) is well-
defined after a corresponding re-denotations. Moreover, the inclusion (217) is
natural with respect to structuring mappings, therefore it is sufficient to
proof the lemma for $K_{d}(N)$.

Projections $p_{b}:K_{d}(N)\to K_{b}(N)$, $p_{a,\dot{{a}}}:K_{d}(N)\to
K_{a\times\dot{{a}}}(N)$, are well-defined, and the classifaing mapping
$\bar{\pi}_{d}$ is the sum of the two considered mappings mappings into the
circle admite the sum operation.

$\bar{\pi}_{d}(N)=\pi_{a,\dot{{a}}}\circ p_{a,\dot{{a}}}+\pi_{b}\circ p_{b},$

where $\pi_{a,\dot{{a}}}$ is defined using (215) by the transfere with respect
to the subgroup ${\bf I}_{a\times\dot{{a}}}\subset C$ of the index $4$,
$\pi_{b}$ is defined from (215) by the tranfere with respect to the subgroup
${\bf I}_{b}\subset C$ of the index $4$. ∎

Let us finish the proof of Lemma 20. Let us prove that the image of the
fundamental class of the polyhedron ${\bf N}_{a\times\dot{{a}},\circ}$ in the
formula (110) is trivial, and the image of the fundamental class of the closed
polyhedron ${\bf N}_{b}$ in the formula (111) by the structuring mapping
$\eta$ is transformed into the generator of the group $H_{n-2k}(K({\bf
D},1);{\mathbb{Z}}/2)$.

The polyhedron ${\bf N}_{b}$ is decomposed into maximal strata, each maximal
stratum is a manifold with singularities in the codimension $2$ and,
therefore, each elementary stratum admits a fundamental class. The structure
of maximal strata on ${\bf N}_{b}$ is induced by the structure of strata of
the polyhedron $K_{b}$. In $K_{b}$ there exists the only anti-diagonal maximal
stratum (all the residues equals to $+{\bf i}$) and a few number of maximal
(regular) strata, which addmits the full collection of imaginary residues.

The polyhedron ${\bf N}_{b}$ is the result of a deformation of $K_{b}$, this
defirmation keeps the structure of strata (each stratum is deformed by a
vertical deformation and the result of the deformation is a subpolyhedron in
$K_{b}$ which is generic with respect to the stratification). Moreover, the
symmetry $\tau$ should be keept by the deformation on each regular strata and
the polyhedron ${\bf N}_{b}\subset K_{b}$ admits such a symmetry outside the
anti-diagonal. The image of the fundamental class $[{\bf N}_{b}]$ by the
structuring mapping does not depend of a deformation from the considered
class.

Let us consider the transfer of the structuring mapping $\eta$ over each
elementary maximal stratum of ${\bf N}_{b}$ separately, and do calculation of
its image by the transfer $\eta_{A\times\mathcal{A}}^{!}$.

With the dimension $n^{\prime}=\frac{n}{2}$, we get that
$\eta_{A\times\mathcal{A};\ast}^{!}([{\bf N}_{d}]+[{\bf
N}_{a\times\dot{{a}},\circ}])$ in the generator in
$H_{n^{\prime}-k}(A;{\mathbb{Z}}/2)$ (the projection $A\times\mathcal{A}\to
A={\mathbb{Z}}/2$ on the first generator).

Let us do the analogous calculation for
$[K_{b}]+[K_{a\times\dot{{a}},\circ}]$. For the maximal anti-diagonal stratum
of the polyhedron $K_{b}$ the image of its fundamental class in
$K(A\times\dot{A},1)$ by the projection in $K(A\times\dot{A},1)\to K(A,1)$ is
mapped into the fundamental class from the group
$H_{n-k}(K(A,1);{\mathbb{Z}}/2)$, $k\equiv 0\pmod{2}$.

An elementary stratum $K_{i}$ of $K_{b}$ of a general type (which is nof the
anti-diagonal) is a 2-sheeted covering over the corresponding stratum
$\hat{K}_{i}$ of the polyhedron $\hat{K}_{b}$.

The transfer on $K_{i}$ with respect to the subgroup
$(A,\mathcal{A})\subset{\bf D}$ is compressed trough the covering in the
topwer $(A,\dot{A},c)\subset C$. On the stratum $\hat{K}_{i}$ the structuring
mapping takes the value in the commutative subgroup $\\{b,c\\}\subset C\\}$ of
the order $2$. Therefore the transfer is considered with respect to the
subgroup $\\{-1\\}\subset\\{b,c\\}$.

The transfer on $K_{i}$ is commpresed trough the covering over $\hat{K}_{i}$
with respect to the subgroup $\\{bc^{-1}\\}\subset\\{b,c\\}$. On the covering
we get the mapping $\hat{\eta}^{!}:\hat{K}_{i}^{!}\to K({\mathbb{Z}}/4,1)$,
${\mathbb{Z}}/4=\\{b\\}$. The image of the fubdamental claass by
$\hat{\eta}^{!}$ has to be estimated, this class is not the generator. This
class is a covering over the class
$p_{\\{bc^{-1}\\}}\circ\hat{\eta}[\hat{K}_{i}]$,
$p_{\\{bc^{-1}\\}}:B({\mathbb{Z}}/2\times{\mathbb{Z}}/4)\to B({\mathbb{Z}}/4)$
is the projection with the kernel $\\{bc^{-1}\\}$. Therefore $\hat{\eta}^{!}$
is compressed trough the double covering and is not a generator.

An analogous argument for an elementary stratum $K_{i}$ of
$K_{a\times\dot{{a}}}$ is satisfied.

For $[N_{b}]+[N_{a\times\dot{{a}},\circ}]$ the calculation analogous (with the
corresponding re-denotations). On the polyhedron ${\bf
N}_{a\times\dot{{a}},\circ}$ calculation can be translated using calculations
for ${\bf N}_{b;\circ}$ by the diffeomorphism (184). This prove that the
component (110) is neglected. Lemma 20 is proved. ∎

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