# Lectures on families of Dirac operators and applications

Francesco Lin  Department of Mathematics, Columbia University
<EMAIL_ADDRESS>
###### Abstract.

These are the notes for a minicourse taught at the 2022 ICTP summer school
Frontiers in Geometry and Topology. The goal is to introduce families of Dirac
operators and how they can be used to study interactions between geometry and
topology. In particular, we discuss the index theorem for families of twisted
Dirac operators, and its applications to metrics of positive scalar curvature
and the three-dimensional Weinstein conjecture.

Dedicated to Tom Mrowka in the occasion of his 60th birthday.

## Introduction

These are the notes for the three and a half hour long minicourse Families of
Dirac operators and applications taught at the 2022 ICTP summer school
Frontiers in Geometry and Topology for an audience of graduate students. The
main goal is to show via concrete examples how families of Dirac operators can
be used as a bridge between the geometry and the topology of manifolds. These
ideas are closely related to current topics of research in gauge theory and
low-dimensional topology, and the hope is to provide a friendly first
introduction to the subject for those working in the field. In particular,
these notes assume the reader to be somehow familiar with the following:

  * •

vector bundles, their classifying spaces, and characteristic classes [MS74];

  * •

basic differential geometry, including connections on vector bundles and the
various notions of curvature of a Riemannian manifold [Lee18];

  * •

basic notions in functional analysis such as separable Hilbert spaces and
bounded operators [Eva10, Appendix D].

The last part of the notes will also assume the reader to be familiar with the
formal properties of either monopole or Heegaard Floer homology (see for
example [Lin16] and [Gre21] respectively for introductions to the topics).

The minicourse consisted of three lectures covering the following:

  1. (1)

an informal introduction to spin Dirac operators; the Atiyah-Singer index
theorem, the Lichnerowicz formula, and applications to metrics of positive
scalar curvature.

  2. (2)

$K$-theory and its relation with Fredholm operators; the index theorem for
families of Dirac operators and the Gromov-Lawson theorem on the torus.

  3. (3)

the space of self-adjoint Fredholm operators; relations with Taubes’ proof of
the Weinstein conjecture in dimension three.

This is a very ambitious syllabus - each lecture could have been the topic of
a minicourse itself. When preparing the lectures, we have tried to highlight
the motivation and key ideas behind the constructions while keeping
technicalities at the bare minimum (some important ones are pointed out in the
footnotes); we provide precise references for more detailed treatments. These
notes follow the content of the lectures very closely, and will have a rather
terse style. At the end of each lecture, the interested reader will find some
exercises.

### Warning.

Throughout the text, we will be very imprecise about the analytic setup. For
example, we will state most index theorems in terms of bounded operators, but
switch to unbounded (differential) ones without further notice when discussing
geometric applications.

### Acknowledgements.

The author thanks the participants of the summer school, whose questions
greatly helped to improve these notes. This work was partially supported by
the Alfred P. Sloan Foundation and NSF grant DMS-2203498.

##  1\. Dirac operators, geometry and topology

###  1.1. Dirac operators on $\mathbb{R}^{n}$

The most fundamental differential operator on $\mathbb{R}^{n}$ is the
Laplacian111Our convention is to use all negative signs so that $\langle
u,\Delta u\rangle_{L^{2}}\geq 0\text{ for all compactly supported $u$}$, where
we consider the $L^{2}$-inner product $\langle u,v\rangle_{L^{2}}=\int
uvd\mathrm{vol}$.

$\Delta:=-\frac{\partial^{2}}{\partial x_{1}^{2}}-\frac{\partial^{2}}{\partial
x_{2}^{2}}-\dots-\frac{\partial^{2}}{\partial x_{n}^{2}}.$

Functions satisfying $\Delta u=0$ are called harmonic; these can be used to
describe many important objects, such as potentials in electromagnetism. While
working on the relativistic quantum mechanics of the electron, Dirac [Dir28]
was led to look for a first order differential operator $D$ such that
$D^{2}=\Delta$.

######  Remark 1.1.

To be precise, Dirac was working with a Lorenzian metric on $\mathbb{R}^{4}$,
so that he considered the d’Alembertian

$\Box=-\frac{\partial^{2}}{\partial x_{1}^{2}}-\frac{\partial^{2}}{\partial
x_{2}^{2}}-\frac{\partial^{2}}{\partial
x_{3}^{2}}+\frac{\partial^{2}}{\partial t^{2}}$

instead of the Laplacian $\Delta$. We will always work with Riemannian metrics
in these notes.

Heuristically, what does such an operator $D$ look like? Let us assume for
simplicity that it takes the form

$D=a_{1}\frac{\partial}{\partial x_{1}}+a_{2}\frac{\partial}{\partial
x_{2}}+\dots+a_{n}\frac{\partial}{\partial x_{n}}$

with the $\\{a_{i}\\}$ are constant quantities (belonging to some algebraic
structure to be decided). Then

$D^{2}=\sum_{i}a_{i}^{2}\frac{\partial^{2}}{\partial
x_{i}^{2}}+\sum_{i,j}a_{i}a_{j}\frac{\partial}{\partial
x_{i}}\frac{\partial}{\partial x_{j}}.$

Hence, as derivatives commute, $D^{2}=\Delta$ is equivalent to the relations

(1) $\begin{cases}a_{i}^{2}&=-1\\\ a_{i}a_{j}+a_{i}a_{j}&=0\text{ if }i\neq
j.\end{cases}$

Let us see how to solve such system of equations in the $\\{a_{i}\\}$. If
$n=1$ we just need $a_{1}^{2}=-1$; of course, we cannot solve this within real
numbers, but if we allow complex valued functions, we can set $a_{1}=i$, and

$D=i\frac{\partial}{\partial x_{1}}$

squares to $\Delta$ (acting now on complex-valued functions).

When $n=2$, the anticommutativity relation $a_{1}a_{2}+a_{2}a_{1}=0$ implies
that we cannot work anymore with complex numbers; on the other hand, we can
solve (1) if we allow the $a_{i}$ to be $2\times 2$ matrices by setting

$a_{1}=\begin{bmatrix}0&-1\\\ 1&0\end{bmatrix}\quad
a_{2}=\begin{bmatrix}0&i\\\ i&0\end{bmatrix},$

so that

(2) $D=a_{1}\frac{\partial}{\partial x_{1}}+a_{2}\frac{\partial}{\partial
x_{2}}=\begin{bmatrix}0&-\frac{\partial}{\partial
x_{1}}+i\frac{\partial}{\partial x_{2}}\\\ \frac{\partial}{\partial
x_{1}}+i\frac{\partial}{\partial x_{2}}&0\end{bmatrix}$

squares to $\Delta$ thought of as an operator on $\mathbb{C}^{2}$ valued
functions. Notice that $D$ is closely related to the holomorphic and anti-
holomorphic derivative operators.

In higher dimensions, the situation is similar; when $n=3$, we can set the
$a_{i}$’s to be the Pauli matrices

$\sigma_{1}=\begin{bmatrix}i&0\\\
0&-i\end{bmatrix}\quad\sigma_{2}=\begin{bmatrix}0&-1\\\
1&0\end{bmatrix}\quad\sigma_{3}=\begin{bmatrix}0&i\\\ i&0\end{bmatrix},$

and the corresponding Dirac operator $D$ is again a square root of $\Delta$
acting on $\mathbb{C}^{2}$ valued functions. When $n=4$, we can set the
$a_{i}$ to be the $4\times 4$ block matrices defined by

$\begin{bmatrix}0&-I_{2}\\\ I_{2}&0\end{bmatrix}\text{ and
}\begin{bmatrix}0&-\sigma_{i}^{*}\\\ \sigma_{i}&0\end{bmatrix}\text{ for
}i=1,2,3$

where the $I_{2}$ is the $2\times 2$ identity matrix and ∗ denotes transpose
conjugate. The corresponding Dirac operator $D$ squares to $\Delta$ acting on
$\mathbb{C}^{4}$ valued functions, and is the (Riemannian version of the)
operator Dirac was looking for. We will leave to the reader to guess the
general recipe to define $D$ on $\mathbb{R}^{n}$ for $n\geq 5$.

######  Remark 1.2.

Of course, we could have made different choices for the matrices $a_{i}$,
which would have lead to different Dirac operators; for example, when $n=1$ we
could have taken $D=i\frac{d}{dx_{1}}$ to be acting on $\mathbb{C}^{m}$-valued
functions. The specific operators we constructed above are ‘minimal’ in a
specific sense (involving the representation theory of Clifford algebras), and
are called the spin Dirac operators.

###  1.2. Dirac operators on a manifold

In this lectures we will be interested in how spin Dirac operators can be used
as a bridge between the geometry and topology of an oriented Riemannian
manifold $(M,g)$; the precise definition of the analogue $D$ is quite
involved, and we refer the reader for example to [Roe98, Ch. 3,4] for more
details. We will only focus on formal properties of such generalization.

First of all, in order to define $D$, we need to choose a spin structure on
$M$; we will not need the precise definition of what this is (see for example
[Kir89, Ch. IV]), but will only remark that this exists if and only the second
Stiefel-Whitney class $w_{2}(TM)$ is zero. For manifolds of dimension $n\geq
3$, this is equivalent to the restriction of $TM$ to the $2$-skeleton of $M$
being trivial. A spin structure affords us an analogue of the target vector
spaces $\mathbb{C}^{m}$ we considered before, which is a hermitian bundle

$S\rightarrow M$

of rank $2^{\lfloor n/2\rfloor}$ equipped with a hermitian connection
$\nabla$, called the spinor bundle. The spin Dirac operator is then a first
order differential operator

$D:\Gamma(S)\rightarrow\Gamma(S)$

with the following key properties:

  * •

it is formally self-adjoint in $L^{2}$, i.e. if $\varphi,\psi\in\Gamma(S)$ are
compactly supported sections (i.e. if $M$ is compact), then

$\langle D\varphi,\psi\rangle_{L^{2}}=\langle\varphi,D\psi\rangle_{L^{2}}.$

  * •

it is elliptic; we will not define exactly what this means, but this implies
for example that harmonic spinors, i.e. solutions to the equation $D\varphi=0$
(which makes sense for $\varphi$ merely $C^{1}$) are automatically smooth.

One should compare the latter with the analogous result for holomorphic and
harmonic functions, which are in the kernel of the elliptic operators
$\overline{\partial}$ and $\Delta$ respectively. We refer the reader to
[War71, Chapter 6] for a nice introduction to the analysis of elliptic
equations on manifolds which only assumes some basic familiarity with Fourier
series.

###  1.3. Dirac operators and topology

Another key consequence of ellipticity is that, when $M$ is compact, the space
of harmonic spinors is finite dimensional. This is surprisingly related to
topological information about $M$ in a way that we now discuss.

Assume that $n$ is even. Looking back at the case of $\mathbb{R}^{n}$, we see
that in this case the matrices corresponding to $a_{i}$ have a simple
structure as block matrices. In the case of a Riemannian manifold, this
manifests in a canonical splitting of the spinor bundle

$S=S^{+}\oplus S^{-}$

for which we can write the Dirac operator in block form

$D=\begin{bmatrix}0&D^{-}\\\ D^{+}&0\end{bmatrix}\text{ where
}D^{\pm}:\Gamma(S^{\pm})\rightarrow\Gamma(S^{\mp}).$

We call $D^{\pm}$ the chiral Dirac operators. As $D$ is self-adjoint,
$D^{\pm}$ are adjoint (i.e. transpose) to each other; in particular, the
analogue in this situation of the relation
$(\mathrm{im}A)^{\perp}=\mathrm{ker}A^{T}$ for a matrix $A$ implies that

$\mathrm{dim}\mathrm{coker}D^{+}=\mathrm{dim}\mathrm{ker}D^{-},$

hence is also finite dimensional222There is a subtlety here: we use that
$\mathrm{im}D^{+}$ is a closed subspace to identify its orthogonal with
$\mathrm{coker}D^{+}$; closedness also follows from ellipticity.. In general,
an operator $T$ between Hilbert spaces with finite dimensional kernel and
cokernel is called Fredholm; its index is defined to be

$\mathrm{ind}(T)=\mathrm{dim}\mathrm{ker}T-\mathrm{dim}\mathrm{coker}T\in\mathbb{Z}.$

The following fundamental result (a special case of the Atiyah-Singer index
theorem, one of the cornerstones of 20th century mathematics) relates the
index of $D^{+}$ (which is an analytical quantity) to the topology of $M$;
this is particularly striking because $\mathrm{dim}\mathrm{ker}D^{+}$ and
$\mathrm{dim}\mathrm{coker}D^{+}$ are not themselves topological invariants
(cf. [Hit74])!

######  Theorem 1.1 ([AS63]).

The index of $D^{+}$ can be computed in terms of the Pontryagin classes
$p_{i}(TM)$. In particular:

  * •

if $n\equiv 2$ modulo $4$, $\mathrm{ind}(D^{+})=0$;

  * •

if $n=4$, $\mathrm{ind}(D^{+})=-\langle p_{1},[M]\rangle/24$;

  * •

if $n=8$, $\mathrm{ind}(D^{+})=\langle-4p_{2}+7p_{1}^{2},[M]\rangle/5760$.

More in general, the index is determined by an expression in the Pontryagin
classes of $TM$ called $A$-hat genus of $M$, and
$\mathrm{ind}(D^{+})=\langle\widehat{\mathcal{A}}(M),[M]\rangle$.

######  Remark 1.3.

When $n=4$, the Hirzebruch signature formula $p_{1}=3\sigma$ implies that we
can also write $\mathrm{ind}(D^{+})=-\sigma(M)/8$, where $\sigma$ denotes the
signature.

For a more thorough discussion of the Atiyah-Singer index theorem and its
various proofs we refer the reader to [Sha78] and [Roe98].

###  1.4. Dirac operators and geometry

Recall that Dirac’s goal was to find a differential operator such that
$D^{2}=\Delta$. The Dirac operator on a spin manifold satisfies an analogous
relation. Indeed, recall that our bundle $S\rightarrow M$ comes with a
hermitian connection

$\nabla:\Gamma(S)\rightarrow\Gamma(T^{*}M\otimes S).$

We denote the formal adjoint in $L^{2}$ of such operator by

$\nabla^{*}:\Gamma(T^{*}M\otimes S)\rightarrow\Gamma(S).$

This is the differential operator determined by the identity

$\langle\nabla\varphi,\alpha\rangle_{L^{2}}=\langle\varphi,\nabla^{*}\alpha\rangle_{L^{2}}\text{
for every }\varphi\in\Gamma(S),\alpha\in\Gamma(T^{*}M\otimes S),$

and is (after changing sign) the bundle analogue of the classical divergence
of a vector field.

The second order operator $\nabla^{*}\nabla$ is called the Bochner Laplacian;
it makes sense on any bundle with Hermitian connection, and corresponds in
$\mathbb{R}^{n}$ to the familiar expression

$\Delta=-\mathrm{div}\circ\mathrm{grad}.$

The fundamental identity relating the Dirac operator and geometry is then the
following.

######  Theorem 1.2 (Lichnerowicz ‘63).

Let $s$ be the scalar curvature of $(M,g)$. Then $D^{2}=\nabla^{*}\nabla+s/4$.

The magic behind the formula is the following: $D^{2}$ and $\nabla^{*}\nabla$
are second order differential operators which by design have the same second
order part; one would expect then their difference to be a first order
differential operator, but it turns out to be a zeroth order one! This has
surprising consequences for the study of the existence of metrics of positive
scalar curvature, which is a quite subtle problem in Riemannian geometry.

######  Proposition 1.3.

Suppose $(M,g)$ is a compact spin manifold with $s>0$ everywhere. Then the
identity $\mathrm{ind}(D^{+})=0$ (which is topological in nature by Theorem
1.1) holds. In particular, if $M$ is $4$-dimensional, $\sigma(M)=0$.

So for example the $K3$ surface, which is a spin $4$-manifold with
$\sigma=-16$ (see [GS99]), does not admit metrics of positive scalar
curvature. Notice that the spin assumption is essential, as the Fubini-Study
metric on $\mathbb{C}P^{2}$ has $s>0$.

To prove the proposition, we just need to show that $D$ (and therefore
$D^{\pm}$) has trivial kernel. To see this, assume $D\varphi=0$ and $\varphi$
is not identically zero. The Lichnerowicz formula then implies

$\nabla^{*}\nabla\varphi+\frac{s}{4}\varphi=0.$

Taking the $L^{2}$-inner product with $\varphi$, we obtain

$0=\langle\nabla^{*}\nabla\varphi,\varphi\rangle_{L^{2}}+\langle\frac{s}{4}\varphi,\varphi\rangle_{L^{2}}>\langle\nabla^{*}\nabla\varphi,\varphi\rangle_{L^{2}}=\langle\nabla\varphi,\nabla\varphi\rangle_{L^{2}}\geq
0$

which is a contradiction.

###  1.5. Twisted Dirac operators

Consider now the case of the torus $T^{n}$, which admits a flat metric, hence
a metric with $s\equiv 0$; it is a natural ask whether it admits a metric with
$s>0$. The answer is negative for $n=2$ by Gauss Bonnet’s theorem. Also,
notice that Myers’ theorem [Lee18, Chapter $12$] implies that $T^{n}$ does not
admit metrics with positive Ricci curvature because it has infinite
fundamental group.

The tangent bundle of ${T}^{n}$ is trivial, so in particular it is spin
($w_{2}=0$) and has vanishing Pontryagin classes; hence
$\mathrm{ind}(D^{+})=0$ and the obstruction of Proposition 1.3 does not apply.

The key idea to obtain more subtle obstructions is to study twisted Dirac
operators. In particular given $M$, consider flat hermitian connections $B$ on
the trivial line bundle $\underline{\mathbb{C}}\rightarrow M$; up to
isomorphism, these are classified by their holonomy around loops, and
correspond bijectively to the $b_{1}(M)$-dimensional torus

$\mathbb{T}_{M}=H^{1}(M;\mathbb{R})/H^{1}(M;2\pi\mathbb{Z}).$

To each of these, one can associate a twisted connection $\nabla_{B}$ on $S$,
and correspondingly a twisted Dirac operator $D_{B}$ on $\Gamma(S)$ which
satisfies the analogues of Theorem 1.1 and 1.2333Here it is important that $B$
is flat.. For example, on $M=S^{1}=\mathbb{R}/2\pi\mathbb{Z}$, the twisted
Dirac operators are of the form

$i\frac{d}{dt}+c\text{ for }c\in\mathbb{R}.$

Two such operators are conjugate exactly when $c-c^{\prime}\in\mathbb{Z}$, so
that we get a $\mathbb{T}_{S^{1}}=S^{1}$-family of operators. In the next
lecture, we will see that the family of Fredholm operators $\\{D_{B}\\}$
carries much more information than a single operator.

###  1.6. Exercises

  1. (1)

The Dirac operator (2) on $\mathbb{R}^{2}$ is translation invariant, and
descends to the Dirac operator on the torus $\mathbb{R}^{2}/\mathbb{Z}^{2}$.
Determine the space of harmonic spinors, and check that the index formula for
$D^{+}$ holds in this case.

  2. (2)

By looking carefully at the argument in Proposition $1.3$, show that if
$(M,g)$ is a spin $4$-manifold with $s\equiv 0$, then $|\sigma(M)|\leq 16$.

  3. (3)

Use Fourier series to compute the spectrum of the twisted Dirac operator
$i\frac{d}{dt}+c$ on $S^{1}$; show that two such operators are conjugate when
$c-c^{\prime}\in\mathbb{Z}$.

##  2\. K-theory and the index theorem for families

Consider a complex separable Hilbert space $H$, e.g. $H=L^{2}(M)$. Recall that
a bounded operator $T:H\rightarrow H$ is Fredholm if it has finite dimensional
kernel and cokernel444Because we are working with Hilbert (hence Banach)
spaces, the image of such $T$ is automatically closed; when working with more
general topological vector spaces the latter needs to be incorporated in the
definition of Fredholm. Denote by $\mathrm{Fred}(H)$ the space of such
operators; one can show that it is an open set of the space of bounded
operators, and the map

$\mathrm{ind}:\mathrm{Fred}(H)\rightarrow\mathbb{Z}$

is locally constant (even though the dimensions of kernel and cokernel are
not!). It turns out that the space $\mathrm{Fred}(H)$ has very interesting
topology, which we now describe.

###  2.1. K-theory

We discuss the essential notions in topological $K$-theory, and refer the
reader to [FF16, Ch. 38] for more details. Consider a finite CW complex $X$,
which will assume for simplicity to be connected (in our applications $X$ will
be the torus $\mathbb{T}_{M}$). Denote by $\mathrm{Vect}(X)$ the set of
isomorphism classes of complex vector bundles on $X$. The operation of direct
sum $\oplus$ and tensor product $\otimes$ make $\mathrm{Vect}(X)$ into a
semiring, i.e. a structure like a ring but without additive inverses. For
example, $\mathbb{N}$ is a semiring, and there is indeed a semiring
homomorphism $\mathrm{Vect}(X)\rightarrow\mathbb{N}$ given by the rank555If
$X$ is not connected, one allows vector bundles with different ranks on each
of them, and this map depends on a choice of connected component.. Notice
that, unlike $\mathbb{N}$, $\mathrm{Vect}(X)$ is not in general cancellative,
i.e. $a+c=b+c$ does not in general imply $a=b$.

Of course, it is nicer in general to work with rings than semirings. Given any
semiring $S$, there is a natural way to associate a ring called the
Grothendieck ring $\mathrm{Gr}(S)$. This follows the usual construction of
$\mathbb{Z}$ from $\mathbb{N}$; one defines $\mathrm{Gr}(S)$ to be the set of
formal differences $a-b$ of elements in $S$, modulo the relation

$a-b\sim a^{\prime}-b^{\prime}\text{ if }a+b^{\prime}+e=a^{\prime}+b+e\text{
for some }e.$

Here we need the extra $e$ to have a good definition for non-cancellative
semirings.

######  Definition 2.1.

For a finite CW complex $X$, the $K$-theory of $X$, denoted by $K(X)$, is the
Grothendieck ring of $\mathrm{Vect}(X)$.

Here are some basic properties:

  * •

An element of $K(X)$ is represented by a formal difference of vector bundles
$E-F$;

  * •

There is a ring homomorphism $K(X)\rightarrow\mathbb{Z}$ given by the rank;

  * •

$K$ gives rise to a contravariant functor from finite CW complexes to rings;
homotopic maps induce the same ring homomorphism.

Denote by $\mathrm{Vect}_{k}(X)$ the isomorphism classes of $k$-dimensional
vector bundles. Recall that we have the identification with the set of
homotopy classes

$\mathrm{Vect}_{k}(X)=[X,BU(k)]$

where, concretely, $BU(k)$ is the union of the Grassmannians $G(n,k)$ of
$k$-planes in $\mathbb{C}^{n}$. Now, for any vector bundle $E\rightarrow X$,
there is a vector bundle $E^{\prime}$ such that $E\oplus E^{\prime}$ is
trivial. This implies that any element in $K(X)$ has a representative of the
form $F-\underline{\mathbb{C}}^{m}$ for some vector bundle $F$ and
$m\in\mathbb{N}$. From this, one shows that

$K(X)=[X,BU\times\mathbb{Z}]$

where $BU=\bigcup_{k\geq 1}BU(k)$, and $\mathbb{Z}$ corresponds to the rank.

###  2.2. Relation with Fredholm operators

It turns out that $K$-theory is the right algebraic tool to study families of
Fredholm operators because of the following.

######  Theorem 2.2 (Atiyah-Jänich).

For a finite CW complex $X$ , there is a natural bijection

$K(X)=[X,\mathrm{Fred}(H)],$

i.e. the space of Fredholm operators is also a classifying space for
$K$-theory. In particular, homotopy classes of families of Fredholm operators
parametrized by $X$ are in bijection with $K(X)$.

As a consequence, this also tells us that the space $\mathrm{Fred}(H)$ has a
very interesting topology.

Following [Ati67, Appendix], let us discuss how to associate to a family
$\\{T_{x}\\}_{x\in X}$ of Fredholm operators $T_{x}:H\rightarrow H$ a class in
$K(X)$. Intuitively, this corresponds to the difference between the kernel and
cokernel at each point of $T_{x}$. Of course, these do not form a vector
bundle (the dimensions might not even be constant!) so we need a different
approach to formalize this. Because $X$ is compact, one can find a finite
dimensional subspace $W\subset H$ such that

$\mathrm{Im}(T_{x})+W=H$

for all $x\in X$. Then, $V=\bigcup T_{x}^{-1}(W)$ is a vector bundle over $X$,
and we associate to $\\{T_{x}\\}$ the element

$\mathrm{ind}\\{T_{x}\\}:=V-\underline{\mathbb{C}}^{\mathrm{dim}W}\in K(X)$

called the index of the family. Notice that the rank map
$K(X)\rightarrow\mathbb{Z}$ sends the index of the family to the index of a
single operator in the family. It turns out that $\mathrm{ind}\\{T_{x}\\}$ is
independent of choices, and that the assignment is a bijection. A key input in
the proof is Kuiper’s theorem, which states that the unitary group of a
Hilbert space is contractible.666This also implies that any Hilbert bundle
over a CW complex is trivial. In particular, after trivializing, a family of
operators on a Hilbert bundle can be considered as a family of operators on
fixed Hilbert space; we implicitly did so for the family of twisted Dirac
operators on $S^{1}$.

######  Remark 2.1.

For our purposes, it is important to point out the following: if $\\{T_{x}\\}$
is a family of isomorphisms, the corresponding element in $K(X)$ is zero. This
is because we can choose $W=\\{0\\}$.

###  2.3. The index theorem for families of twisted Dirac operators

Consider now the family of twisted Dirac operators $\\{D_{B}^{+}\\}$ on $M$
parametrized by the torus $\mathbb{T}_{M}$, and the corresponding element

$\mathrm{ind}\\{D_{B}^{+}\\}\in K(\mathbb{T}_{M}).$

Theorem 1.1 gives us a formula for the image of such element under the
dimension map $K(\mathbb{T}_{M})\rightarrow\mathbb{Z}$; we would like to
obtain more detail information. Of course, $K$-theory is a quite complicated
object to deal with, so it would be desirable to work with something simpler
such as cohomology. Luckily, one can use Chern classes $c_{i}$ to define a
ring homomorphism

$\mathrm{ch}:K(X)\rightarrow H^{\mathrm{even}}(X;\mathbb{Q})$

called the Chern character; this is given on a vector bundle $E$ by

$\mathrm{ch}(E)=\mathrm{dim}E+c_{1}(E)+\frac{c_{1}^{2}-2c_{2}}{2}+\dots;$

in particular, the degree zero part of $\mathrm{ch}$ is exactly the dimension.
It turns out that $\mathrm{ch}$ is a rational isomorphism.

######  Remark 2.2.

On the other hand, the torsion subgroups of $K(X)$ and
$H^{\mathrm{even}}(X;\mathbb{Z})$ are in general very different.

The index theorem for families will provide a formula for
$\mathrm{ch}(\mathrm{ind}\\{D_{B}^{+}\\})\in
H^{\mathrm{even}}(\mathbb{T}_{M};\mathbb{Q})$ in terms of the topology of $M$.
Recall that $\mathbb{T}_{M}=H^{1}(M;\mathbb{R})/H^{1}(M;2\pi\mathbb{Z})$, so
that we can identify

$H_{1}(\mathbb{T}_{M};\mathbb{Z})=H^{1}(M;\mathbb{Z}).$

Via this identification, a basis $x_{1},\dots,x_{m}$ of $H^{1}(M;\mathbb{Z})$
determines a dual basis $y_{1},\dots,y_{m}$ of
$H^{1}(\mathbb{T}_{M};\mathbb{Z})=H_{1}(\mathbb{T}_{M};\mathbb{Z})^{*}$. We
can then consider the element

$\Omega=\sum x_{i}\otimes y_{i}\in H^{1}(M;\mathbb{Z})\otimes
H^{1}(\mathbb{T}_{M};\mathbb{Z})\subset
H^{2}(M\times\mathbb{T}_{M};\mathbb{Z}),$

which is independent of the choice of basis777More intrinsically, $\Omega$ is
the first Chern class of the so-called Poincaré line bundle on
$M\times\mathbb{T}_{M}$.. Recall that the A-hat genus is the combination of
Pontryagin classes

$\widehat{\mathcal{A}}(M)=1-\frac{p_{1}}{24}+\frac{7p_{1}^{2}-4p_{2}}{5760}+\dots,$

which is an element of $H^{\mathrm{even}}(M;\mathbb{Q})$.

######  Theorem 2.3 ([AS71]).

Consider an even dimensional spin manifold $M$. Then

$\mathrm{ch}(\mathrm{ind}\\{D_{B}^{+}\\})=\langle
e^{\Omega}\cdot\widehat{\mathcal{A}}(M),[M]\rangle\in
H^{\mathrm{even}}(\mathbb{T}_{M}.\mathbb{Q}).$

Here $e^{\Omega}=\sum_{n\geq 0}\Omega^{n}/n!$ and
$e^{\Omega}\cdot\widehat{\mathcal{A}}(M)$ defines an element in
$H^{\mathrm{even}}(M\times\mathbb{T}_{M};\mathbb{Q})$, which can be evaluated
on the fundamental class $[M]$ to obtain an element in
$H^{\mathrm{even}}(\mathbb{T}_{M};\mathbb{Q})$.

We refer the reader to [LM89] for a more detailed discussion of this result
and its generalizations.

###  2.4. Metrics of positive scalar curvature on the torus

We are now ready to prove the following theorem of Gromov and Lawson.

######  Theorem 2.4 ([GL80]).

The torus $T^{n}$ does not admit a metric of positive scalar curvature.

######  Remark 2.3.

In fact, they prove the stronger result that a metric on $T^{n}$ with $s\geq
0$ is necessarily flat.

To see this, we can assume $n$ is even (otherwise we can consider $S^{1}\times
T^{n}$). If $s>0$, then the argument of Proposition 1.3 shows that $D_{B}^{+}$
has trivial kernel and cokernel, hence is an isomorphism, for all $B$. By
Remark 2.1, this implies that $\mathrm{ch}(\mathrm{ind}\\{D_{B}^{+}\\})=0$. As
$\widehat{\mathcal{A}}(T^{n})=1$, the index formula simplifies to

$\displaystyle\mathrm{ch}(\mathrm{ind}\\{D_{B}^{+}\\})$ $\displaystyle=\langle
e^{\Omega},[T^{n}]\rangle=\langle\frac{\Omega^{n}}{n!},[T^{n}]\rangle=$
$\displaystyle=\pm\langle x_{1}\cup\cdots\cup x_{n}\otimes y_{1}\cup\cdots\cup
y_{n},[T^{n}]\rangle=\pm y_{1}\cup\cdots\cup y_{n}\neq 0$

because the cup products $x_{1}\cup\cdots\cup x_{n}$ and $y_{1}\cup\cdots\cup
y_{n}$ evaluate to $\pm 1$ on the fundamental classes $[T^{n}]$ and
$[\mathbb{T}_{T^{n}}]$ of the tori respectively, and we obtain a
contradiction.

###  2.5. Exercises

  1. (1)

Recall that $BU$ is simply connected and has
$H^{2}(BU;\mathbb{Z})=\mathbb{Z}$, generated by the first Chern class $c_{1}$.
Use this to compute $K(S^{2})$, and describe explicitly the generators.

  2. (2)

We will discuss a very concrete example of Kuiper’s theorem. Consider the
family of unitary maps $T_{\vartheta}$ on $l^{2}$ parametrized by
$\vartheta\in S^{1}$ given in the standard basis $\\{e_{i}\\}$ by

$T(e_{i})=\begin{cases}e^{i\vartheta}e_{i}\text{ if }i=0\\\ e_{i}\text{ if
}i>0.\end{cases}$

Show that this this family is homotopically trivial by showing that the family

$\begin{bmatrix}e^{i\vartheta}&0\\\
0&e^{-i\vartheta}\end{bmatrix}\in\mathrm{SU}(2)$

is trivial, and applying the Mazur swindle

$1=1+(-1+1)+(-1+1)+\dots=(1-1)+(1-1)+\dots=0.$

  3. (3)

Show that an orientable surface $\Sigma$ is spin. Use the index theorem for
families to show that on any Riemannian surface $(M,g)$ of genus at least one
there exists a twisted Dirac operator $D_{B}^{+}$ which has non-trivial
kernel.

##  3\. Dirac operators and the Weinstein conjecture in dimension three

###  3.1. Self-adjoint operators

In our previous examples we focused on the case of even dimensional manifolds,
as in this case we have the decomposition of $D_{B}$ into two chiral operators
$D^{\pm}_{B}$ adjoint to each other. In odd dimensions, there is no such
canonical splitting, so we can only work with the Dirac operators $D_{B}$
themselves. Notice that as these are self-adjoint,
$\mathrm{coker}D_{B}=\mathrm{ker}D_{B}$ so they all have automatically index
zero, and hence there is no interesting analogue of Theorem 1.1. On the other
hand, it turns out that it is interesting to study families of them. More
abstractly, consider again a complex separable Hilbert space $H$. Recall that
an operator $T$ is self-adjoint if

$\langle Tx,y\rangle=\langle x,Ty\rangle\text{ for all }x,y\in H.$

This is the infinite dimensional analogue of a being symmetric, but one has to
be careful as the usual spectral theorem in general does not hold: in fact,
$T$ might not even have eigenvalues at all. On the other hand, the
differential operators we are interested in (e.g. $\Delta$ and $D_{B}$) are
always diagonalizable on a compact manifold, so for simplicity we will only
discuss their case888In general, one needs to rephrase the discussion that
follows in terms of the quadratic form $x\mapsto\langle x,Tx\rangle$.. It
turns out that the spectral theory of such operators can have very different
qualitative behavior; for example, looking at complex-valued functions on the
circle $S^{1}$:

  * •

for $c\in\mathbb{R}$ the shifted Laplacian $\Delta+c=-\frac{d^{2}}{dx^{2}}+c$
has eigenvalues $n^{2}+c$ for $n\in\mathbb{Z}$, hence only finitely many
negative eigenvalues and infinitely many positive ones. We call operators like
this essentially positive;

  * •

on the other hand $-\Delta+c$ has infinitely many negative eigenvalues and
finitely many positive ones, and we call it essentially negative;

  * •

finally, the twisted Dirac operator $i\frac{d}{dx}+c$ has eigenvalues $n+c$
for $n\in\mathbb{Z}$ hence it has infinitely many negative eigenvalues and
infinitely many positive eigenvalues. We call operators like this indefinite.

We then have the following.

######  Theorem 3.1 ([AS69]).

The space $\mathrm{Fred}^{\mathrm{sa}}(H)$ of Fredholm self-adjoint operators
on $H$ has three connected components

$\mathrm{Fred}^{\mathrm{sa}}(H)=\\{\text{essentially
positive}\\}\coprod\\{\text{essentially
negative}\\}\coprod\\{\text{indefinite}\\}.$

The first two components are contractible, while the last component has the
homotopy type of $U=\bigcup_{n\geq 1}U(n)$.

######  Remark 3.1.

One should not confuse $U$ with the unitary group $U(H)$ of a Hilbert space
$H$; the latter is contractible by Kuiper’s theorem, while for example
$\pi_{1}(U)=\mathbb{Z}$.

As in the case of the circle $S^{1}$, a twisted Dirac operator $D_{B}$ on a
compact manifold $M$ is an indefinite operator, hence the family $\\{D_{B}\\}$
gives rise to a map

$\mathbb{T}_{M}\rightarrow U,$

well-defined up to homotopy.

Now, for a finite connected CW complex $X$, the set $[X,U]$ can be identified
with $K^{1}(X)$ the odd $K$-theory of $X$; we also denote $K(X)$ by
$K^{0}(X)$, and call it even $K$-theory.

######  Remark 3.2.

More concretely, $K^{1}(X)$ is usually defined as the the kernel of the rank
homomorphism $K^{0}(\Sigma X)\rightarrow\mathbb{Z}$, where $\Sigma X$ is the
reduced suspension of $X$. This is equivalent to the definition above because

$\mathrm{ker}\left(K^{0}(\Sigma X)\rightarrow\mathbb{Z}\right)=[\Sigma
X,BU]=[X,\Omega BU]=[X,U],$

where we used that reduced suspension $\Sigma$ is the adjoint of loopspace
$\Omega$, and $\Omega BG$ is equivalent to $G$ for any $G$.

In particular, a family $\\{S_{x}\\}$ of indefinite self-adjoint operators
parametrized by $X$ determines an index element in $\mathrm{ind}\\{S_{x}\\}\in
K^{1}(X)$. As in the even case, there is a Chern character map

$\mathrm{ch}:K^{1}(X)\rightarrow H^{\mathrm{odd}}(X;\mathbb{Q})$

which is a rational isomorphism. We are then ready to state the analogue of
Theorem 2.3 in the case of odd dimensional manifolds.

######  Theorem 3.2 ([APS76]).

Consider an odd dimensional spin manifold $M$. Then

$\mathrm{ch}(\mathrm{ind}\\{D_{B}\\})=\langle
e^{\Omega}\cdot\widehat{\mathcal{A}}(M),[M]\rangle\in
H^{\mathrm{odd}}(\mathbb{T}_{M};\mathbb{Q})$

where again $e^{\Omega}=\sum\Omega^{n}/n!$.

For example, when $M=S^{1}$, this shows that the family of twisted Dirac
operators $i\frac{d}{dt}+c$ corresponds to a generator of
$\pi_{1}(U)=\mathbb{Z}$.

######  Remark 3.3.

In fact, $\pi_{1}(U)=\mathbb{Z}$ is very closely related to the spectral flow
of a family of operators, a key concept in Floer theory.

###  3.2. The Weinstein conjecture

We want to discuss how families of Dirac operators are related to a dynamical
problem in contact geometry that we now recall (see [CdS01, Part IV] for a
more detailed discussion). Recall that, given an oriented $(2n-1)$-dimensional
manifold $Y$, a contact form is a $1$-form $\alpha$ such that

$\alpha\wedge(d\alpha)^{n-1}>0\text{ at each point.}$

The basic example is $\mathbb{R}^{2n-1}$ with coordinates
$(x_{1},y_{1},\dots,x_{n-1},y_{n-1},z)$ and $\alpha=dz-\sum y_{i}dx_{i}$. Now,
at each point $d\alpha$ is a skew-symmetric bilinear form with maximal rank,
so there exists a unique vector field $R_{\alpha}$, called the Reeb vector
field, for which

$d\alpha(R_{\alpha},-)=0\text{ and }\alpha(R_{\alpha})=1.$

For example, in the case of $\mathbb{R}^{n}$, $R_{\alpha}=\frac{d}{dz}$. The
following conjecture is motivated by questions in classical (Hamiltonian)
mechanics.

######  Conjecture 3.3 (Weinstein).

If $(Y,\alpha)$ is a closed contact manifold, then the Reeb vector field
$R_{\alpha}$ admits a closed orbit.

Of course, the example of $\mathbb{R}^{n}$ above shows the assumption that $M$
is closed is necessary. The Weinstein conjecture has been proved in many
cases, under assumptions with very different nature; we refer for [Hut10] for
a more thorough discussion of the literature. Our goal here is to discuss how
families of Dirac operators enter in the proof of the following celebrated
result of Taubes.

######  Theorem 3.4 ([Tau07]).

The Weinstein conjecture holds in dimension three.

The rough idea is the following: one can perturb the (curvature part of the)
Seiberg-Witten equations on $Y$ [KM07, Ch. 4] by a very large multiple of the
contact form $\alpha$; in this limit, solutions to the Seiberg-Witten
equations naturally give rise to certain union of orbits of $R_{\alpha}$
(including the empty orbit). Because the count of solutions is independent of
the perturbation (in a suitable sense), one can use computations done purely
in Seiberg-Witten theory to show that there are closed Reeb orbits.

In fact, the following result (which is really is a consequence of the
subsequently proved result that embedded contact homology coincides with
monopole Floer homology, see [Hut10]) holds.

######  Theorem 3.5 (Taubes).

If $(Y^{3},\alpha)$ has no closed Reeb orbit, then for the monopole Floer
homology999This is phrased in terms of cohomology in [Hut10]; the statement
here follows from the universal coefficient theorem. groups of $Y$ we have

$\widehat{\mathit{HM}}_{*}(Y,\mathfrak{s})=\begin{cases}\mathbb{Z}\text{ if
}\mathfrak{s}\text{ is the spin${}^{c}$ structure corresponding to
}\mathrm{ker}(\alpha)\\\ 0\text{ for all other spin${}^{c}$ structures on
$Y$.}\end{cases}$

Here, $\mathbb{Z}$ corresponds to the empty Reeb orbit.

The key step is then the following non-triviality result of Kronheimer and
Mrowka.

######  Theorem 3.6 ([KM07], Chapter 35).

For any $Y$, if $\mathfrak{s}$ is a torsion spinc structure,
$\widehat{\mathit{HM}}_{*}(Y,\mathfrak{s})$ is non-vanishing (and in fact has
positive rank) in infinitely many gradings.

This concludes the argument because every three-manifold admits a torsion
spinc structure (for example one induced by a spin structure). Recall that for
a torsion spinc structure $\widehat{\mathit{HM}}_{*}(Y,\mathfrak{s})$
correponds to the Heegaard Floer homology group $HF^{-}(Y,\mathfrak{s})$ (see
[CGH11], [KLT20] and subsequent papers).

###  3.3. Relations with the triple cup product

Theorem 3.6 is proved by showing that the bar version of monopole Floer
homology $\overline{\mathit{HM}}_{*}(Y,\mathfrak{s})$ has positive rank in
infinitely many gradings. Recall that using the $\mathbb{Z}[U]$-module
structure, this can be thought as the localization

$\widehat{\mathit{HM}}_{*}(Y,\mathfrak{s})\otimes_{\mathbb{Z}[U]}\mathbb{Z}[U,U^{-1}],$

and for a torsion spinc structure it corresponds to
$HF^{\infty}(Y,\mathfrak{s})$ in Heegaard Floer homology.

The bar version turns out to be a much simpler object to study. Indeed,
Kronheimer and Mrowka provide in [KM07] a complete computation of
$\overline{\mathit{HM}}_{*}(Y,\mathfrak{s};\mathbb{Q})$. Later, in [Lid12]
Lidman provided a computation with $\mathbb{Z}/2$-coefficients using surgery
techniques in Heegaard Floer homology. Finally, in recent joint work with
Miller Eismeier [LME21], we built on the work of Kronheimer and Mrowka to
provide a full computation with $\mathbb{Z}$-coefficients using techniques
from homotopical algebra. We will give the idea of proof of the following
statement, which is an important step in the proof of Theorem 3.6.

######  Theorem 3.7 ([KM07], Ch. 35).

For any $Y$, if $\mathfrak{s}$ is a torsion spinc structure,
$\overline{\mathit{HM}}_{*}(Y,\mathfrak{s})$ only depends on the triple cup
product map

$\displaystyle\cup^{3}_{Y}$
$\displaystyle:\Lambda^{3}H^{1}(Y;\mathbb{Z})\rightarrow\mathbb{Z}$
$\displaystyle\alpha\wedge\beta\wedge\gamma$
$\displaystyle\mapsto\langle\alpha\cup\beta\cup\gamma,[Y]\rangle.$

In particular, $\overline{\mathit{HM}}_{*}(Y,\mathfrak{s})$ only depends on
the algebraic topology of $Y$, while
$\widehat{\mathit{HM}}_{*}(Y,\mathfrak{s})$ is a much subtler invariant of
$Y$. Intuitively, this is because while
$\widehat{\mathit{HM}}_{*}(Y,\mathfrak{s})$ involves both irreducible and
reducible solutions to the Seiberg-Witten equations,
$\overline{\mathit{HM}}_{*}(Y,\mathfrak{s})$ only involves the reducible ones.
Because of this, it can be described purely in terms of the family of Dirac
operators $\\{D_{B}\\}$ parametrized by the torus $\mathbb{T}_{Y}$.

######  Remark 3.4.

This is a manifestation in Floer homology of the classical localization
theorem in $S^{1}$-equivariant cohomology, see for example [Tu20].

More specifically, associated with $(Y,\mathfrak{s})$ there is a family of
Dirac operators $\\{D_{B}\\}$ parametrized by $\mathbb{T}_{Y}$ which
generalizes the family of twisted Dirac operators associated to a spin
structure.101010Here one should think of $\mathbb{T}_{Y}$ as the space of
reducible solutions to the Seiberg-Witten equations. After choosing a Morse
function $f:\mathbb{T}_{Y}\rightarrow\mathbb{R}$, the chain complex computing
$\overline{\mathit{HM}}_{*}(Y,\mathfrak{s})$ is generated by pairs $(B_{*},E)$
where $B_{*}$ is a critical point of $f$, and $E$ is an eigenspace of
$D_{B_{*}}$. The differential counts solutions to equations combining
flowlines $B(t)\in\mathbb{T}_{Y}$ of $-\nabla f$ and the family of operators
$\\{D_{B(t)}\\}$.

Then the key point is that this construction can be generalized to any family
of indefinite self-adjoint Fredholm operators
$\mathbb{T}_{Y}\rightarrow\mathrm{Fred}^{\mathrm{sa}}(H)$ (and not just the
family of Dirac operators); the result, called coupled Morse homology, only
depends on the homotopy class of the family. In our case this is the index

$\mathrm{ind}\\{D_{B}\\}\in[\mathbb{T}_{Y},U]=K^{1}(\mathbb{T}_{Y}).$

For dimensional reasons we have $\widehat{A}(Y)=1$, so the index formula in
Theorem 3.2 (which also holds in the slightly more general situation of
torsion spinc structures) gives the computation

$\mathrm{ch}(\mathrm{ind}\\{D_{B}\\})=\cup^{3}_{Y}\in
H^{\mathrm{odd}}(\mathbb{T}_{Y})$

under the identification

$H^{3}(\mathbb{T}_{Y};\mathbb{Z})=\Lambda^{3}H^{1}(\mathbb{T}_{Y};\mathbb{Z})=\left(\Lambda^{3}H^{1}(Y;\mathbb{Z})\right)^{*}.$

Finally, one can show that $K^{1}(\mathbb{T}_{Y})$ is torsion-free, so that
the Chern character (which is a rational isomorphism) is injective and
$\mathrm{ind}\\{D_{B}\\}$ is determined by its Chern character.

In fact, the same computation can be used to show that the classifying map
$\mathbb{T}_{Y}\rightarrow U$ for the family $\\{D_{B}\\}$ factors through
$\mathbb{T}_{Y}\rightarrow SU(2)$. This is a key ingredient in the proof of
Theorem 3.6 because for these simpler families one can provide a concrete
description of coupled Morse homology in terms of suitably ‘twisted’ versions
of more familiar theories. In particular, Theorem 3.6 is proved by identifying
$\overline{\mathit{HM}}^{*}(Y,\mathfrak{s})$ over the reals with a version of
twisted de Rham cohomology for which the underlying vector space is
$\Omega^{*}(\mathbb{T}_{Y})\otimes\mathbb{R}[U,U^{-1}]$111111Here we are
working with cohomology, so the action of $U$ has degree $+2$. with
differential

$x\mapsto dx+(\zeta\wedge x)U^{-1}$

for a closed $3$-form $\zeta$ with $[\zeta]=\cup^{3}_{Y}\in
H^{3}(\mathbb{T}_{Y};\mathbb{R})$ [KM07, Chapter $34.3$]. Notice that this is
a differential because $\zeta\wedge\zeta=0$.

###  3.4. Exercises

  1. (1)

Let $f:S^{1}\rightarrow\mathbb{R}$ a non-constant continuous function. Show
that the map on $L^{2}(S^{1})$ (complex valued functions) given by
multiplication by $f$ is self-adjoint but does not have eigenvalues.

  2. (2)

Show that the family of operators $T_{\vartheta}$ in Exercise $2.2$ defines a
loop in $U=\bigcup U(n)$, and in fact represents a generator of $\pi_{1}(U)$.
Why does the Mazur swindle argument not apply in this situation?

  3. (3)

Consider a link $L\subset S^{3}$. Show that the branched double cover
$\Sigma(L)$ has vanishing triple cup product by considering the action of the
covering involution on cohomology. Use this to show that the three-torus
$T^{3}$ is not a branched double cover of $S^{3}$.

## References

  * [APS76] M. F. Atiyah, V. K. Patodi, and I. M. Singer.  Spectral asymmetry and Riemannian geometry. III.  Math. Proc. Cambridge Philos. Soc., 79(1):71–99, 1976.
  * [AS63] M. F. Atiyah and I. M. Singer.  The index of elliptic operators on compact manifolds.  Bull. Amer. Math. Soc., 69:422–433, 1963.
  * [AS69] M. F. Atiyah and I. M. Singer.  Index theory for skew-adjoint Fredholm operators.  Inst. Hautes Études Sci. Publ. Math., (37):5–26, 1969.
  * [AS71] M. F. Atiyah and I. M. Singer.  The index of elliptic operators. IV.  Ann. of Math. (2), 93:119–138, 1971.
  * [Ati67] M. F. Atiyah.  $K$-theory.  W. A. Benjamin, Inc., New York-Amsterdam, 1967.  Lecture notes by D. W. Anderson.
  * [CdS01] Ana Cannas da Silva.  Lectures on symplectic geometry, volume 1764 of Lecture Notes in Mathematics.  Springer-Verlag, Berlin, 2001.
  * [CGH11] Vincent Colin, Paolo Ghiggini, and Ko Honda.  Equivalence of Heegaard Floer homology and embedded contact homology via open book decompositions.  Proc. Natl. Acad. Sci. USA, 108(20):8100–8105, 2011.
  * [Dir28] P. A. M. Dirac.  The Quantum Theory of the Electron.  Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 117 (778):610–624, 1928.
  * [Eva10] Lawrence C. Evans.  Partial differential equations, volume 19 of Graduate Studies in Mathematics.  American Mathematical Society, Providence, RI, second edition, 2010.
  * [FF16] Anatoly Fomenko and Dmitry Fuchs.  Homotopical topology, volume 273 of Graduate Texts in Mathematics.  Springer, [Cham], second edition, 2016.
  * [GL80] Mikhael Gromov and H. Blaine Lawson, Jr.  Spin and scalar curvature in the presence of a fundamental group. I.  Ann. of Math. (2), 111(2):209–230, 1980.
  * [Gre21] Joshua Evan Greene.  Heegaard Floer homology.  Notices Amer. Math. Soc., 68(1):19–33, 2021.
  * [GS99] Robert E. Gompf and András I. Stipsicz.  $4$-manifolds and Kirby calculus, volume 20 of Graduate Studies in Mathematics.  American Mathematical Society, Providence, RI, 1999.
  * [Hit74] Nigel Hitchin.  Harmonic spinors.  Advances in Math., 14:1–55, 1974.
  * [Hut10] Michael Hutchings.  Taubes’s proof of the Weinstein conjecture in dimension three.  Bull. Amer. Math. Soc. (N.S.), 47(1):73–125, 2010.
  * [Kir89] Robion C. Kirby.  The topology of $4$-manifolds, volume 1374 of Lecture Notes in Mathematics.  Springer-Verlag, Berlin, 1989.
  * [KLT20] Çağatay Kutluhan, Yi-Jen Lee, and Clifford Henry Taubes.  $\rm HF{=}HM$, I: Heegaard Floer homology and Seiberg-Witten Floer homology.  Geom. Topol., 24(6):2829–2854, 2020.
  * [KM07] Peter Kronheimer and Tomasz Mrowka.  Monopoles and three-manifolds, volume 10 of New Mathematical Monographs.  Cambridge University Press, Cambridge, 2007.
  * [Lee18] John M. Lee.  Introduction to Riemannian manifolds, volume 176 of Graduate Texts in Mathematics.  Springer, Cham, 2018.  Second edition of [ MR1468735].
  * [Lid12] Tye Lidman.  Triple Cup Products in Heegaard Floer Homology.  ProQuest LLC, Ann Arbor, MI, 2012.  Thesis (Ph.D.)–University of California, Los Angeles.
  * [Lin16] Francesco Lin.  Lectures on monopole Floer homology.  In Proceedings of the Gökova Geometry-Topology Conference 2015, pages 39–80. Gökova Geometry/Topology Conference (GGT), Gökova, 2016.
  * [LM89] H. Blaine Lawson, Jr. and Marie-Louise Michelsohn.  Spin geometry, volume 38 of Princeton Mathematical Series.  Princeton University Press, Princeton, NJ, 1989.
  * [LME21] Francesco Lin and Mike Miller Eismeier.  Monopoles, twisted integral cohomology, and Hirsch algebras.  preprint, arXiv:2108.00984, 2021.
  * [MS74] John W. Milnor and James D. Stasheff.  Characteristic classes.  Annals of Mathematics Studies, No. 76. Princeton University Press, Princeton, N. J.; University of Tokyo Press, Tokyo, 1974.
  * [Roe98] John Roe.  Elliptic operators, topology and asymptotic methods, volume 395 of Pitman Research Notes in Mathematics Series.  Longman, Harlow, second edition, 1998.
  * [Sha78] Patrick Shanahan.  The Atiyah-Singer index theorem, volume 638 of Lecture Notes in Mathematics.  Springer, Berlin, 1978.  An introduction.
  * [Tau07] Clifford Henry Taubes.  The Seiberg-Witten equations and the Weinstein conjecture.  Geom. Topol., 11:2117–2202, 2007.
  * [Tu20] Loring W. Tu.  Introductory lectures on equivariant cohomology, volume 204 of Annals of Mathematics Studies.  Princeton University Press, Princeton, NJ, 2020.  With appendices by Tu and Alberto Arabia.
  * [War71] Frank W. Warner.  Foundations of differentiable manifolds and Lie groups.  Scott, Foresman & Co., Glenview, Ill.-London, 1971.