# Families of relatively exact Lagrangians, free loop spaces and generalised
homology

Noah Porcelli

###### Abstract

We prove that (under appropriate orientation conditions, depending on $R$) a
Hamiltonian isotopy $\psi^{1}$ of a symplectic manifold $(M,\omega)$ fixing a
relatively exact Lagrangian $L$ setwise must act trivially on $R_{*}(L)$,
where $R_{*}$ is some generalised homology theory. We use a strategy inspired
by that of Hu, Lalonde and Leclercq ([HLL11]), who proved an analogous result
over $\mathbb{Z}/2$ and over $\mathbb{Z}$ under stronger orientation
assumptions. However the differences in our approaches let us deduce that if
$L$ is a homotopy sphere, $\psi^{1}|_{L}$ is homotopic to the identity. Our
technical set-up differs from both theirs and that of Cohen, Jones and Segal
([CJS95, Coh09]).

We also prove (under similar conditions) that $\psi^{1}|_{L}$ acts trivially
on $R_{*}(\mathcal{L}L)$, where $\mathcal{L}L$ is the free loop space of $L$.
From this we deduce that when $L$ is a surface or a $K(\pi,1)$,
$\psi^{1}|_{L}$ is homotopic to the identity.

Using methods of [LM03], we also show that given a family of Lagrangians all
of which are Hamiltonian isotopic to $L$ over a sphere or a torus, the
associated fibre bundle cohomologically splits over $\mathbb{Z}/2$.

###### Contents

  1. 1 Introduction
    1. 1.1 Background
    2. 1.2 Extensions to Generalised Homology Theories
    3. 1.3 Extensions to the Free Loop Space
    4. 1.4 Extensions to Other Bases
    5. 1.5 Acknowledgements
  2. 2 Spectra and Pontryagin-Thom Maps
    1. 2.1 Spectra
    2. 2.2 Thom Spectra
    3. 2.3 Pontryagin-Thom Collapse Maps
    4. 2.4 Fundamental Classes
  3. 3 Index Bundles and Their Orientations
    1. 3.1 Cauchy-Riemann Operators
    2. 3.2 Index Bundles
    3. 3.3 Orientations
  4. 4 The Moduli Spaces $\mathcal{P}$ and $\mathcal{W}$
    1. 4.1 Construction of the Moduli Spaces
    2. 4.2 Orientations on the Moduli Space
  5. 5 Monodromy in the Looped Case
    1. 5.1 General Set-Up
    2. 5.2 A Chas-Sullivan Type Product
    3. 5.3 The Moduli Spaces $\mathcal{M}$, $\mathcal{N}$ and $\mathcal{Q}$
    4. 5.4 Bundle Pairs on the Moduli Spaces
    5. 5.5 Gluing and Orientations of the Moduli Spaces
    6. 5.6 Composition of the Moduli Spaces
    7. 5.7 Proof of Theorem 1.16
  6. 6 Families Over Other Bases
  7. A de Silva’s Theorem
  8. B An Example in $K$-Theory

##  1 Introduction

###  1.1 Background

Let $(M^{2n},\omega)$ be a symplectic manifold of dimension $2n$, and
$L^{n}\subseteq M$ a Lagrangian submanifold. Let $\psi^{t}$ be a Hamiltonian
isotopy of $M$ such that its time-1 flow preserves $L$ setwise,
$\psi^{1}(L)=L$. We can consider the _Hamiltonian monodromy group_
$\mathcal{G}_{L}\subseteq\operatorname{Diff}(L)$ of diffeomorphisms of $L$
arising in this way. A natural question is:

######  Question 1.1.

What is $\mathcal{G}_{L}$?

An elementary argument using the Weinstein neighbourhood theorem shows that if
$f$ and $g$ are isotopic diffeomorphisms of $L$ and $g$ lies in
$\mathcal{G}_{L}$, then $f$ is too. This implies that $\mathcal{G}_{L}$ is a
union of connected components in $\operatorname{Diff}(L)$, and hence to study
$\mathcal{G}_{L}$, it suffices to study its image in the mapping class group
$\pi_{0}\operatorname{Diff}(L)$.

The subgroup $\mathcal{G}_{L}$ was first studied by Yau in [Yau09], who proved
the following theorem.

######  Theorem 1.2 ([Yau09]).

Let $T$ and $T^{\prime}$ be the standard monotone Clifford and Chekanov tori
in $\mathbb{C}^{2}$ with the same monotonicity constant. Then in both cases,
$\pi_{0}\mathcal{G}_{L}\cong\mathbb{Z}/2$.

However, there is no isomorphism $H_{1}(T)\cong H_{1}(T^{\prime})$ which
respects the $\mathbb{Z}/2$ action on both and also preserves the Maslov index
homomorphism.

This provides a new proof that these two Lagrangians are not Hamiltonian
isotopic.

Question 1.1 has been studied in few other places: by Hu, Lalonde and Leclercq
in [HLL11], by Mangolte and Welschinger in [MW12], by Ono in [Ono15], by
Keating in [Kea21], and by Augustynowicz, Smith and Wornbard in [ASW22]. Out
of these, only Hu, Lalonde and Leclercq focus on the relatively exact setting,
which is where we will focus.

We now assume (for the rest of the paper) the following:

######  Assumption 1.3.

  1. 1.

$M$ is a product of symplectic manifolds which are either compact or
Liouville.

  2. 2.

$L$ is compact, connected and relatively exact, i.e. $\omega$ vanishes on the
relative homotopy group $\pi_{2}(M,L)$.

  3. 3.

$\psi^{t}$ is compactly supported for all $t$ and constant in $t$ near 0 and
1.

Note that given $\psi^{t}$ not satisfying (3), we can always deform it so that
it does, without changing $\psi^{1}|_{L}$.

$\mathcal{G}_{L}$ was studied by Hu, Lalonde and Leclercq in this more
restrictive setting in [HLL11], where they proved the following:

######  Theorem 1.4 ([HLL11]).

  1. 1.

$\psi^{1}|_{L}$ acts as the identity on $H_{*}(L;\mathbb{Z}/2)$.

  2. 2.

If $L$ admits a relative spin structure in $M$ which $\psi^{1}$ preserves,
$\psi^{1}|_{L}$ acts as the identity on $H_{*}(L;\mathbb{Z})$.

It follows from Yau’s results that relative exactness is a necessary condition
here.

We will recover this as a special case of Theorem 1.7. Note that in the case
that $L$ is a homotopy sphere, this theorem does not give any information on
the homotopy class of $\psi^{1}|_{L}$: if $\psi^{1}|_{L}$ were not homotopic
to the identity, it would not preserve any orientation of $L$ and hence would
not preserve any relative spin structure.

Our goals will be to weaken the orientation assumptions, extend this result to
some other generalised cohomology theories, and extend this result to the free
loop space of $L$. From this, we will fully compute $\mathcal{G}_{L}$ in the
case that $n=2$. Our technical set-up will be very different to theirs but our
general approach is inspired by theirs.

######  Remark 1.5.

Hu, Lalonde and Leclercq prove this using Morse and quantum cohomology (as in
[BC09]), and one possible approach to extending this theorem to other
generalised cohomology theories would be to recreate their proof using the
methods of Cohen, Jones and Segal ([CJS95, Coh09]). However one may need to
require stronger orientation hypotheses than Assumption 1.6.

###  1.2 Extensions to Generalised Homology Theories

We will define a space $\mathcal{D}_{0}$ which is roughly the space of smooth
maps $D^{2}\rightarrow M$ sending 1 to $L$, with moving Lagrangian boundary
conditions on the rest of the boundary. This admits a map
$\pi\mathrel{\mathop{\mathchar 58\relax}}\mathcal{D}_{0}\rightarrow L$ by
evaluating at 1. We will construct a virtual vector bundle called the _index
bundle_ , $\mathrm{Ind}$, on any finite CW complex in $\mathcal{D}_{0}$
(compatible with restriction). After picking a (generic) almost complex
structure $J$ on $M$ and some specific choices of moving Lagrangian boundary
conditions, the tangent bundle of the moduli space of $J$-holomorphic discs
with these boundary conditions will be stably isomorphic to $\mathrm{Ind}$.

We fix some ring spectrum $R$, and will recall definitions of ring spectra and
$R$-orientability in Section 2. We will make the following assumption, and in
Section 4.2, find conditions under which it holds.

######  Assumption 1.6.

The virtual vector bundle $\mathrm{Ind}-\pi^{*}TL$ is $R$-orientable.

We will show:

######  Theorem 1.7.

Under Assumption 1.6, the map

$\psi^{1}|_{L}\mathrel{\mathop{\mathchar 58\relax}}\Sigma^{\infty}_{+}L\wedge
R\rightarrow\Sigma^{\infty}_{+}L\wedge R$

is homotopic to the identity as a map of $R$-modules.

######  Remark 1.8.

Our proof of this will use a minimal amount of technical machinery: we will
only use standard Gromov compactness results and standard transversality
results, and we will not need any form of gluing. In fact, it is possible to
prove this without invoking any transversality results, and instead only use
the fact that certain operators are Fredholm, using ideas in [Hof88].

From Corollaries 2.21 and 2.20, we deduce:

######  Corollary 1.9.

Under Assumption 1.6,

  1. 1.

$\psi^{1}|_{L}$ induces the identity map on $R_{*}(L)$.

  2. 2.

If $TL$ is $R$-orientable, $\psi^{1}|_{L}$ induces the identity map on
$R^{*}(L)$.

In Section 4, we will define a moduli space $\mathcal{P}$ of holomorphic discs
in $M$, with moving boundary conditions, lying naturally inside
$\mathcal{D}_{0}$. Its tangent space $T\mathcal{P}$ will by stably isomorphic
to $\mathrm{Ind}$. This will satisfy the following:

######  Lemma 1.10.

$\mathcal{P}$ is a closed smooth manifold of dimension $n$.

and

######  Lemma 1.11.

The following diagram commutes up to homotopy:

${L}$${\mathcal{P}}$${L}$$\scriptstyle{\psi^{1}}$$\scriptstyle{\pi}$$\scriptstyle{\pi}$

Furthermore, by varying the boundary conditions in a 1-parameter family, we
will construct a cobordism $\pi\mathrel{\mathop{\mathchar
58\relax}}\mathcal{W}\to L$ from $Id\mathrel{\mathop{\mathchar
58\relax}}L\rightarrow L$ to $\pi\mathrel{\mathop{\mathchar
58\relax}}\mathcal{P}\rightarrow L$ over $L$, such that
$T\mathcal{W}-\pi^{*}TL$ will be $R$-orientable under Assumption 1.6. A
standard application of the Pontryagin-Thom construction (see Section 2) will
prove the following lemma:

######  Lemma 1.12.

The induced map $\pi\mathrel{\mathop{\mathchar
58\relax}}\Sigma^{\infty}_{+}\mathcal{P}\wedge
R\rightarrow\Sigma^{\infty}_{+}L\wedge R$ admits a section of $R$-module maps
(up to homotopy of $R$-module maps).

###### Proof.

This follows from the construction of $\mathcal{W}$ in Section 4, along with
Lemma 2.19. ∎

From these three lemmas, Theorem 1.7 follows.

In Section 4.2, we will show the following:

######  Proposition 1.13.

Assumption 1.6 holds when:

  1. 1.

$R=H\mathbb{Z}/2$ representing mod-2 singular homology.

  2. 2.

$R=H\mathbb{Z}$ representing integral singular homology, and $w_{2}(L)(x)=0$
for $x$ in the image of the composition

$\pi_{3}(M,L)\rightarrow\pi_{2}(L)\rightarrow H_{2}(L;\mathbb{Z})$

  3. 3.

$R=KU$ representing complex $K$-theory, and $L$ admits a spin structure.

  4. 4.

$R=KO$ representing real $K$-theory, and $TL$ admits a stable trivialisation
over a 3-skeleton of $L$ which extends (after applying
$\cdot\otimes\mathbb{C}$) to a stable trivialisation of $TM$ over a 4-skeleton
of $M$.

  5. 5.

$R$ is a complex-orientable generalised cohomology theory and $L$ is stably
parallelisable.

  6. 6.

$R=\mathbb{S}$ is the sphere spectrum and there is a stable trivialisation of
$TL$ which (after applying $\cdot\otimes\mathbb{C}$) extends to a stable
trivialisation of $TM$.

######  Remark 1.14.

Our methods can be used to show that when $R=H\mathbb{Z}$, Assumption 1.6
holds if $L$ is spin. We will deduce the stronger statement (2) from [Geo13,
Theorem 1.1].

######  Corollary 1.15.

When $L$ is a homotopy sphere, $\psi^{1}|_{L}$ is homotopic to the identity.

###### Proof.

$L$ admits a spin structure, so $\psi^{1}$ acts as the identity on integral
homology and so is homotopic to the identity. ∎

It is nontrivial to find examples of self-diffeomorphisms of spin manifolds
which act trivially on integral cohomology but non-trivially on complex
$K$-theory, so we provide an example in Appendix B, following a suggestion of
Randal-Williams ([RW]).

###  1.3 Extensions to the Free Loop Space

Fix a ring spectrum $R$. We will prove an analogue of Theorem 1.7 for the free
loop space $\mathcal{L}L$ of $L$, the space of maps $S^{1}\rightarrow L$.

######  Theorem 1.16.

Assume Assumption 5.9 holds. Then the induced map

$\psi^{1}|_{L}\mathrel{\mathop{\mathchar
58\relax}}\Sigma^{\infty}_{+}\mathcal{L}L\wedge
R\rightarrow\Sigma^{\infty}_{+}\mathcal{L}L\wedge R$

is homotopic to the identity.

###### Idea of proof.

To prove this, we will define $\mathcal{L}_{1}$ to be the space of free loops
in the mapping torus $L_{\psi^{1}}$ of $\psi^{1}|_{L}$ which have winding
number one over $S^{1}$, and whose basepoint lies over $L$. $\psi$ determines
an automorphism $\Psi^{1}\mathrel{\mathop{\mathchar
58\relax}}\mathcal{L}_{1}\rightarrow\mathcal{L}_{1}$, which one should think
of as parallel transporting once around this mapping torus.

We will construct a map

$p\circ\left(\left[\mathcal{N}\right]\cdot\right)\mathrel{\mathop{\mathchar
58\relax}}\bigvee\limits_{j}\Sigma^{\infty+j}_{+}\mathcal{L}_{1}\wedge
R\rightarrow\Sigma_{+}^{\infty}\mathcal{L}L\wedge R$

This will use a moduli space of holomorphic discs with moving boundary
conditions $\mathcal{N}$, as well as Cohen and Jones’ version of the Chas-
Sullivan product, constructed similarly to [CJ02]. We will then show:

  1. 1.

$p\circ\left(\left[\mathcal{N}\right]\cdot\right)$ intertwines the actions of
$\Psi^{1}$ and $\psi^{1}|_{L}$.

  2. 2.

$\Psi^{1}$ acts as the identity on $\mathcal{L}_{1}$ up to homotopy.

  3. 3.

$p\circ\left(\left[\mathcal{N}\right]\cdot\right)$ admits a section.

Together, these will imply Theorem 1.16. ∎

######  Proposition 1.17.

Assumption 5.9 holds when

  1. 1.

$R=H\mathbb{Z}/2$.

  2. 2.

$R=H\mathbb{Z}$ and $L$ admits a spin structure which is preserved by
$\psi^{1}$.

  3. 3.

$R=KU$ and $L$ admits a homotopy class of stable trivialisations over a
2-skeleton which is preserved under $\psi^{1}|_{L}$.

  4. 4.

$R$ is any complex-orientable generalised cohomology theory and $TL$ admits a
homotopy class of stable trivialisations which is preserved under
$\psi^{1}|_{L}$.

######  Corollary 1.18.

Assume Assumption 5.9 holds. Then the map

$\left(\psi^{1}|_{L}\right)_{*}\mathrel{\mathop{\mathchar
58\relax}}R_{*}(\mathcal{L}L)\rightarrow R_{*}(\mathcal{L}L)$

is the identity.

From this, we can obtain strong resctricions on $\mathcal{G}_{L}$ for certain
$L$ that we could not get from Theorem 1.7:

######  Corollary 1.19.

  1. 1.

If $L$ is a $K(\pi_{1}(L),1)$, $\psi^{1}|_{L}$ is homotopic to the identity.

  2. 2.

If $n=2$, $\psi^{1}|_{L}$ is isotopic to the identity, and hence
$\mathcal{G}_{L}$ is trivial.

###### Proof.

Recall that a diffeomorphism of a closed surface is homotopic to the identity
iff it is isotopic to the identity.

It follows from Corollary 1.15 that if $L=S^{2}$, then $\psi^{1}|_{L}$ is
isotopic to the identity. Furthermore, since the mapping class group of
$\mathbb{R}\mathbb{P}^{2}$ is trivial, we can assume that $L$ is a
$K(\pi_{1}(L),1)$.

Note that $H_{0}(\mathcal{L}L;\mathbb{Z}/2)$ is the free $\mathbb{Z}/2$-module
generated by homotopy classes of free loops in $L$. Therefore by Theorem 1.16,
$\psi^{1}|_{L}$ acts as the identity on this set of generators. Then by [CV20,
Theorem 2.4], $\psi^{1}|_{L}$ must be homotopic to the identity. ∎

###  1.4 Extensions to Other Bases

In this subsection and Section 6, we restrict to the case $R=H\mathbb{Z}/2$.

######  Definition 1.20.

Given a fibre bundle $E\twoheadrightarrow B$, we say it _c-splits_ if the
inclusion of a fibre into $E$ induces an injection on mod-2 singular homology,
for any fibre of this bundle.

We define $Lag_{L}$ to be the space of (unparametrised) Lagrangian
submanifolds of $M$ which are Hamiltonian isotopic to $L$. There is a natural
fibre bundle $\mathcal{E}$ over $Lag_{L}$, where the fibre over $K$ in
$Lag_{L}$ is $K$. This is naturally a subbundle of $Lag_{L}\times M$. Given a
map $\gamma\mathrel{\mathop{\mathchar 58\relax}}S^{1}\rightarrow Lag_{L}$,
Theorem 1.4 implies that the pullback bundle $\gamma^{*}\mathcal{E}$ c-splits.
We will use techniques in [LM03] to deduce the following from Theorem 1.4:

######  Theorem 1.21.

  1. (i).

Given a map $\gamma\mathrel{\mathop{\mathchar 58\relax}}(S^{1})^{i}\rightarrow
Lag_{L}$, the pullback bundle
$\gamma^{*}\mathcal{E}\twoheadrightarrow(S^{1})^{i}$ c-splits.

  2. (ii).

Given a map $\gamma\mathrel{\mathop{\mathchar 58\relax}}S^{i}\rightarrow
Lag_{L}$, the pullback bundle $\gamma^{*}\mathcal{E}\twoheadrightarrow S^{i}$
c-splits.

###  1.5 Acknowledgements

The author would like to thank Amanda Hirschi, Jack Smith, Nick Nho and Oscar
Randal-Williams for helpful conversations regarding this project, as well as
Ailsa Keating, Mohammed Abouzaid, Oscar Randal-Williams and Thomas Kragh for
useful comments on an earlier draft. The author would also like to express his
deep gratitude to his supervisor Ivan Smith for all his support, motivation,
productive discussions and ability to endure endless questions over the past
two and a half years. The author would also like to thank the EPSRC for
providing funding for his PhD.

##  2 Spectra and Pontryagin-Thom Maps

We assume all spaces that we work with are Hausdorff, paracompact and homotopy
equivalent to CW complexes. For an unbased space $X$, we will write $X_{+}$
for $X$ with a disjoint basepoint added.

We will use the category of spectra described in [Rud98], whose construction
and properties we will sketch here.

###  2.1 Spectra

######  Definition 2.1.

A _spectrum_ $X$ is a sequence $\\{X_{n}\\}_{n\in\mathbb{N}}$ of based spaces
along with _structure maps_ $\Sigma X_{n}\rightarrow X_{n+1}$.

A _function_ $f$ from $X$ to $Y$ is a family of based maps of spaces
$f_{n}\mathrel{\mathop{\mathchar 58\relax}}X_{n}\rightarrow Y_{n}$ such that
the square

${\Sigma X_{n}}$${\Sigma Y_{n}}$${X_{n+1}}$${Y_{n+1}}$$\scriptstyle{\Sigma
f_{n}}$$\scriptstyle{f_{n+1}}$

commutes.

We will not define a morphism of spectra (see [Rud98, §II.1] for details), but
any function of spectra is a morphism of spectra, and all morphisms of spectra
that we will consider will arise in this way.

######  Remark 2.2.

Rudyak actually defines a spectrum in a slightly different way but shows the
two are equivalent in [Rud98, Lemma II.1.19].

######  Definition 2.3.

Given a spectrum $X$ and a based space $Z$, there is a spectrum $X\wedge Z$
with $(X\wedge Z)_{n}=X_{n}\wedge Z$.

A _homotopy_ between two functions $f,g\mathrel{\mathop{\mathchar
58\relax}}X\rightarrow Y$ is a function

$H\mathrel{\mathop{\mathchar 58\relax}}X\wedge[0,1]_{+}\rightarrow Y$

restricting to $f$ over $\\{0\\}_{+}$ and to $g$ over $\\{1\\}_{+}$.

One can extend this definition to homotopies of morphisms of spectra, as in
[Rud98, Definition II.1.9], and for homotopic morphisms $f$ and $g$ we write
$f\simeq g$. Then for spectra $X$ and $Y$, we define $[X,Y]$ to be the set of
homotopy classes of morphisms $X\rightarrow Y$. There is a natural structure
of an abelian group on this set.

From this definition of a homotopy we can define a notion of _(homotopy)
equivalence_ as with spaces, which we denote by $\simeq$.

There is a natural functor $\Sigma^{\infty}$ from based spaces to $Sp$ sending
a space $X$ to the spectrum with $(\Sigma^{\infty}X)_{n}=\Sigma^{n}X$, with
structure maps the identity. We write $\Sigma^{\infty}_{+}$ for the functor
from unbased spaces to $Sp$ sending $X$ to $\Sigma^{\infty}X_{+}$. These both
send homotopies to homotopies. We will denote $\Sigma^{\infty}_{+}\\{*\\}$ by
$\mathbb{S}$, and call this the _sphere spectrum_.

For any $N$ in $\mathbb{Z}$, there is an endofunctor $\Sigma^{N}$ of $Sp$
which sends $X$ to the spectrum $\Sigma^{N}X$ with

$(\Sigma^{N}X)_{n}=\begin{cases}X_{n+N}&\text{ if }n+N\geq 0\\\ \\{*\\}&\text{
Otherwise}\end{cases}$

This satisfies $\Sigma^{N}\Sigma^{M}X\simeq\Sigma^{N+M}X$ for all $X$, and
$[X,Y]=[\Sigma^{N}X,\Sigma^{N}Y]$ for all $X,Y$. We write
$\Sigma^{\infty+N}_{+}X$ for $\Sigma^{N}\Sigma^{\infty}_{+}X$.

######  Definition 2.4.

Given a family $\\{X_{\lambda}\\}_{\lambda\in\Lambda}$ of spectra, we can
define their _wedge product_ to be the spectrum
$\bigvee\limits_{\lambda}X_{\lambda}$ with

$\left(\bigvee\limits_{\lambda}X_{\lambda}\right)_{n}=\bigvee\limits_{\lambda}X_{n}$

######  Lemma 2.5 ([Rud98, Proposition II.1.16]).

For any spectrum $Y$, there are natural isomorphisms

$\left[\bigvee\limits_{\lambda}X_{\lambda},Y\right]\cong\prod\limits_{\lambda}\left[X_{\lambda},Y\right]$

and

$\left[Y,\bigvee\limits_{\lambda}X_{\lambda}\right]\cong\bigoplus\limits_{\lambda}\left[Y,X_{\lambda}\right]$

We can extend the smash product $\wedge$ to two spectra, functorially in each
argument, as in [Ada74]. We will not need explicit construction but will use
some of its properties:

######  Theorem 2.6 ([Rud98, Theorem II.2.1]).

There are equivalences

  1. 1.

$(X\wedge Y)\wedge Z\simeq X\wedge(Y\wedge Z)$

  2. 2.

$X\wedge Y\simeq Y\wedge X$

  3. 3.

$X\wedge\mathbb{S}\simeq X$

  4. 4.

$\Sigma X\wedge Y\simeq\Sigma(X\wedge Y)$

such that all natural diagrams made up of these equivalences commute up to
homotopy.

######  Definition 2.7.

A _ring spectrum_ $R$ is a spectrum equipped with a unit morphism
$\eta\mathrel{\mathop{\mathchar 58\relax}}\mathbb{S}\rightarrow R$ and product
morphism $\mu\mathrel{\mathop{\mathchar 58\relax}}R\wedge R\rightarrow R$,
satisfying appropriate associativity and unitality conditions up to homotopy.

A _module_ over a ring spectrum is defined similarly.

A _map_ of $R$-modules $X\rightarrow Y$ is a map of spectra $X\rightarrow Y$
which commutes with the actions of $R$ on $X$ and $Y$ up to homotopy.

A _homotopy_ of $R$-module maps between $f$ and $g\mathrel{\mathop{\mathchar
58\relax}}X\rightarrow Y$ is a homotopy of spectra
$X\wedge[0,1]_{+}\rightarrow Y$ between $f$ and $g$, which is also a map of
$R$-modules.

######  Example 2.8.

The sphere spectrum $\mathbb{S}$ is naturally a ring spectrum, and every
spectrum is naturally a module spectrum over it.

######  Example 2.9.

For a ring spectrum $R$, there is a functor from spectra to $R$-module
spectra, given by $\cdot\wedge R$.

######  Definition 2.10.

We define the _stable homotopy groups_ of $X$ to be

$\pi_{i}X\mathrel{\mathop{\mathchar 58\relax}}=[\Sigma^{i}\mathbb{S},X]$

This is covariantly functorial in $X$.

Given a spectrum $R$, we define $R_{i}(X)$ to be $\pi_{i}(X\wedge R)$ and
$R^{i}(X)$ to be $[X,\Sigma^{i}R]$. These are functorial in $X$, covariantly
and contravariantly respectively.

Given a space $Z$, we define $R_{*}(Z)$ to be $R_{*}(\Sigma^{\infty}_{+}Z)$
and $R^{*}(Z)$ to be $R^{*}(\Sigma^{\infty}_{+}Z)$. These are functorial in
$Z$, covariantly and contravariantly respectively.

By Brown’s representability theorem ([Hat02, Theorem 4E.1]), for an abelian
group $G$ there is a (unique up to homotopy equivalence) spectrum $HG$ such
that $HG_{*}$ and $HG^{*}$ are homology and cohomology with co-efficients in
$G$ respectively, and when $G$ is a ring these are in fact ring spectra.
Similarly there are (unique up to homotopy equivalence) ring spectra $KO$ and
$KU$ such that $KO^{*}$ and $KU^{*}$ are real and complex $K$-theory
respectively.

###  2.2 Thom Spectra

Let $\xi\mathrel{\mathop{\mathchar 58\relax}}E\rightarrow X$ be a vector
bundle of rank $n$ over a space $X$. We let $DE$ and $SE$ denote the unit disc
and unit sphere bundles of $E$ respectively, with respect to some choice of
metric.

######  Definition 2.11.

We define the _Thom space_ of $E$, denoted $X^{E}_{u}$, to be the quotient
$DE/SE$. This is a based space with basepoint given by the image of $SE$.

We define the _Thom spectrum_ of $E$, denoted $X^{E}$, to be the spectrum
$\Sigma^{\infty}X^{E}_{u}$.

If $E^{\prime}$ is a virtual vector bundle which can be written as
$E^{\prime}\cong E-\mathbb{R}^{m}$ for some $m$ (in particular, this includes
any virtual vector bundle pulled back from one over a compact space), we
define the _Thom spectrum_ of $E^{\prime}$ to be $\Sigma^{\infty-m}X_{u}^{E}$.

######  Remark 2.12.

None of these depend on the choice of metric up to homotopy equivalence, and
furthermore the Thom spectrum only depends on the stable isomorphism class of
the virtual vector bundle by [Rud98, Lemma IV.5.14].

Now let $\xi\mathrel{\mathop{\mathchar 58\relax}}E\rightarrow X$ be a virtual
vector bundle.

######  Definition 2.13.

$E$ is _orientable_ with respect to $R$ if there is a morphism
$U\mathrel{\mathop{\mathchar 58\relax}}X^{E}\rightarrow\Sigma^{n}R$, called
the _Thom class_ , whose restriction to a fibre (which is a copy of
$\Sigma^{n}\mathbb{S}$) represents plus or minus the unit in $R$.

An _orientation_ is a homotopy class of such morphisms.

It follows from Remark 2.12 that being $R$-orientable only depends on the
stable isomorphism class of $E$.

######  Lemma 2.14 ([Rud98, Proposition V.1.10 and Examples V.1.23]).

  1. 1.

Any virtual vector bundle is canonically oriented with respect to
$H\mathbb{Z}/2$.

  2. 2.

A virtual vector bundle is orientable with respect to $H\mathbb{Z}$ iff it is
orientable in the usual sense, and there is a natural bijection between
$H\mathbb{Z}$-orientations and orientations in the usual sense.

  3. 3.

A trivial virtual vector bundle is orientable with respect to any $R$.

  4. 4.

If $E$ is oriented with respect to $R$ and $f\mathrel{\mathop{\mathchar
58\relax}}Y\rightarrow X$ is any map, the pullback bundle $f^{*}E$ admits a
natural $R$-orientation.

  5. 5.

If $F$ is another virtual vector bundle over $X$ and any two of $E$, $F$ and
$E\oplus F$ are $R$-oriented, then the third admits a natural $R$-orientation.

We will need a stable version of the Thom isomorphism theorem:

######  Theorem 2.15 ([Rud98, Theorem V.1.15 and Exercise V.1.28]).

An $R$-orientation of $E$ induces an equivalence of $R$-module spectra

$\Sigma^{\infty+n}_{+}X\wedge R\simeq X^{E}\wedge R$

More generally, if $F\rightarrow X$ is another virtual vector bundle, then an
$R$-orientation of $E$ induces an equivalence of $R$-module spectra

$\Sigma^{n}X^{F}\wedge R\simeq X^{E\oplus F}\wedge R$

###  2.3 Pontryagin-Thom Collapse Maps

Let $i\mathrel{\mathop{\mathchar 58\relax}}X\hookrightarrow Y$ be a smooth
embedding of manifolds with $X$ compact, such that $i(\partial
X)\subseteq\partial Y$. Let $\nu$ be its normal bundle, with unit disc bundle
$D\nu$ and unit sphere bundle $S\nu$ (with respect to some choice of metric).
Furthermore we fix a tubular neighbourhood of $X$, i.e. we pick an embedding
$j\mathrel{\mathop{\mathchar 58\relax}}D\nu\hookrightarrow Y$ extending $i$,
and not touching $\partial Y$ apart from over $\partial X$.

We will denote the one-point compactification of $Y$ by $Y_{\infty}$, viewed
as a based space with basepoint at infinity. If $Y$ is already compact then
this is $Y_{+}$.

######  Definition 2.16.

We define the _Pontryagin-Thom collapse map_
$i_{!,u}\mathrel{\mathop{\mathchar 58\relax}}Y_{\infty}\rightarrow
X^{\nu}_{u}$ by

$i_{!,u}(x)=\begin{cases}j^{-1}(x)&\textit{ if }x\in j(D\nu)\\\ S\nu&\textit{
otherwise}\end{cases}$

We denote the stabilisation $\Sigma^{\infty}i_{!,u}$ by $i_{!}$.

The map $i_{!,u}$ is continuous, and both the space $X^{\nu}_{u}$ and the map
$i_{!,u}$ are independent of the choices of metric and tubular neighbourhood
up to homotopy.

Now let $f\mathrel{\mathop{\mathchar 58\relax}}X^{m}\rightarrow Y^{n}$ be any
smooth map of manifolds.

######  Definition 2.17.

We define the _stable normal bundle_ of $f$, $\nu_{f}$, to be the virtual
vector bundle $f^{*}TY-TX$.

When $f$ is an embedding, this is stably isomorphic to the normal bundle of
the embedding, defined in the usual sense.

Now assume $X$ and $Y$ are closed, and choose a smooth embedding

$i\mathrel{\mathop{\mathchar 58\relax}}X\hookrightarrow\mathbb{R}^{N}$

for some $N$. Now $\nu_{f}\oplus\mathbb{R}^{N}$ is stably isomorphic to the
normal bundle $\nu_{f\times i}$ of the embedding

$f\times i\mathrel{\mathop{\mathchar 58\relax}}X\hookrightarrow
Y\times\mathbb{R}^{N}$

We can consider the map

$(f\times i)_{!}\mathrel{\mathop{\mathchar
58\relax}}\Sigma^{N}\Sigma^{\infty}Y_{\infty}\simeq\Sigma^{\infty}\left(Y\times\mathbb{R}^{N}\right)_{\infty}\rightarrow
X^{\nu_{f\times i}}\simeq\Sigma^{N}X^{\nu_{f}}$

######  Lemma 2.18.

Up to homotopy the morphism
$\Sigma^{\infty+N}Y_{\infty}\rightarrow\Sigma^{N}X^{\nu_{f}}$ does not depend
on the choice of $i$ or $N$.

###### Proof.

First, observe that if we compose $i$ with the standard embedding
$\mathbb{R}^{N}\hookrightarrow\mathbb{R}^{N+1}$, the resulting map is the same
up to suspension.

Assume $i$, $i^{\prime}$ are two choices of embedding
$X\hookrightarrow\mathbb{R}^{N}$. Pick an embedding

$I\mathrel{\mathop{\mathchar
58\relax}}X\times[0,1]\hookrightarrow\mathbb{R}^{N}\times[0,1]$

over $[0,1]$ restricting to $i$ and $i^{\prime}$ over $0$ and $1$
respectively, increasing $N$ if necessary as above. Then $(f\times I)_{!,u}$
provides a homotopy between $(f\times i)_{!,u}$ and $(f\times
i^{\prime})_{!,u}$. ∎

Now that we know the desuspended map $\Sigma^{\infty}Y_{\infty}\rightarrow
X^{\nu_{f}}$ doesn’t depend on these choices (up to homotopy), we denote it by
$f_{!}$.

We fix a ring spectrum $R$. Given an $R$-orientation of $\nu_{f}$, we consider
the following composition, which is a map of $R$-modules:

$p_{f}\mathrel{\mathop{\mathchar 58\relax}}\Sigma^{\infty}Y_{\infty}\wedge
R\xrightarrow{f_{!}}X^{\nu_{f}}\wedge
R\xrightarrow{\simeq}\Sigma^{\infty+d}_{+}X\wedge
R\xrightarrow{f}\Sigma^{\infty+d}Y_{\infty}\wedge R$

where $d=n-m$. By construction, this factors through
$\Sigma^{\infty+d}_{+}X\wedge R$.

Now suppose that $n=m$ (so $d=0$), $Y$ is compact (so $Y_{\infty}=Y_{+}$), and
there is a cobordism $W$ from $X$ to $Y$ along with a map
$F\mathrel{\mathop{\mathchar 58\relax}}W\rightarrow Y$ restricting to the
identity on $Y$ and to $f$ on $X$, as in the following diagram:

${X}$${W}$${Y}$${Y}$$\scriptstyle{f}$$\scriptstyle{Id}$$\scriptstyle{F}$

We assume that $\nu_{F}$ admits an orientation with respect to $R$, which, by
restriction, induces one for $\nu_{f}$.

######  Lemma 2.19.

After applying $\Sigma^{\infty}_{+}\cdot\wedge R$, $f$ admits a section of
$R$-module maps up to homotopy (of $R$-module maps).

###### Proof.

We will show that the map $p_{f}$ constructed above is an equivalence, then
$f_{!}\circ p_{f}^{-1}$ will be the desired section.

We pick an embedding

$I\mathrel{\mathop{\mathchar
58\relax}}W\hookrightarrow\mathbb{R}^{N}\times[0,1]$

for some $N$, such that $I^{-1}\left(\mathbb{R}^{N}\times\\{0\\}\right)=X$ and
$I^{-1}\left(\mathbb{R}^{N}\times\\{1\\}\right)=Y$. Then choosing an
orientation of $\nu_{F}$ and performing the above construction to $F$ gives us
a homotopy from $p_{f}$ to $p_{Id}$, where $p_{Id}$ is constructed similarly
to $p_{f}$ but for $Id\mathrel{\mathop{\mathchar 58\relax}}Y\rightarrow Y$,
using some choice of trivialisation on the trivial vector bundle over $Y$. But
$p_{Id}$ is a composition of three equivalences and hence $p_{f}$ is an
equivalence.

All the maps and homotopies are constructed either by applying $\cdot\wedge R$
to a map of spectra or by applying the Thom isomorphism theorem over $R$, so
they are all $R$-module maps. ∎

######  Corollary 2.20.

The map $f_{*}\mathrel{\mathop{\mathchar 58\relax}}R_{*}X\rightarrow R_{*}Y$
is surjective.

It follows from [Rud98, Lemma II.2.4 and Theorem V.2.3] that

######  Corollary 2.21.

If $W$ is $R$-orientable (and hence $X$ and $Y$ are too), the map
$f^{*}\mathrel{\mathop{\mathchar 58\relax}}R^{*}Y\rightarrow R^{*}X$ is
injective.

We can also define Pontryagin-Thom collapse maps without smoothness
assumptions. Let $i\mathrel{\mathop{\mathchar 58\relax}}X\hookrightarrow Y$ be
an embedding of spaces.

######  Definition 2.22.

We say $X$ admits a _tubular neighbourhood_ in $Y$ with _normal bundle_ $\nu$
if there is a vector bundle $\nu$ over $X$ and an open neighbourhood $U$ of
$X$ in $Y$, such that there is a homeomorphism $\phi\mathrel{\mathop{\mathchar
58\relax}}U\rightarrow DE$ between $U$ and the unit disc bundle $DE$ of $\nu$
(for some choice of fibrewise metric on $\nu$), sending $X$ to the zero
section.

######  Definition 2.23.

Let $E\rightarrow Y$ be another vector bundle. We define the _Pontryagin-Thom
collapse map_ $i_{!,u}\mathrel{\mathop{\mathchar
58\relax}}Y^{E}_{u}\rightarrow X^{\nu\oplus E}_{u}$ by

$(x,v)\mapsto\begin{cases}(\phi(x),v)&\mathrm{if}\,x\in U\\\ S(\nu\oplus
E)&\mathrm{otherwise}\end{cases}$

We write $i_{!}$ for the induced map on Thom spectra.

This definition extends to the case when $E$ is a virtual vector bundle of the
form $E^{\prime}-\mathbb{R}^{N}$, as we can apply this construction to
$E^{\prime}$ and desuspend. Similarly to before, up to homotopy
$i_{!}\mathrel{\mathop{\mathchar 58\relax}}=\Sigma^{\infty}i_{!,u}$ then only
depends on the virtual vector bundle up to stable isomorphism.

######  Example 2.24.

When $X\hookrightarrow Y$ is a proper embedding of smooth manifolds, $X$
always admits a tubular neighbourhood in $Y$, and we recover the earlier
construction.

###  2.4 Fundamental Classes

Let $X^{n}$ be a closed manifold of dimension $n$. Choose an embedding

$i\mathrel{\mathop{\mathchar
58\relax}}X\hookrightarrow\mathbb{R}^{N}\hookrightarrow S^{N}$

for some $N$, with normal bundle $\nu$. Then as virtual vector bundles, there
is a natural isomorphism $\nu-\mathbb{R}^{N}\cong-TX$. Consider the map
$i_{!,u}\mathrel{\mathop{\mathchar 58\relax}}S^{N}\rightarrow X^{\nu}$.
Stabilising and desuspending gives us a well-defined (up to homotopy) map of
spectra

$[X]\mathrel{\mathop{\mathchar 58\relax}}\mathbb{S}\rightarrow X^{-TX}$

Given a ring spectrum $R$ and an $R$-orientation of $TX$ (which induces one on
$-TX$), the Thom isomorphism theorem gives us a well-defined (up to homotopy)
map of spectra

$[X]\mathrel{\mathop{\mathchar
58\relax}}\mathbb{S}\rightarrow\Sigma^{\infty-n}_{+}X\wedge R$

representing an element $[X]$ in $R_{n}(X)$. More generally, given a space
$Z$, a map of spaces $f\mathrel{\mathop{\mathchar 58\relax}}X\rightarrow Z$, a
vector bundle $E\rightarrow Z$, and an $R$-orientation of $TX-f^{*}E$ (which
we assume to have rank $d$), we can use the Thom isomorphism theorem to obtain
a well-defined (up to homotopy) map of spectra

$\mathbb{S}\rightarrow X^{-TX}\wedge R\simeq\Sigma^{-d}X^{-f^{*}E}\wedge
R\rightarrow\Sigma^{-d}Z^{-E}\wedge R$

which we also denote by $[X]$.

We call all of these maps $[X]$ _fundamental classes_.

######  Lemma 2.25.

Let $Y$ be another closed manifold of dimension $n$. Suppose
$f\mathrel{\mathop{\mathchar 58\relax}}X\rightarrow Z$ and
$g\mathrel{\mathop{\mathchar 58\relax}}Y\rightarrow Z$ are two maps, such that
there is a cobordism $W$ from $X$ to $Y$, and a map
$F\mathrel{\mathop{\mathchar 58\relax}}W\rightarrow Z$ extending $f$ and $g$,
as shown below.

${X}$${W}$${Z}$${Y}$$\scriptstyle{f}$$\scriptstyle{g}$$\scriptstyle{F}$

Let $E\rightarrow Z$ be a vector bundle and assume there is an $R$-orientation
of $TW-F^{*}E$, which we assume to have rank $d+1$. Then the two fundamental
classes $[X],[Y]$ in $R_{d}(Z^{-E})$ (defined using the orientations of
$TX-f^{*}E$ and $TY-g^{*}E$ given by restricting the orientation of
$TW-F^{*}E$) agree.

###### Proof.

Pick an embedding

$I\mathrel{\mathop{\mathchar 58\relax}}W\rightarrow\mathbb{R}^{N}\times[0,1]$

for some $N$, such that $W^{-1}\left(\mathbb{R}^{N}\times\\{0\\}\right)=X$ and
$W^{-1}\left(\mathbb{R}^{N}\times\\{1\\}\right)=Y$. Then the composition

$\mathbb{S}\wedge[0,1]_{+}\rightarrow W^{\mathbb{R}-TW}\wedge
R\simeq\Sigma^{-d}W^{-F^{*}E}\rightarrow\Sigma^{-d}Y^{-E}$

defines a homotopy between $[X]$ and $[Y]$. ∎

######  Remark 2.26.

If $X=\bigsqcup\limits_{i\geq 0}X^{i}$ is a compact manifold consisting of
components $X^{i}$ of dimension $i$, we can consider the fundamental classes
of these independently and obtain a fundamental class $[X]$ in
$\bigoplus\limits_{i}R_{i}\left(X^{-TX}\right)$. We will use this later in
Section 5.

##  3 Index Bundles and Their Orientations

###  3.1 Cauchy-Riemann Operators

Let $B$ be a space and $A\subseteq B$ some subspace.

######  Definition 3.1.

A _bundle pair_ over the pair $(B,A)$ is a complex vector bundle $E$ over $B$
along with a totally real subbundle $F$ of $E|_{A}$. This is written as

$(E,F)\rightarrow(B,A)$

We can perform certain operations on bundle pairs, analogously to vector
bundles.

  * •

Given a real vector bundle $G$ over $B$, we can define $(E,F)\otimes G$ to be
the bundle pair

$(E\otimes_{\mathbb{R}}G,F\otimes_{\mathbb{R}}G|_{A})$

  * •

Given $(E^{\prime},F^{\prime})\rightarrow(B,A)$ another bundle pair, we can
define ${(E,F)\oplus(E^{\prime},F^{\prime})}$ to be the bundle pair

$(E\oplus E^{\prime},F\oplus F^{\prime})$

  * •

A _virtual bundle pair_ is a formal difference $(E,F)-(E^{\prime},F^{\prime})$
of two bundle pairs, and we say two virtual bundle pairs
$(E,F)-(E^{\prime},F^{\prime})$ and ${(G,H)-(G^{\prime},H^{\prime})}$ are
_stably isomorphic_ if there is an isomorphism of bundle pairs

$(E,F)\oplus(G^{\prime},H^{\prime})\oplus(\mathbb{C}^{N},\mathbb{R}^{N})\cong(E^{\prime},F^{\prime})\oplus(G,H)\oplus(\mathbb{C}^{N},\mathbb{R}^{N})$

for some $N$.

  * •

The _rank_ of a bundle pair $(E,F)$ is $\mathrm{Rank}_{\mathbb{C}}E$, and the
_(virtual) rank_ of a virtual bundle pair $(E,F)-(E^{\prime},F^{\prime})$ is
$\mathrm{Rank}_{\mathbb{C}}E-\mathrm{Rank}_{\mathbb{C}}E^{\prime}$.

  * •

Given a bundle pair $(E,F)\rightarrow(B,A)$ and a map of pairs
${f\mathrel{\mathop{\mathchar
58\relax}}(B^{\prime},A^{\prime})\rightarrow(B,A)}$, we define the bundle pair
$f^{*}(E,F)\rightarrow(B^{\prime},A^{\prime})$ to be the bundle pair
$(f^{*}E,f^{*}F)$.

  * •

We define a _section_ of $(E,F)$ to be a section of $E$ whose restriction to
$A$ lies in $F$. We write $\Gamma(E,F)$ for the space of sections.

Similarly to the case of real or complex vector bundles, when $B$ is a finite
CW complex and $A$ a subcomplex, every virtual bundle pair over $(B,A)$ is
stably isomorphic to $(E,F)-(\mathbb{C}^{N},\mathbb{R}^{N})$ for some $(E,F)$
and some $N$. Similarly the pullbacks of a bundle pair under homotopic maps
are isomorphic.

We will usually take $(B,A)$ to be $(D^{2},\partial D^{2})\times X$ for some
space $X$, and we will assume this to be the case for the rest of Section 3.

For the rest of Section 3.1, we will assume that $X$ is a point, so $(E,F)$ is
a bundle pair over $(D^{2},\partial D^{2})$.

######  Definition 3.2.

A (real) Cauchy-Riemann operator on $(E,F)$ is an $\mathbb{R}$-linear first
order differential operator

$D\mathrel{\mathop{\mathchar 58\relax}}\Gamma(E,F)\rightarrow\Omega^{0,1}(E)$

satisfying the Leibniz rule:

$D(f\eta)=(\bar{\partial}f)\eta+fD\eta$

for $f$ in $C^{\infty}(D^{2})$ and $\eta$ in $\Gamma(E,F)$.

######  Lemma 3.3 ([MS12, Remark C.1.2]).

The space of Cauchy-Riemann operators on $(E,F)$ is contractible (and in fact,
convex).

For a choice of Hermitian metric on $E$ and a real number $q>2$ (which we
fix), a Cauchy-Riemann operator induces an operator

$\hat{D}\mathrel{\mathop{\mathchar 58\relax}}W^{1,q}(E,F)\rightarrow
W^{q}\left(\Lambda^{0,1}T^{*}D^{2}\otimes E\right)$

where these spaces are appropriate Sobolev completions of the above spaces of
smooth sections. By [MS12, Theorem C.1.10], this operator $\hat{D}$ is
Fredholm, and in fact $\mathrm{Ker\,}\hat{D}=\mathrm{Ker\,}D$.

###  3.2 Index Bundles

Let $X$ be a finite CW complex with basepoint $x_{0}$, and let
${(E,F)\rightarrow(D^{2},\partial D^{2})\times X}$ be a bundle pair.

By Lemma 3.3, we can choose a Cauchy-Riemann operator $D_{x}$ on

$\left(E|_{D^{2}\times\\{x\\}},F|_{\partial D^{2}\times\\{x\\}}\right)$

for each $x$, varying continuously in $x$. The space of such choices is
contractible.

######  Definition 3.4.

Assume $\hat{D}_{x}$ is surjective for all $x$. The index bundle of $(E,F)$ is
the vector bundle $\mathrm{Ind}(E,F)$ over $X$, with fibre at a point $x$
given by $\mathrm{Ker\,}\hat{D}_{x}$.

If $\hat{D}_{x}$ is not always surjective, we can still define the index
bundle, following [Ati68]. Since $X$ is compact, we can find, for some finite
$N$, a continuous family of linear maps

$\phi_{x}\mathrel{\mathop{\mathchar 58\relax}}\mathbb{R}^{N}\rightarrow
W^{q}\left(\Lambda^{0,1}T^{*}D^{2}\otimes E|_{D^{2}\times\\{x\\}}\right)$

for $x$ in $X$, such that the _stabilised operator_

$T_{x}\mathrel{\mathop{\mathchar
58\relax}}=\hat{D}_{x}+\phi_{x}\mathrel{\mathop{\mathchar
58\relax}}W^{1,q}\left(E|_{D^{2}\times\\{x\\}},F|_{\partial
D^{2}\times\\{x\\}}\right)\oplus\mathbb{R}^{N}\rightarrow
W^{q}\left(\Lambda^{0,1}T^{*}D^{2}\otimes E|_{D^{2}\times\\{x\\}}\right)$

is surjective. In this situation we call $\phi$ a _stabilisation_ of _rank_
$N$. We then define $\mathrm{Ind}(E,F)+\mathbb{R}^{N}$ to be the vector bundle
with fibre over $x$ given by $\mathrm{Ker\,}T_{x}$.

######  Lemma 3.5.

The vector bundle $\mathrm{Ind}(E,F)$ is well-defined up to (unique up to
weakly contractible choice) stable isomorphism.

###### Proof.

Firstly we note that stabilising $\phi$ by adding another copy of $\mathbb{R}$
which is sent to 0 does not change the index bundle up to canonical
isomorphism.

We choose a metric on $E$, and note that the space of such choices is
contractible. This defines a $W^{0,2}$ inner product on $W^{1,q}(E,F)$.

Given two choices of family of Cauchy-Riemann operators and stabilisation
$(D,\phi)$ and $(D^{\prime},\phi^{\prime})$, by the above we can assume that
both stabilisations are of the same rank. If $(D,\phi)$ and
$(D^{\prime},\phi^{\prime})$ have distance less than 1 with respect to the
operator norm, orthogonal projection defines an isomorphism between their
kernels. Note that since the space of such operators is convex, this open ball
is contractible.

In general, we can pick a path in the space of such $(D,\phi)$, and iterate
this process along the path. Because the spaces of such pairs $(D,\phi)$ (up
to stabilisation) are weakly contractible, the uniqueness result follows from
a similar argument. ∎

We will require some elementary properties of index bundles.

######  Lemma 3.6.

If $X^{\prime}$ is another finite CW complex and $f\mathrel{\mathop{\mathchar
58\relax}}X^{\prime}\rightarrow X$ is any map, then there is a natural
isomorphism of vector bundles

$f^{*}\mathrm{Ind}(E,F)\cong\mathrm{Ind}f^{*}(E,F)$

where we write $f^{*}(E,F)$ for $(Id_{D^{2}}\times f)^{*}(E,F)$.

###### Proof.

All of the choices made to define the index bundle are compatible under
pullbacks. ∎

######  Lemma 3.7.

If $(E,F)$ and $(E^{\prime},F^{\prime})$ are bundle pairs on $(D^{2},\partial
D^{2})\times X$, then there is a natural isomorphism of virtual vector bundles

$\mathrm{Ind}(E,F)\oplus\mathrm{Ind}(E^{\prime},F^{\prime})\cong\mathrm{Ind}(E\oplus
E^{\prime},F\oplus F^{\prime})$

###### Proof.

All of the choices made to define the index bundle are compatible under direct
sums. ∎

######  Corollary 3.8.

If $(E,F)$ is stably isomorphic to $(E^{\prime},F^{\prime})$, then
$\mathrm{Ind}(E,F)$ is stably isomorphic to
$\mathrm{Ind}(E^{\prime},F^{\prime})$.

######  Lemma 3.9.

Let $G\rightarrow X$ be a real virtual vector bundle over $X$. Then there is a
stable isomorphism

$\left(\mathrm{Ind}(E,F)\right)\otimes G\cong\mathrm{Ind}\left((E,F)\otimes
G\right)$

###### Proof.

We assume $G$ is an actual vector bundle, and observe that the general case
follows from Lemma 3.7.

Let $D$ be a family of Cauchy-Riemann operators on $(E,F)$, and $\phi$ a
stabilisation of rank $N$, corresponding to the family of stabilised operators
$T$. We pick another vector bundle $G^{\prime}$ over $X$, along with an
isomorphism $G\oplus G^{\prime}\cong\mathbb{R}^{M}$ for some $M$.

For each $x$ in $X$, we consider the Cauchy-Riemann operator

$D_{x}\otimes Id_{G_{x}}\mathrel{\mathop{\mathchar
58\relax}}\Gamma\left(\left(E|_{D^{2}\times\\{x\\}},F|_{\partial
D^{2}\times\\{x\\}}\right)\otimes
G_{x}\right)\rightarrow\Omega^{0,1}\left(E_{D^{2}\times\\{x\\}}\otimes
G_{x}\right)$

along with the stabilisation of rank $NM$

$\theta_{x}\mathrel{\mathop{\mathchar 58\relax}}(\mathbb{R}^{N}\otimes
G_{x})\oplus(\mathbb{R}^{N}\otimes G_{x}^{\prime})\rightarrow
W^{q}\left(\Lambda^{0,1}T^{*}D^{2}\otimes E|_{D^{2}\times\\{x\\}}\otimes
G_{x}\right)$

sending $(a\otimes u,b\otimes v)$ to $\phi_{x}(a)\otimes u$.

Then $\mathrm{Ind}\left(\left(E,F\right)\otimes
G\right)_{x}\oplus\mathbb{R}^{NM}$ is equal to $\mathrm{Ker\,}S_{x}$ by
definition, where $S_{x}$ is the stabilised operator

$\left(\widehat{D_{x}\otimes Id_{G}}\right)\oplus\theta_{x}$

But $\mathrm{Ker\,}S_{x}$ is isomorphic to

$\left(\left(\mathrm{Ker\,}T_{x}\right)\otimes
G_{x}\right)\oplus\left(\mathbb{R}^{N}\otimes G^{\prime}_{x}\right)$

which is isomorphic to

$\left(\left(\mathrm{Ind}(E,F)\right)_{x}\otimes
G_{x}\right)\oplus\mathbb{R}^{N,M}$

∎

There is a standard bundle pair over a point,
$H=(\mathbb{C},\delta)\rightarrow(D^{2},\partial D^{2})$, where
$\delta(z)=\sqrt{z}\mathbb{R}$ for $z$ in $\partial D^{2}$.

We can pick a trivialisation $E\cong\mathbb{C}^{n}$ over $D^{2}\times\\{x\\}$
for a point $x$ in $X$. Then $F|_{\partial D^{2}\times\\{x\\}}$ determines a
loop in the space of totally real subspaces of $\mathbb{C}^{n}$, which is
isomorphic to $U(n)/O(n)$. There are isomorphisms

$\pi_{1}U(n)/O(n)\cong\mathbb{Z}$

for all $n$, compatibly with stabilisation in $n$. There are two such choices
of isomorphism, and we fix the one such that
$H\oplus(\mathbb{C}^{n-1},\mathbb{R}^{n-1})$ is sent to 1.

######  Definition 3.10.

We define the _Maslov index_ of $(E,F)$, denoted $\mu(E,F)$, to be the image
of $F$ under the isomorphism $\pi_{1}U(n)/O(n)\rightarrow\mathbb{Z}$ after
choosing a trivialisation of $E$ over a point. This is well-defined on each
connected component of $X$.

Using the index theorem, one can compute the virtual rank of the index bundle.

######  Lemma 3.11 ([MS12, Theorem C.1.10]).

The virtual rank of $\mathrm{Ind}(E,F)$ is

$n+\mu(E,F)$

If $(E,F)$ is a virtual bundle pair over $(D^{2},\partial D^{2})\times Y$ for
some (possibly non-compact) space $Y$, for any map from a finite CW complex
$X$ to $Y$, $\mathrm{Ind}(E,F)|_{X}$ is well-defined, and furthermore this is
compatible with restriction.

######  Definition 3.12.

We say that $\mathrm{Ind}(E,F)$ is _$R$ -orientable_ if
$\mathrm{Ind}(E,F)|_{X}$ is $R$-orientable for all such $X$.

An _$R$ -orientation_ on $\mathrm{Ind}(E,F)$ is a choice of $R$-orientation on
$\mathrm{Ind}(E,F)|_{X}$ for all $X$, compatible with restriction.

###  3.3 Orientations

Our goal in this section will be to establish the following two propositions,
the first of which is clear from Corollary 3.8. These will be used later to
find conditions under which the tangent bundles to the moduli spaces
constructed in Sections 4.1 and 5.3 are $R$-orientable for various ring
spectra $R$.

######  Proposition 3.13.

A stable trivialisation of $(E,F)$ induces a stable trivialisation of
$\mathrm{Ind}(E,F)$, and hence an $R$-orientation for any ring spectrum $R$,
compatibly with direct sums and pullbacks.

###### Proof.

By Corollary 3.8, it suffices to prove the result for
$(E,F)=(\mathbb{C}^{n},\mathbb{R}^{n})$. Over every point $x$ in the base $X$,
we pick the standard Cauchy-Riemann operator

$\bar{\partial}\mathrel{\mathop{\mathchar
58\relax}}\Gamma\left(\mathbb{C}^{n},\mathbb{R}^{n}\right)\rightarrow\Gamma\left(\mathbb{C}^{n}\right)$

on sections over the disc $D^{2}\times\\{x\\}$.

Evaluation at any fixed point $z$ in $\partial D^{2}$ defines a map
$\mathrm{Ker\,}\bar{\partial}\rightarrow\mathbb{R}^{n}$, which is an
isomorphism (and which does not depend on $z$). Then because the index of this
Cauchy-Riemann operator is $n$, this Cauchy-Riemann operator must be
surjective. This gives us the required trivialisation of $\mathrm{Ind}(E,F)$.
∎

The following proposition seems well-known to experts, but to the author’s
knowledge a proof has not yet appeared in the literature.

######  Proposition 3.14.

A stable trivialisation of $F$ induces a stable complex structure of
$\mathrm{Ind}(E,F)$, compatibly with direct sums and pullbacks.

###### Proof.

Assume $X$ is connected. Recall that a based vector bundle (or bundle pair)
over $X$ is a vector bundle (or bundle pair) over $X$ (or $(D^{2},\partial
D^{2})\times X$) with a fixed trivialisation over the basepoint of $X$.

Let $\mu=\mu(E,F)$ be the Maslov index. Then since $F$ is stably trivial,
$\mu$ must be even. Therefore the real part of $H^{\oplus\mu}$ is also stably
trivial. Furthermore, $\mathrm{Ind}(H^{\mu})$ is canonically trivial and hence
admits a canonical stable complex structure.

Note that the virtual bundle pair $(E,F)-H^{\oplus\mu}$ has Maslov index 0.
Therefore it suffices to assume $\mu=0$, which we now do.

The stable trivialisation of $F$ induces a stable trivialisation of $E$ which
restricts to that of $F$ over $\\{1\\}\times X$ (but not necessarily over the
rest of $\partial D^{2}\times X$). Furthermore because $\mu=0$ we can pick
this stable trivialisation to extend $F$ over $\partial
D^{2}\times\\{x_{0}\\}$ too. Thus $F$ determines a based map $\Sigma
X\rightarrow U(n)/O(n)$ and, stabilising, a based map

$\psi\mathrel{\mathop{\mathchar 58\relax}}\Sigma X\rightarrow U/O$

whose based homotopy class, along with the rank $n$, classifies $(E,F)$ up to
stable, based isomorphism.

The stable trivialisation of $F$ determines a nullhomotopy of the composition

$\Sigma X\rightarrow U/O\xrightarrow{\zeta}BO$

where the second map $\zeta$ is the forgetful map, sending a totally real
subspace $V$ in $\mathbb{C}^{N}$, to $V$ viewed as in $\mathbb{R}^{2N}$. After
applying $[\Sigma X,\cdot]_{*}$, this sends a bundle pair to its real part.
Taking adjoints determines a nullhomotopy of the adjoint map

$X\rightarrow\Omega BO$

We define a map

$[\Sigma X,U/O]_{*}\rightarrow[X,BO]_{*}$

sending a based bundle pair $(E,F)$ of rank $n$ to the classifying map of
$\mathrm{Ind}(E,F)-\mathbb{R}^{n}$. This was proved to be a bijection by de
Silva in [dS98], and we recap his proof in Appendix A. de Silva constructed an
explicit inverse to this map, which we call

$\beta\mathrel{\mathop{\mathchar 58\relax}}[X,BO]_{*}\rightarrow[\Sigma
X,U/O]_{*}$

sending a based vector bundle $E$ of rank $n$ to the map $\Sigma X\rightarrow
U/O$ corresponding to the based virtual bundle pair of rank $n$:

$H\otimes E-(\mathbb{C},\mathbb{R})\otimes E-H\otimes\mathbb{R}^{n}$

We define a map

$\varepsilon\mathrel{\mathop{\mathchar 58\relax}}[X,BO]_{*}\rightarrow[\Sigma
X,BO]_{*}$

to send a based vector bundle $E$ of rank $n$ to

$\lambda\otimes E-\mathbb{R}\otimes E-\lambda\otimes\mathbb{R}^{n}$

where $\lambda\rightarrow S^{1}$ is the Möbius line bundle. Then it is clear
that the following diagram commutes:

${\left[X,BO\right]_{*}}$${\left[\Sigma X,BO\right]_{*}}$${\left[\Sigma
X,U/O\right]_{*}}$${\left[\Sigma X,BO\right]_{*}=\left[X,\Omega
BO\right]_{*}}$$\scriptstyle{\varepsilon}$$\scriptstyle{\beta}$$\scriptstyle{=}$$\scriptstyle{\zeta}$

Consider the fibration sequence

$BU\rightarrow BO\rightarrow\Omega BO$

where the first arrow corresponds to the realification of a vector bundle and
the second to $\zeta$. Then the given contraction of our map
$X\rightarrow\Omega BO$ induces a lift to a map $X\rightarrow BU$, which is
the stable complex structure we desired.

It it clear from construction that everything is compatible with direct sums
and pullbacks. ∎

##  4 The Moduli Spaces $\mathcal{P}$ and $\mathcal{W}$

###  4.1 Construction of the Moduli Spaces

Let $C$ be a convex domain in the upper half plane in $\mathbb{C}$, with
smooth boundary $\partial C$ containing 0. Let $f\mathrel{\mathop{\mathchar
58\relax}}\partial C\rightarrow[0,1]$ be a smooth map sending $0$ to $0$, and
let $J$ be an $\omega$-tame almost complex structure on $M$ which is convex at
infinity.

######  Definition 4.1.

We define $\mathcal{D}_{f,C}$ to be the space of smooth maps
$u\mathrel{\mathop{\mathchar 58\relax}}C\rightarrow M$ such that for all $z$
in $\partial C$, $u(z)$ lies in $\psi^{f(z)}(L)$.

We define $\pi$ to be the natural evaluation map
$\pi\mathrel{\mathop{\mathchar 58\relax}}\mathcal{D}_{f,C}(J)\rightarrow L$
sending $u$ to $u(0)$.

######  Definition 4.2.

We define $\mathcal{U}_{f,C}(J)$ to be the space of maps $u$ in
$\mathcal{D}_{f,C}$ which are $J$-holomorphic, namely

$J\circ du=du\circ j$

where $j$ is the complex structure on $C\subseteq\mathbb{C}$.

We call the triple $(C,f,J)$ the _moduli data_.

Let $G$ be a fixed convex domain in $\mathbb{C}$ with smooth boundary, such
that both $(-\eta,\eta)$ and $i+(-\eta,\eta)$ are contained in $\partial G$
for some $\eta>0$. Let $G_{\pm}$ be $G\cap\mathbb{C}_{\pm\mathrm{Re\,}\geq
0}$.

For $l\geq 0$, define $Z_{l}$ to be $[0,1]i+[-l,l]$, and $G_{l}$ to be

${Z_{l}\cup(G_{+}+l)\cup(G_{-}-l)}$

as shown below.

$0$ $i$ $l$ $-l$ $Z_{l}\mathchar 24634\relax$ $0$ $i$ $l$ $-l$ $G_{l}\mathchar
24634\relax$

We now define a 1-parameter family of moduli data $(C_{l},f_{l},J_{l})$ as
follows.

$C_{l}\mathrel{\mathop{\mathchar 58\relax}}=\begin{cases}G_{l-1}&\text{ if
}l\geq 1\\\ G_{0}&\text{ if }l\leq 1\end{cases}$
$f_{l}(z)\mathrel{\mathop{\mathchar
58\relax}}=\begin{cases}\mathrm{Im\,}z&\textit{ if }l\geq 1\\\
l\mathrm{Im\,}z&\textit{ if }l\leq 1\end{cases}$

As a 1-parameter family of domains in $\mathbb{C}$ and maps to $[0,1]$, both
of these are continuous everywhere in $l$ and vary smoothly in $l$ except at
1, so we pick a small smoothing of these families supported near $l=1$.

Using relative exactness of $L$ in $M$, the proof of [Hof88, Lemma 2] shows
that

######  Lemma 4.3.

There exists some $A$ in $\mathbb{R}$ such that for any $l\geq 0$ and any
smooth map $u$ in $\mathcal{D}_{f_{l},C_{l}}$, the topological energy
$\int_{u}u^{*}\omega$ is bounded above by $A$.

We now pick a Riemannian metric on $L$, with injectivity radius $\varepsilon$
and distance function $d$. Using this energy bound, [Hof88, Proposition 3]
then directly implies the following result, which should be viewed as a form
of Gromov compactness. Roughly, he shows that for large $l$, these moduli
spaces, when restricted to $Z_{l}$, live close to holomorphic strips with
boundary on $L$ and $\psi^{1}(L)$, which we know are constant since
$L=\psi^{1}(L)$ is relatively exact.

######  Lemma 4.4.

There is some $c>1$ such that for all $l\geq c$ and $u$ in
$\mathcal{U}_{f_{l},C_{l}}(J_{l})$,

$d(u(i),u(0))<\varepsilon$

Standard techniques in [MS12] show that for a generic ($l$-dependent)
perturbation of $J_{l}$, $\mathcal{W}^{\prime}\mathrel{\mathop{\mathchar
58\relax}}=\bigsqcup\limits_{l\geq 0}\mathcal{U}_{f_{l},C_{l}}(J_{l})$ is a
smooth manifold with boundary $\mathcal{U}_{f_{0},C_{0}}(J_{0})$, which
consists only of constant maps and hence $\pi\mathrel{\mathop{\mathchar
58\relax}}\mathcal{U}_{f_{0},C_{0}}(J_{0})\rightarrow L$ is a diffeomorphism.
If we choose the perturbation small enough then Lemma 4.4 still holds, by
Lemma 4.3 along with Gromov compactness.

Let $\mathcal{U}_{l}=\mathcal{U}_{f_{l},C_{l}}(J_{l})$, and let
$p\mathrel{\mathop{\mathchar
58\relax}}\mathcal{W}^{\prime}\rightarrow\mathbb{R}$ send $\mathcal{U}_{l}$ to
$l$.

######  Lemma 4.5.

Each $\mathcal{U}_{l}$ is compact and $p$ is a proper map.

###### Proof.

We will show each $\mathcal{U}_{l}$ is compact, a similar argument shows that
$p$ is a proper map.

Let $u_{i}$ be a sequence in $\mathcal{U}_{l}$. Gromov compactness as in
[FZ15] implies that this contains a convergent subsequence, but only after
precomposing with a sequence of holomorphic automorphisms of the domain.

Let $\tilde{M}$ be $M\times\mathbb{C}$, with almost complex structure $J_{l}$
times the standard complex structure on $\mathbb{C}$. Let
$\tau\mathrel{\mathop{\mathchar 58\relax}}\tilde{M}\rightarrow\mathbb{C}$ be
the projection map onto the second factor. Let $\tilde{L}$ be

$\bigcup\limits_{z\in\partial C_{l}}\psi^{f_{l}(z)}(L)\times\\{z\\}$

a totally real submanifold of $\tilde{M}$.

We let $\tilde{\mathcal{U}}_{l}$ be the moduli space of holomorphic discs in
$\tilde{M}$ with boundary in $\tilde{L}$. There is an embedding
$\theta\mathrel{\mathop{\mathchar
58\relax}}\mathcal{U}_{l}\hookrightarrow\tilde{\mathcal{U}}_{l}$ sending $u$
to the map $\tilde{u}$ which sends $z$ to $(u(z),z)$. This clearly is a
bijection with the subset of $\tilde{\mathcal{U}}_{l}$ consisting of sections
of $\tau|_{C_{l}}$.

For $u$ in $\mathcal{U}_{l}$, the energy of $\tilde{u}$ is equal to the energy
of $u$ plus the area of $C_{l}$, so in particular by Lemma 4.3 the energies of
the $\tilde{u}_{i}$ are uniformly bounded. Therefore by Gromov compactness,
there is a subsequence (which we will continue to call $\tilde{u_{i}}$, by
abuse of notation) and a sequence of holomorphic automorphisms $\phi_{i}$ of
$C_{l}$, such that $\tilde{u}_{i}\circ\phi_{i}$ lives in
$\tilde{\mathcal{U}}_{l}$ for all $i$, and $\tilde{u}_{i}\circ\phi_{i}$
converges to some $v$ in $\tilde{\mathcal{U}}_{l}$.

$\tau\circ\tilde{u}_{i}\circ\phi_{i}=\phi_{i}$ also converges to
$\phi\mathrel{\mathop{\mathchar 58\relax}}=\tau\circ v$ which is a holomorphic
automorphism of $C_{l}$, so $\tilde{u}_{i}$ converges to $v\circ\phi^{-1}$,
which lies in the image of $\theta$ and so $v=\tilde{u}$ for some $u$ in
$\mathcal{U}_{l}$. Therefore $u_{i}$ converges to $u$ in $\mathcal{U}_{l}$. ∎

We pick $c^{\prime}>c$ (where $c$ is from Lemma 4.4) a regular value of $p$.
We then define $\mathcal{W}$ to be the path component of $\mathcal{U}_{0}=L$
in $p^{-1}[0,c^{\prime}]$, and $\mathcal{P}$ to be

$\mathcal{W}\cap\mathcal{P}_{c^{\prime}}$

Then, by construction, $\pi\mathrel{\mathop{\mathchar
58\relax}}\mathcal{W}\rightarrow L$ is a cobordism from
$Id\mathrel{\mathop{\mathchar 58\relax}}L\rightarrow L$ to
$\pi\mathrel{\mathop{\mathchar 58\relax}}\mathcal{P}\rightarrow L$ over $L$.
Furthermore whilst $\mathcal{W}^{\prime}$ may have had different path
components of different dimensions, $\mathcal{W}$ is a manifold (with
boundary) of dimension $n+1$.

###### Proof of Lemma 1.11.

We define $\pi^{\prime}\mathrel{\mathop{\mathchar
58\relax}}\mathcal{P}\rightarrow L$ to send $u$ to $u(i)$. Then by Lemma 4.4,
this is homotopic to $\pi$. We now write down a homotopy
$H\mathrel{\mathop{\mathchar 58\relax}}\mathcal{P}\times[0,1]\rightarrow L$
from $\pi^{\prime}$ to $\psi^{1}\circ\pi$.

Let $\gamma\mathrel{\mathop{\mathchar 58\relax}}[0,1]\rightarrow\partial
C_{c^{\prime}}$ be some fixed path from $0$ to $i$. For $u$ in $\mathcal{P}$
and $t$ in $[0,1]$, we define $H(u,t)$ to be

$\psi^{1}\left(\left(\psi^{\mathrm{Im\,}\gamma(t)}\right)^{-1}\left(u\left(\gamma(t)\right)\right)\right)$

∎

######  Remark 4.6.

[Hof88, Theorem 1] shows that even without any genericity assumption on the
almost complex structure, the map $\pi\mathrel{\mathop{\mathchar
58\relax}}\mathcal{P}\rightarrow L$ induces an injective map

$H^{*}(L;\mathbb{Z}/2)\cong\check{H}^{*}(L;\mathbb{Z}/2)\rightarrow\check{H}^{*}(\mathcal{P};\mathbb{Z}/2)$

where $\check{H}^{*}$ denotes Čech cohomology. The statement of Lemma 4.4
still holds and hence so does Lemma 1.11. From this it follows that
$\psi^{1}|_{L}$ induces the identity map on $H^{*}(L;\mathbb{Z}/2)$.

###  4.2 Orientations on the Moduli Space

We define $\mathcal{D}_{0}$ to be the space of triples $(f,C,u)$, where $C$ is
a convex domain in the upper half plane in $\mathbb{C}$ with smooth boundary
$\partial C$ containing 0, ${f\mathrel{\mathop{\mathchar 58\relax}}\partial
C\rightarrow[0,1]}$ is a smooth map sending 0 to 0, and $u$ is an element of
$\mathcal{D}_{f,C}$. Note that each inclusion
$\mathcal{D}_{f,C}\hookrightarrow\mathcal{D}_{0}$ is a homotopy equivalence.

There is a tautological smooth fibre bundle over $\mathcal{D}_{0}$ with fibre
over $(f,C,u)$ given by $C$. The structure group of this bundle is the group
of diffeomorphisms of $(D^{2},\partial D^{2})$ which send 1 to itself. This
group is contractible, so we can pick a trivialisation $\Phi$ from
$D^{2}\times\mathcal{D}_{0}$ to this bundle, preserving 0. Note the space of
such choices is contractible.

There is an evaluation map

$ev\mathrel{\mathop{\mathchar 58\relax}}\left(D^{2},\partial
D^{2}\right)\times\mathcal{D}_{0}\rightarrow(M,L)$

sending $(z,u)$ in $D^{2}\times\mathcal{D}_{f,C}$ to

$\left(\psi^{\tilde{f}}(x)\right)^{-1}\left(u\left(\Phi(z\right))\right)$

where $\tilde{f}$ is some smooth extension of $f$ to the entirety of $C$.
Since for each $(f,C)$ the space of such extensions is contractible, we can
make some consistent choice of such things to define $ev$, and therefore this
defines $ev$ uniquely up to a contractible space of choices.

There is a bundle pair over $\left(D^{2},\partial
D^{2}\right)\times\mathcal{D}_{0}$ given by $ev^{*}\left(TM,TL\right)$. For a
map from a CW complex $g\mathrel{\mathop{\mathchar
58\relax}}X\rightarrow\mathcal{D}_{0}$, we write $\mathrm{Ind}$ for the index
of the pullback of this bundle pair along $g$. Note that the virtual rank of
$\mathrm{Ind}$ may be different on different components of $X$ if the
evaluation map lands in different components of $\mathcal{D}_{0}$.

By construction, if $J$ is regular, $T\mathcal{U}_{f,C}(J)$ is stably
isomorphic to the index bundle $\mathrm{Ind}$, and similarly $T\mathcal{W}$ is
stably isomorphic to $\mathrm{Ind}\oplus\mathbb{R}$.

Let $X$ be a finite CW complex and $g\mathrel{\mathop{\mathchar
58\relax}}X\rightarrow\mathcal{D}_{0}$ some map. In the rest of this section
we will find conditions under which $\mathrm{Ind}$ is $R$-orientable over $X$
for various ring spectra $R$.

######  Proposition 4.7.

If there is a stable trivialisation $TL\cong\mathbb{R}^{n}$ which (after
applying $\cdot\otimes\mathbb{C}$) extends to a stable trivialisation
$TM\cong\mathbb{C}^{n}$, then $\mathrm{Ind}-\pi^{*}TL$ is stably trivial and
hence $R$-orientable for any ring spectrum $R$.

###### Proof.

Follows from Proposition 3.8. ∎

######  Proposition 4.8.

If there is a stable trivialisation $TL\cong\mathbb{R}^{n}$ over the
3-skeleton of $L$ which (after applying $\cdot\otimes\mathbb{C}$) extends to a
stable trivialisation $TM\cong\mathbb{C}^{n}$ over the 4-skeleton of $M$, then
$\mathrm{Ind}$ and hence $\mathrm{Ind}-\pi^{*}TL$ are $KO$-orientable.

###### Proof.

We pick a 3-skeleton $L_{3}$ of $L$, a 4-skeleton $M_{4}$ of $M$ containing
$L_{3}$, and a 2-skeleton $X_{2}$ of $X$. Then the map of pairs
$ev\mathrel{\mathop{\mathchar 58\relax}}(D^{2},\partial D^{2})\times
X_{2}\rightarrow(M,L)$ can be homotoped to land in $(M_{4},L_{3})$, and we can
apply Proposition 3.8 to see that $\mathrm{Ind}$ admits a stable
trivialisation over a 2-skeleton. Then by [ABS64, Theorem 12.3],
$\mathrm{Ind}$ is $KO$-orientable. ∎

######  Lemma 4.9.

If there is a stable trivialisation $TL\cong\mathbb{R}^{n}$ over the
$i+1$-skeleton of $L$, $\mathrm{Ind}$ admits a stable complex structure over
the $i$-skeleton of $X$.

###### Proof.

The real part of the bundle pair
$(ev^{*}TM,ev^{*}TL)\rightarrow(D^{2},\partial D^{2})\times X$ is pulled back
from a map $\partial D^{2}\times X\rightarrow L$. Letting $X_{i}$ be some
choice of $i$-skeleton of $X$, $ev\mathrel{\mathop{\mathchar
58\relax}}\partial D^{2}\times X_{i}\rightarrow L$ can be homotoped to have
image in the $i+1$-skeleton of $L$, so by Proposition 3.14, $\mathrm{Ind}$
admits a stable complex structure over $X_{i}$. ∎

######  Corollary 4.10.

  1. 1.

If $L$ admits a spin structure, $\mathrm{Ind}$ and $\mathrm{Ind}-\pi^{*}TL$
are orientable.

  2. 2.

If $L$ admits a spin structure, $\mathrm{Ind}$ and $\mathrm{Ind}-\pi^{*}TL$
are $KU$-orientable.

###### Proof.

  1. 1.

If $L$ admits a spin structure, $TL$ is stably trivial over the 2-skeleton of
$L$, and so $\mathrm{Ind}$ admits a complex structure over the 1-skeleton of
$X$ and hence is orientable.

  2. 2.

A vector bundle over a finite CW complex $X$ is stably trivial over an
$i$-skeleton $X_{i}$ iff the classifying map $X\rightarrow BO$ is
nullhomotopic when restricted to $X_{i}$. Since $\pi_{3}BO=0$, if
$X_{2}\rightarrow BO$ is nullhomotopic, so is $X_{3}\rightarrow BO$. So if $L$
admits a spin structure, $TL$ admits a stable trivialisation over any
3-skeleton. Therefore $\mathrm{Ind}$ admits a spinc structure, i.e. it admits
a stable complex structure over any 2-skeleton. Then by [ABS64, Theorem 12.3],
$\mathrm{Ind}$ is $KU$-orientable.

∎

######  Lemma 4.11.

If $w_{2}(L)(x)=0$ for $x$ in the image of the composition

$\pi_{3}(M,L)\rightarrow\pi_{2}(L)\rightarrow H_{2}(L;\mathbb{Z})$

then $\mathrm{Ind}-\pi^{*}TL$ is orientable.

###### Proof.

$w_{1}(\mathrm{Ind}-\pi^{*}TL)$ vanishes by [Geo13, Theorem 1.1]. ∎

##  5 Monodromy in the Looped Case

###  5.1 General Set-Up

We define $L_{\psi^{1}}$ to be the space of pairs $(t,x)$, where $t$ lies in
$S^{1}=\mathbb{R}/\mathbb{Z}$ and $x$ lies in $\psi^{t}(L)$. We let
$q\mathrel{\mathop{\mathchar 58\relax}}L_{\psi^{1}}\rightarrow S^{1}$ be the
projection map to the first co-ordinate. This is a model for the mapping torus
of $\psi^{1}$. We denote its vertical tangent space by $T^{v}L_{\psi^{1}}$.

The Hamiltonian flow induces a natural family of maps which we denote by
$\Psi^{t}\mathrel{\mathop{\mathchar 58\relax}}L_{\psi^{1}}\rightarrow
L_{\psi^{1}}$ for $t$ in $\mathbb{R}$, which live over the identity map on
$S^{1}$ for $t$ in $\mathbb{Z}$. More explicitly, pick $x$ in $\psi^{s}(L)$
and $t$ in $\mathbb{R}$, and write $t+s=m+\tau$, where $m$ is in $\mathbb{Z}$
and $\tau$ is in $[0,1)$. Then we define $\Psi^{t}(x)$ to be

$\psi^{\tau}\left(\left(\psi^{1}\right)^{m}\left(\left(\psi^{s}\right)^{-1}\left(x\right)\right)\right)$

Then $\Psi^{0}$ is the identity map and $\Psi^{t}\circ\Psi^{s}=\Psi^{t+s}$ for
all $t,s$ in $\mathbb{R}$. Furthermore $\Psi^{1}|_{q^{-1}(\\{0\\})}=\psi^{1}$.
$\Psi^{t}$ should be thought of as parallel transporting by a distance $t$
around the base of the fibre bundle $L_{\psi^{1}}\twoheadrightarrow S^{1}$.

The free loop space $\mathcal{L}L_{\psi^{1}}$ of $L_{\psi^{1}}$ is naturally a
fibre bundle over $L_{\psi^{1}}$. We define $\mathcal{L}$ to be the
restriction of this to the fibre over $0$ in $S^{1}$. Then $\mathcal{L}$ is
naturally a fibre bundle over $L$ and splits into a disjoint union

$\mathcal{L}=\bigsqcup\limits_{j\in\mathbb{Z}}\mathcal{L}_{j}$

where $\mathcal{L}_{j}$ is the space of loops in $\mathcal{L}$ with winding
number $j$ about $S^{1}$.

$\Psi^{1}$ induces a map $\mathcal{L}\rightarrow\mathcal{L}$ which preserves
winding numbers, so $\Psi^{1}$ induces an automorphism of $\mathcal{L}_{i}$
for all $i$, which we will also denote by $\Psi^{1}$.

We define $\mathcal{L}^{\prime}_{j}$ to be subspace of $\mathcal{L}_{j}$ given
by loops $\gamma$ such that the composition $S^{1}\rightarrow
L_{\psi^{1}}\rightarrow S^{1}$ is given by $z\mapsto jz$, and
$\mathcal{L}^{\prime}$ to be the disjoint union of all
$\mathcal{L}^{\prime}_{j}$. Then $\Psi^{1}$ sends $\mathcal{L}^{\prime}$ to
$\mathcal{L}^{\prime}$.

######  Lemma 5.1.

The inclusion $\mathcal{L}^{\prime}\hookrightarrow\mathcal{L}$ is a homotopy
equivalence.

###### Proof.

We first define a continuous map $f\mathrel{\mathop{\mathchar
58\relax}}\mathcal{L}_{j}\rightarrow\left\\{\mathrm{continuous\,maps}\,[0,1]\rightarrow\mathbb{R}\right\\}$.
For all $\gamma$ in $\mathcal{L}_{j}$, $f_{\gamma}$ is uniquely determined by
the following properties.

  1. 1.

$f_{\gamma}(0)=0$

  2. 2.

$f_{\gamma}(1)=j$

  3. 3.

The composition

$[0,1]\rightarrow S^{1}\xrightarrow{\gamma}L_{\psi^{1}}\xrightarrow{q}S^{1}$

agrees with the composition

$[0,1]\xrightarrow{f_{\gamma}}\mathbb{R}\rightarrow S^{1}$

We define a map $F\mathrel{\mathop{\mathchar
58\relax}}\mathcal{L}\rightarrow\mathcal{L}^{\prime}$ by

$F(\gamma)(t)=\Psi^{jt-f_{\gamma}(t)}(\gamma(t))$

for $t$ in $[0,1]$. Then this is a homotopy inverse to the inclusion
$\mathcal{L}^{\prime}\hookrightarrow\mathcal{L}$. ∎

By Lemma 5.1, $\mathcal{L}_{0}\simeq\mathcal{L}L$ and it is clear that the
action of $\Psi^{1}$ on $\mathcal{L}_{0}$ is the same as the action of
$\psi^{1}$ on $\mathcal{L}L$.

######  Lemma 5.2.

$\Psi^{1}$ acts as the identity on $\mathcal{L}_{\pm 1}$ up to homotopy.

###### Proof.

We prove this in the case of $+1$, the case of $-1$ is similar. Note by Lemma
5.1 it suffices to prove $\Psi^{1}$ acts as the identity on
$\mathcal{L}^{\prime}_{1}$.

We define $H\mathrel{\mathop{\mathchar
58\relax}}\mathcal{L}^{\prime}_{1}\times[0,1]\rightarrow\mathcal{L}^{\prime}_{1}$
by

$H(\gamma,s)(t)=\Psi^{s}(\gamma(t-s))$

Then $H(\cdot,0)$ is the identity and $H(\cdot,1)$ is $\Psi^{1}$. ∎

###  5.2 A Chas-Sullivan Type Product

We first note that by [Mil59], all spaces considered so far in this section
are homotopy equivalent to CW complexes, and furthermore they are all
Hausdorff and paracompact.

We denote the pullback of the virtual vector bundle $-TL\rightarrow L$ to
$\mathcal{L}$ also by $-TL$. We will define ring and module structures on the
spectra $\mathcal{L}^{-TL}$ and $\mathcal{L}$, following [CJ02].

We have a commutative diagram:

${\mathcal{L}_{j}\times_{L}\mathcal{L}_{k}}$${\mathcal{L}_{j}\times\mathcal{L}_{k}}$${\mathcal{L}_{j+k}}$$\scriptstyle{\Delta}$$\scriptstyle{c}$

where $\mathcal{L}_{j}\mathop{\times}_{L}\mathcal{L}_{k}$ is the fibre product
of $\mathcal{L}_{j}$ and $\mathcal{L}_{k}$ over $L$, $\Delta$ is the natural
inclusion map into $\mathcal{L}_{j}\times\mathcal{L}_{k}$, and $c$ is a
concatenation map: given $(\gamma,\delta)$ in
$\mathcal{L}_{j}\mathop{\times}_{L}\mathcal{L}_{k}$ and $t$ in $[0,1]$,

$c(\gamma,\delta)(t)=\begin{cases}\gamma(2t)&\mathrm{if}\,t\leq\frac{1}{2}\\\
\delta(2t-1)&\mathrm{if}\,t\geq\frac{1}{2}\\\ \end{cases}$

Then $\mathcal{L}_{j}\mathop{\times}_{L}\mathcal{L}_{k}$ admits a tubular
neighbourhood in $\mathcal{L}_{j}\times\mathcal{L}_{k}$ with normal bundle
$TL$, and by applying Definition 2.23 with the virtual vector bundles
$-TL\boxplus-TL$ and $-TL\boxplus 0$ over
$\mathcal{L}_{j}\times\mathcal{L}_{k}$, we obtain two maps, both of which we
denote by $\mu$:

$\mu\mathrel{\mathop{\mathchar
58\relax}}\mathcal{L}_{j}^{-TL}\wedge\mathcal{L}_{k}^{-TL}\xrightarrow{\Delta_{\mathchar
24609\relax}}\left(\mathcal{L}_{j}\times_{L}\mathcal{L}_{k}\right)^{-TL}\xrightarrow{c}\mathcal{L}_{j+k}^{-TL}$
$\mu\mathrel{\mathop{\mathchar
58\relax}}\mathcal{L}_{j}^{-TL}\wedge\Sigma^{\infty}_{+}\mathcal{L}_{k}\xrightarrow{\Delta_{\mathchar
24609\relax}}\Sigma^{\infty}_{+}\left(\mathcal{L}_{j}\times_{L}\mathcal{L}_{k}\right)\xrightarrow{c}\Sigma^{\infty}_{+}\mathcal{L}_{j+k}$

which fit together to give maps

$\mu\mathrel{\mathop{\mathchar
58\relax}}\mathcal{L}^{-TL}\wedge\mathcal{L}^{-TL}\rightarrow\mathcal{L}^{-TL}$
$\mu\mathrel{\mathop{\mathchar
58\relax}}\mathcal{L}^{-TL}\wedge\Sigma^{\infty}_{+}\mathcal{L}\rightarrow\Sigma^{\infty}_{+}\mathcal{L}$

It follows from homotopy coassociativity of $\Delta$ that these define a
homotopy associative product on $\mathcal{L}^{-TL}$ and a left module action
of this on $\Sigma^{\infty}_{+}\mathcal{L}$.

Let $i\mathrel{\mathop{\mathchar 58\relax}}L\hookrightarrow\mathcal{L}$ be the
inclusion of constant loops into $\mathcal{L}_{0}$. The composition

$[L]\mathrel{\mathop{\mathchar
58\relax}}\mathbb{S}\mathop{\rightarrow}^{[L]}L^{-TL}\mathop{\rightarrow}^{i}\mathcal{L}^{-TL}$

defines a unit for $\mathcal{L}^{-TL}$, similarly to [CJ02]. Together, we have

######  Proposition 5.3.

$\mathcal{L}^{-TL}$ is a ring spectrum, and $\Sigma^{\infty}_{+}\mathcal{L}$
is a left module over it.

Furthermore if $R$ is a ring spectrum, $\mathcal{L}^{-TL}\wedge R$ is a ring
spectrum, and $\Sigma^{\infty}_{+}\mathcal{L}\wedge R$ is a left module over
it.

It follows from construction that

######  Lemma 5.4.

All of the maps $\mu$ constructed in this subsection commute with the map
$\Psi^{1}$ up to homotopy.

###  5.3 The Moduli Spaces $\mathcal{M}$, $\mathcal{N}$ and $\mathcal{Q}$

We fix $G$ a convex domain in $\mathbb{C}$ with smooth boundary, such that
$\partial G$ contains $(-\eta,\eta)$ and $(-\eta,\eta)+i$ for some $\eta>0$,
and $G$ is symmetric in the lines $i\mathbb{R}$ and $\frac{i}{2}+\mathbb{R}$.
We define

$D_{+}=(\mathbb{R}_{\leq 0}+i[0,1])\cup G\sqcup\\{-\infty\\}$

and

$D_{-}=(\mathbb{R}_{\geq 0}+i[0,1])\cup G\sqcup\\{+\infty\\}$

viewed as the one-point compactifications of half-infinite strips. Both are
compact Riemann surfaces with boundary, biholomorphic to a disc. We fix
explicit identifications between $D_{\pm}$ and the standard disc $D^{2}$: we
fix biholomorphisms $\phi_{\pm}\mathrel{\mathop{\mathchar
58\relax}}D^{2}\rightarrow D_{\pm}$ which send $\mp 1$ to $\mp\infty$,
satisfying

$\phi_{-}(z)=i-\phi_{+}(-z)$

for all $z$, as shown below.

$\longrightarrow$$\cong$$\phi_{-}$$D^{2}$$+1$$...$$+\infty$$D_{-}$$\longrightarrow$$\cong$$\phi_{+}$$D^{2}$$-1$$...$$-\infty$$D_{+}$

######  Definition 5.5.

We define $\mathcal{R}_{\pm}$ to be the space of smooth maps
$w\mathrel{\mathop{\mathchar 58\relax}}D_{\pm}\rightarrow M$ such that for all
$z\neq\mp\infty$ in $\partial D_{\pm}$, $w(z)$ lies in
$\psi^{\mathrm{Im\,}z}(L)$.

Fix $J$ an $\omega$-tame almost complex structure on $M$ which is convex at
infinity.

######  Definition 5.6.

We define $\mathcal{M}$ to be the moduli space of $J$-holomorphic maps in
$\mathcal{R}_{-}$, and $\mathcal{N}$ to be the moduli space of $J$-holomorphic
maps in $\mathcal{R}_{+}$.

There are natural evaluation maps $\pi\mathrel{\mathop{\mathchar
58\relax}}\mathcal{R}_{\pm}\rightarrow L$ sending $u$ to $u(\mp\infty)$. We
define $\mathcal{Q}$ to be the fibre product
$\mathcal{N}\times_{L}\mathcal{M}$ with respect to these evaluation maps.

For generic $J$ these moduli spaces are naturally smooth manifolds, and a
similar argument to Section 4 shows that they are compact. Furthermore, for
generic $J$ the two maps $\pi\mathrel{\mathop{\mathchar
58\relax}}\mathcal{M}\rightarrow L$ and $\pi\mathrel{\mathop{\mathchar
58\relax}}\mathcal{N}\rightarrow L$ are transverse and hence $\mathcal{Q}$ is
also a compact smooth manifold.

$\mathcal{M}$ and $\mathcal{N}$ may have different connected components which
have different dimensions. We write

$\mathcal{M}=\bigsqcup\limits_{i\geq 0}\mathcal{M}^{i}$

where $\mathcal{M}^{i}$ is the component of $\mathcal{M}$ lying in dimension
$i$, and we write

$\mathcal{N}=\bigsqcup\limits_{i\geq 0}\mathcal{N}^{i}$

similarly. Note that since both these moduli spaces are compact, only finitely
many of these components are non-empty.

###  5.4 Bundle Pairs on the Moduli Spaces

We define $\Sigma$ to be the space

$\left(D_{-}\cup[-1,+1]\cup D_{+}\right)/\sim$

where $+\infty\sim-1$ and $+1\sim-\infty$, as shown below (drawing $D_{\pm}$
as discs).

$\Sigma\mathchar 24634\relax$ $D_{-}$$D_{+}$$+\infty$$-\infty$

We define $\partial\Sigma\subseteq\Sigma$ to be the subspace

$\left(\partial D_{-}\cup[-1,+1]\cup\partial D_{+}\right)/\sim$

There are natural collapse maps

$\xi_{\pm}\mathrel{\mathop{\mathchar
58\relax}}(\Sigma,\partial\Sigma)\rightarrow(D_{\pm},\partial D_{\pm})$

collapsing $D_{\mp}$ and $[-1,1]$ to $\mp\infty$.

We define a submanifold $\tilde{L}_{\psi^{1}}\subseteq\mathbb{C}\times M$ to
be the space of pairs $(z,x)$ such that $|z|=1$ and if $z=1$ then $x$ lies in
$L$, otherwise $x$ lies in

$\psi^{\mathrm{Im\,}\phi_{-}(z)}(L)$

This is diffeomorphic to $L_{\psi^{1}}$. We let $T^{v}\tilde{L}_{\psi^{1}}$ be
its vertical tangent bundle.

There are natural maps

$ev_{\pm}\mathrel{\mathop{\mathchar 58\relax}}\left(D_{\pm},\partial
D_{\pm}\right)\times\mathcal{R}_{\pm}\rightarrow\left(\mathbb{C}\times
M,\tilde{L}_{\psi^{1}}\right)$

defined by

$ev_{+}(z,w)=\left(-\phi_{+}^{-1}(i+\bar{z}),w(z)\right)$

and

$ev_{-}(z,w)=\left(\phi_{-}^{-1}(z),w(z)\right)$

We define a map

$C\mathrel{\mathop{\mathchar
58\relax}}(\Sigma,\partial\Sigma)\times\mathcal{Q}\rightarrow\left(\mathbb{C}\times
M,\tilde{L}_{\psi^{1}}\right)$

by

$C(z,(v,u))\mathrel{\mathop{\mathchar
58\relax}}=\begin{cases}\left(ev_{-}(z,u)\right)&\mathrm{if}\,z\in D_{-}\\\
\left(ev_{+}(z,v)\right)&\mathrm{if}\,z\in D_{+}\\\
\left(1,u(+\infty)\right)&\mathrm{if}\,z\in[-1,+1]\end{cases}$

We will consider the bundle pairs

$\xi_{\pm}^{*}ev_{\pm}^{*}\left(TM,T^{v}\tilde{L}_{\psi^{1}}\right)$
$C^{*}\left(TM,T^{v}\tilde{L}_{\psi^{1}}\right)$

and

$\xi_{\pm}^{*}\pi^{*}\left(TM,T^{v}\tilde{L}_{\psi^{1}}\right)$

By construction, the fibres of the complex parts of these bundle pairs are
given as follows.

$\left(\xi^{*}_{-}ev_{-}^{*}TM\right)_{(z,(v,u))}=\begin{cases}T_{u(z)}M&\mathrm{if}\,z\in
D_{-}\\\ T_{u(+\infty)}M&\mathrm{if}\,z\in D_{+}\\\
T_{u(+\infty)}M&\mathrm{if}\,z\in[-1,1]\end{cases}$
$\left(\xi_{+}^{*}ev_{+}^{*}TM\right)_{(z,(v,u))}=\begin{cases}T_{u(+\infty)}M&\mathrm{if}\,z\in
D_{-}\\\ T_{v(z)}M&\mathrm{if}\,z\in D_{+}\\\
T_{u(+\infty)}M&\mathrm{if}\,z\in[-1,1]\end{cases}$
$\left(C^{*}TM\right)_{(z,(v,u))}=\begin{cases}T_{u(z)}M&\mathrm{if}\,z\in
D_{-}\\\ T_{v(z)}M&\mathrm{if}\,z\in D_{+}\\\
T_{u(+\infty)}M&\mathrm{if}\,z\in[-1,1]\end{cases}$
$\left(\xi^{*}_{\pm}\pi^{*}TM\right)_{(z,(v,u))}=T_{u(+\infty)}M$

The fibres of the real parts are given as follows, where $z$ now lies in
$\partial\Sigma$.

$\left(\xi^{*}_{-}ev_{-}^{*}T^{v}\tilde{L}_{\psi^{1}}\right)_{(z,(v,u))}=\begin{cases}T_{u(z)}\psi^{\mathrm{Im\,}z}(L)&\mathrm{if}\,z\in\partial
D_{-}\\\ T_{u(+\infty)}L&\mathrm{if}\,z\in\partial D_{+}\\\
T_{u(+\infty)}L&\mathrm{if}\,z\in[-1,1]\end{cases}$
$\left(\xi_{+}^{*}ev_{+}^{*}T^{v}\tilde{L}_{\psi^{1}}\right)_{(z,(v,u))}=\begin{cases}T_{u(+\infty)}L&\mathrm{if}\,z\in\partial
D_{-}\\\ T_{v(z)}\psi^{\mathrm{Im\,}z}(L)&\mathrm{if}\,z\in\partial D_{+}\\\
T_{u(+\infty)}L&\mathrm{if}\,z\in[-1,1]\end{cases}$
$\left(C^{*}T^{v}\tilde{L}_{\psi^{1}}\right)_{(z,(v,u))}=\begin{cases}T_{u(z)}\psi^{\mathrm{Im\,}z}(L)&\mathrm{if}\,z\in\partial
D_{-}\\\ T_{v(z)}\psi^{\mathrm{Im\,}z}(L)&\mathrm{if}\,z\in\partial D_{+}\\\
T_{u(+\infty)}L&\mathrm{if}\,z\in[-1,1]\end{cases}$
$\left(\xi^{*}_{\pm}\pi^{*}T^{v}\tilde{L}_{\psi^{1}}\right)_{(z,(v,u))}=T_{u(+\infty)}L$

######  Lemma 5.7.

There is an isomorphism of bundle pairs $F$ from

$\xi_{-}^{*}ev_{-}^{*}\left(TM,T^{v}\tilde{L}_{\psi^{1}}\right)\oplus\xi_{+}^{*}ev_{+}^{*}\left(TM,T^{v}\tilde{L}_{\psi^{1}}\right)$

to

$C^{*}\left(TM,T^{v}\tilde{L}_{\psi^{1}}\right)\oplus\xi_{+}^{*}\pi^{*}\left(TM,T^{v}\tilde{L}_{\psi^{1}}\right)$

###### Proof.

We define $F_{(z,(v,u))}$ explicitly as follows.

If $z$ lies in $D_{-}$, then

$F_{(z,(v,u))}\mathrel{\mathop{\mathchar 58\relax}}T_{u(z)}M\oplus
T_{u(+\infty)}M\rightarrow T_{u(z)}M\oplus T_{u(+\infty)}M$

is given by the identity.

If $z$ lies in $D_{+}$, then

$F_{(z,(v,u))}\mathrel{\mathop{\mathchar 58\relax}}T_{u(+\infty)}M\oplus
T_{v(z)}M\rightarrow T_{v(z)}M\oplus T_{u(+\infty)}M$

is given by the matrix

$\begin{pmatrix}0&1\\\ -1&0\end{pmatrix}$

with respect to the decomposition above.

If $z$ lies in $[-1,+1]$, then

$F_{(z,(v,u))}\mathrel{\mathop{\mathchar 58\relax}}T_{u(+\infty)}M\oplus
T_{u(+\infty)}M\rightarrow T_{u(+\infty)}M\oplus T_{u(+\infty)}M$

is given by the matrix

$\begin{pmatrix}\cos\left(\frac{z+1}{2}\pi\right)&\sin\left(\frac{z+1}{2}\pi\right)\\\
-\sin\left(\frac{z+1}{2}\pi\right)&\cos\left(\frac{z+1}{2}\pi\right)\end{pmatrix}$

with respect to the decomposition above.

Note that this map $F$ respects the totally real subbundles when $z$ lies in
$\partial D$. ∎

Note that

$\mathrm{Ind}\,\pi^{*}\left(TM,T^{v}\tilde{L}_{\psi^{1}}\right)$

is isomorphic to $\pi^{*}TL$, over both $\mathcal{R}_{\pm}$.

Now pick a “pinch” map $r\mathrel{\mathop{\mathchar
58\relax}}\left(D^{2},\partial D^{2}\right)\rightarrow(\Sigma,\partial\Sigma)$
such that the compositions $\xi_{\pm}\circ r$ induce maps of degree 1 from
$\partial D^{2}$ to $\partial D_{\pm}$. This determines $r$ up to homotopy,
but later we will make a more specific choice of $r$.

######  Lemma 5.8.

There is an isomorphism of bundle pairs $r^{*}F$ from

$r^{*}\xi_{-}^{*}ev_{-}^{*}\left(TM,T^{v}\tilde{L}_{\psi^{1}}\right)\oplus
r^{*}\xi_{+}^{*}ev_{+}^{*}\left(TM,T^{v}\tilde{L}_{\psi^{1}}\right)$

to

$r^{*}C^{*}\left(TM,T^{v}\tilde{L}_{\psi^{1}}\right)\oplus
r^{*}\xi_{+}^{*}\pi^{*}\left(TM,T^{v}\tilde{L}_{\psi^{1}}\right)$

###  5.5 Gluing and Orientations of the Moduli Spaces

Let $\mathcal{U}_{\infty}=\mathcal{Q}$. Then as in Section 4, for a generic
($l$-dependent) perturbation of the almost complex structure, the space

$\mathcal{V}^{\prime}\mathrel{\mathop{\mathchar
58\relax}}=\bigsqcup\limits_{0\leq l<\infty}\mathcal{U}_{l}$

is a smooth manifold such that the natural map
$\mathcal{V}^{\prime}\rightarrow\mathbb{R}$ is proper, and it follows from the
proof of gluing in [ES16] that

$\mathcal{V}\mathrel{\mathop{\mathchar 58\relax}}=\bigsqcup\limits_{0\leq
l\leq\infty}\mathcal{U}_{l}$

is naturally a compact smooth manifold with boundary $L\sqcup\mathcal{Q}$.
This uses in an important way the fact that there is only one type of disk
breaking.

The vertical tangent bundle of this cobordism restricted to $\mathcal{Q}$ is
then naturally isomorphic to

$\mathrm{Ind}\,r^{*}C^{*}\left(TM,T^{v}\tilde{L}_{\psi^{1}}\right)$

Therefore an $R$-orientation of $\mathrm{Ind}$ induces an $R$-orientation of
$T\mathcal{Q}$.

By construction, $T\mathcal{M}$ and $T\mathcal{N}$ are naturally stably
isomorphic to

$\mathrm{Ind}\,r^{*}\xi_{\pm}^{*}ev_{\pm}^{*}\left(TM,T^{v}L_{\psi^{1}}\right)$

respectively. Therefore $R$-orientations on both of these as well as $TL$
induce one on $T\mathcal{Q}$, since by construction,

$T\mathcal{Q}\cong T\mathcal{N}+T\mathcal{M}-\pi^{*}TL$

######  Assumption 5.9.

There are $R$-orientations of $T\mathcal{N}$, $T\mathcal{M}$, $\mathrm{Ind}$
and $L$, such that the two induced $R$-orientations on $T\mathcal{Q}$ above
agree under the isomorphism $r^{*}F$ constructed in Lemma 5.7. Furthermore the
restriction of $\mathrm{Ind}$ to $L$ is the given orientation of $L$.

######  Lemma 5.10.

Under Assumption 5.9, the cobordism $\mathcal{V}$ admits an $R$-orientation,
restricting to the given ones on $L$ and $\mathcal{Q}$.

###### Proof.

We need to check that the natural bundle isomorphism which comes from gluing

$\mathrm{Ind}\cong T\mathcal{Q}\cong T\mathcal{N}+T\mathcal{M}-\pi^{*}TL$

from above agrees with the one induced by $F$, up to homotopy of bundle
isomorphisms.

We will recall a sketch of how Ekholm and Smith, in [ES16], construct their
diffeomorphism from $\mathcal{Q}$ to some $\mathcal{P}_{l+1}$. Fix some very
large $l>0$. We define $D_{\pm,l}$ to be the subset of $D_{\pm}$ given by

$D_{\pm}\cap\left\\{\pm\mathrm{Re\,}\geq-(l-1)\right\\}$

which we view as subsets of $G_{l}=C_{l+1}$ by translating by $\pm l$. We let
$H$ be the closure of the complement in $G_{l}$ of the union of these two
regions:

$H\mathrel{\mathop{\mathchar 58\relax}}=[-1,1]+i[0,1]$

Ekholm and Smith construct a pre-gluing embedding

$PG\mathrel{\mathop{\mathchar
58\relax}}\mathcal{Q}\hookrightarrow\mathcal{D}_{f_{l+1},C_{l+1}}$

sending $(v,u)$ to a map $(u\\#_{c}v)\mathrel{\mathop{\mathchar
58\relax}}C_{l+1}\rightarrow M$ which agrees with $u$ and $v$ on $D_{\pm,l}$
respectively, and is $C^{1}$-small when restricted to $H$. Roughly, this is
constructed by picking a metric on $M$, picking $l$ large enough so that
outside $D_{\pm,l}$ the images of both $u$ and $v$ lie inside a single
geodesically convex neighbourhood of $M$, and cutting them off with a bump
function.

Then there is a diffeomorphism $\rho\mathrel{\mathop{\mathchar
58\relax}}PG(\mathcal{Q})\rightarrow\mathcal{P}_{l+1}$ such that $PG$ and
$\rho\circ PG$ are $C^{1}$ close. Therefore the derivative $d(\rho\circ PG)$
sends a pair of sections $(s_{\pm})$ of the bundle pairs
$r^{*}\xi_{\pm}^{*}ev_{\pm}^{*}\left(TM,T^{v}L_{\psi^{1}}\right)$ lying in
$T\mathcal{Q}$, to a section $s$ of the bundle pair $ev^{*}(TM,TL)$ over
$\mathcal{P}_{l+1}$, such that $s$ is $C^{1}$-small on $H$ and $C^{1}$-close
to $u$ and $v$ when restricted to $D_{\pm,l}$.

We make some choice of “pinch” map $r\mathrel{\mathop{\mathchar
58\relax}}C_{l+1}\rightarrow\Sigma$ which sends $\partial C_{l+1}$ to
$\partial\Sigma$, and which restricts to the identity on $D_{\pm,l}$. Note
that $ev\simeq C\circ r$.

By construction, we see that the map induced by $r^{*}F$ on sections of these
bundle pairs sends a pair of sections $(s_{\pm})$, whose evaluations at
$\mp\infty$ agree, to a section $s$, whose restrictions to $D_{\pm,l}$ agree
with $s_{\pm}$. Consider the composition $A\mathrel{\mathop{\mathchar
58\relax}}$

${T\mathcal{Q}}$${\Gamma\left(r^{*}\xi_{-}^{*}ev_{-}^{*}\left(TM,T^{v}\tilde{L}_{\psi^{1}}\right)\oplus
r^{*}\xi_{+}^{*}ev_{+}^{*}\left(TM,T^{v}\tilde{L}_{\psi^{1}}\right)\right)}$${\Gamma\left(r^{*}C^{*}\left(TM,T^{v}\tilde{L}_{\psi^{1}}\right)\oplus
r^{*}\xi_{+}^{*}\pi^{*}\left(TM,T^{v}\tilde{L}_{\psi^{1}}\right)\right)}$${\Gamma\left(r^{*}C^{*}\left(TM,T^{v}\tilde{L}_{\psi^{1}}\right)\right)}$${T\mathcal{P}_{l+1}=\mathrm{Ind}(r^{*}C^{*}\left(TM,T^{v}\tilde{L}_{\psi^{1}}\right)}$$\scriptstyle{r^{*}F}$$\scriptstyle{\mathrm{Projection\,to\,the\,first\,factor}}$$\scriptstyle{\mathrm{Orthogonal\,projection}}$

where we weight the metric on the bundle pair to be small on $H$. Note that
since $r^{*}F$ respects the direct sum away from $H$, if the weight is
sufficiently small on $H$, any Cauchy-Riemann operator on the bundle pair in
either the second or third term which respects the direct sum is of distance
less than 1 to a Cauchy-Riemann operator on the other which respects the
direct sum. Therefore the map $T\mathcal{Q}\rightarrow T\mathcal{P}_{l+1}$
coming from Lemma 3.5 agrees up to homotopy with $A$, and so under Assumption
5.9, $A$ respects the $R$\- orientations.

$A$ is close to $d(\rho\circ PG)$ and so the two maps are homotopic
isomorphisms of vector bundles, which is what we wanted. ∎

######  Lemma 5.11.

If $T^{v}L_{\psi^{1}}$ admits a stable trivialisation over an $i+1$-skeleton
of $L_{\psi^{1}}$, then the induced stable trivialisations over an
$i+1$-skeleton of $\mathcal{Q}$ of the bundle pairs appearing in Lemma 5.8
agree up to homotopy, under $r^{*}F$.

###### Proof.

The two stable trivialisations are related by a map
$\eta\mathrel{\mathop{\mathchar 58\relax}}\partial
D^{2}\times\mathcal{Q}\rightarrow O$, where $O$ is the infinite orthogonal
group. We will use the choice of $r$ from the proof of Lemma 5.10 for
convenience, noting that this is unique up to homotopy.

From the construction of $F$, for any $y$ in $\mathcal{Q}$ and $x$ in
$r|_{\partial D^{2}}^{-1}\partial D_{-}$, $\eta(x,y)$ is given by the identity
matrix. For $x$ in $r|_{\partial D^{2}}^{-1}\partial D_{+}$, $\eta(x,y)$ is
given by (the stabilisation of) the block matrix

$\begin{pmatrix}0&1\\\ -1&0\end{pmatrix}$

Then along the top part of $\partial D^{2}$, $\eta(\cdot,y)$ is given by some
homotopy between these two matrices, and travelling in the opposite direction
along the bottom produces the reverse of this homotopy. Therefore for a fixed
$y$, $\eta(\cdot,y)$ is a contractible loop. However these contractions and
loops are independent of $y$, so $\eta$ is contractible. ∎

Then from Proposition 3.14 it follows that

######  Corollary 5.12.

  1. 1.

If $T^{v}L_{\psi^{1}}$ admits a stable trivialisation over a 2-skeleton of
$L_{\psi^{1}}$, then Assumption 5.9 holds for $R=H\mathbb{Z}$.

  2. 2.

If $T^{v}L_{\psi^{1}}$ admits a stable trivialisation over a 3-skeleton of
$L_{\psi^{1}}$, then Assumption 5.9 holds for $R=KU$.

  3. 3.

If $T^{v}L_{\psi^{1}}$ admits a stable trivialisation, then Assumption 5.9
holds when $R$ is any complex-oriented cohomology theory.

###### Proof of Proposition 1.17.

We will show that if $TL$ admits a homotopy class of stable trivialisations
over an $i$-skeleton $L_{i}$ of $L$ which is preserved by $\psi^{1}$ and can
be extended to an $i+1$-skeleton $L_{i+1}$, then $T^{v}L_{\psi^{1}}$ admits a
stable trivialisation over an $i$-skeleton of $L_{\psi^{1}}$.

Let $\tilde{\psi^{1}}\mathrel{\mathop{\mathchar 58\relax}}L\rightarrow L$ be a
map homotopic to $\psi^{1}|_{L}$ which sends $L_{i}$ to itself. Then
$L_{\psi^{1}}$ is homotopy equivalent to

$L\times[0,1]/\\{(0,x)\sim(1,\tilde{\psi^{1}})$

and an $i+1$-skeleton of this is given by

$\left(L_{i}\times[0,1]\cup L_{i+1}\right)$

Then by assumption $T^{v}L_{\psi^{1}}$, admits a stable trivialisation over
this $i+1$-skeleton. ∎

###  5.6 Composition of the Moduli Spaces

Fix some diffeomorphism $\tilde{L}_{\psi^{1}}\cong L_{\psi^{1}}$ covering some
orientation-preserving diffeomorphism $\partial D^{2}\cong S^{1}$ sending
$1\in\partial D^{2}$ to $0\in S^{1}=\mathbb{R}/\mathbb{Z}$. There are natural
evaluation maps

$S^{1}\times\mathcal{R}_{\pm}\xrightarrow{\exp(2\pi\cdot)}\partial
D^{2}\times\mathcal{R}_{\pm}\xrightarrow{\phi_{\pm}(\mp\cdot)}\partial
D_{\pm}\times\mathcal{R}_{\pm}\xrightarrow{ev_{\pm}}\tilde{L}_{\psi^{1}}\xrightarrow{\cong}L_{\psi^{1}}$

whose adjoints define maps

$\sigma_{\pm}\mathrel{\mathop{\mathchar
58\relax}}\mathcal{R}_{\pm}\rightarrow\mathcal{L}_{\pm 1}$

Therefore $\mathcal{Q}$ admits a natural evaluation map

$\sigma\mathrel{\mathop{\mathchar
58\relax}}\mathcal{Q}\rightarrow\mathcal{L}_{0}$

sending $(v,u)$ to the concatenation $c(\sigma_{-}(v),\sigma_{+}(u)$. Then
choices of $R$-orientations of $L$ and all the moduli spaces allow allow us to
use $\sigma_{\pm}$ and $\sigma$ to define fundamental classes $[\mathcal{M}]$,
$[\mathcal{N}]$ and $[\mathcal{Q}]$ in
$\bigoplus\limits_{j}R_{j}\left(\mathcal{L}_{\pm 1}^{-TL}\right)$ and
$\bigoplus\limits_{j}R_{j}\left(\mathcal{L}_{0}^{-TL}\right)$ respectively.

Our goal in this subsection is to prove the following two lemmas.

######  Lemma 5.13.

$[\mathcal{N}]\cdot[\mathcal{M}]=\left[\mathcal{Q}\right]$

in $\bigoplus\limits_{j}R_{j}\left(\mathcal{L}_{0}^{-TL}\right)$.

######  Lemma 5.14.

$\left[\mathcal{Q}\right]=[L]$

in $\bigoplus\limits_{j}R_{j}\left(\mathcal{L}_{0}^{-TL}\right)$, for some
choice of $R$-orientation used to define $[\mathcal{N}]$ and $[\mathcal{M}]$.

From these, we deduce

######  Lemma 5.15.

The composition
${\Sigma^{\infty}_{+}\mathcal{L}_{0}\wedge
R}$${\bigvee\limits_{j}\Sigma^{\infty+j}_{+}\mathcal{L}_{1}\wedge
R}$${\bigvee\limits_{j}\Sigma^{\infty+j}_{+}\mathcal{L}_{0}\wedge
R}$${\mathcal{L}_{0}\wedge
R}$$\scriptstyle{[\mathcal{M}]\cdot}$$\scriptstyle{[\mathcal{N}]\cdot}$$\scriptstyle{p}$
is an equivalence, where $p\mathrel{\mathop{\mathchar
58\relax}}\bigvee\limits_{j}\Sigma^{\infty+j}_{+}\mathcal{L}_{0}\rightarrow\Sigma^{\infty}_{+}\mathcal{L}_{0}$
is the natural projection map.

###### Proof of Lemma 5.13.

This follows from commutativity of the following diagram:

${\mathbb{S}}$${\mathcal{N}^{-T\mathcal{N}}\wedge
R\wedge\mathcal{M}^{-T\mathcal{M}}\wedge
R}$${\mathcal{Q}^{-\left(T\mathcal{N}+T\mathcal{M}-\pi^{*}TL\right)}\wedge
R}$${\bigvee\limits_{j}\Sigma^{j}\mathcal{N}^{-\pi^{*}TL}\wedge
R\wedge\mathcal{M}^{-\pi^{*}TL}\wedge
R}$${\bigvee\limits_{j}\Sigma^{j}\mathcal{Q}^{-\pi^{*}TL}\wedge
R}$${\bigvee\limits_{j}\Sigma^{j}\mathcal{L}_{-1}^{-TL}\wedge
R\wedge\mathcal{L}_{+1}^{-TL}\wedge
R}$${\bigvee\limits_{j}\Sigma^{j}\mathcal{L}_{0}^{-TL}\wedge
R}$$\scriptstyle{[\mathcal{N}]\wedge[\mathcal{M}]}$$\scriptstyle{\left[\mathcal{Q}\right]}$$\scriptstyle{\mathrm{Thom}}$$\scriptstyle{i_{\mathchar
24609\relax}}$$\scriptstyle{\mathrm{Thom}}$$\scriptstyle{\sigma_{-}\wedge\sigma_{+}}$$\scriptstyle{i^{\prime}_{\mathchar
24609\relax}}$$\scriptstyle{\sigma}$$\scriptstyle{\mu}$

where the two arrows labelled $\mathrm{Thom}$ are the isomorphisms from the
Thom isomorphism theorem for our choices of $R$-orientations followed by
inclusion into this wedge product, and $i_{!}$ and $i^{\prime}_{!}$ are
obtained by applying Definition 2.23 to the embedding

$\mathcal{Q}\hookrightarrow\mathcal{N}\times\mathcal{M}$

using the virtual vector bundles $-T\mathcal{N}\boxplus-T\mathcal{M}$ and
$-\pi^{*}TL\boxplus-\pi^{*}TL$ respectively. ∎

###### Proof of Lemma 5.14.

By Lemma 5.10, there exists an $R$-orientable cobordism $\mathcal{V}$ from
$\mathcal{Q}$ to $L$, with respect to the given $R$-orientations on both ends.
Furthermore, evaluating along the boundary allows us to extend the maps
$\sigma\mathrel{\mathop{\mathchar
58\relax}}\mathcal{Q}\rightarrow\mathcal{L}_{0}$ and
$L\hookrightarrow\mathcal{L}_{0}$ to the entirety of $\mathcal{V}$. Therefore
the result follows from Lemma 2.25.

∎

###  5.7 Proof of Theorem 1.16

###### Proof of Theorem 1.16.

Lemma 5.4 implies that the following diagram commutes up to homotopy:

${\Sigma^{\infty}_{+}\mathcal{L}_{0}\wedge
R}$${\bigvee\limits_{j}\Sigma^{\infty+j}_{+}\mathcal{L}_{1}\wedge
R}$${\bigvee\limits_{j}\Sigma^{\infty+j}_{+}\mathcal{L}_{0}\wedge
R}$${\Sigma^{\infty}_{+}\mathcal{L}_{0}\wedge
R}$${\Sigma^{\infty}_{+}\mathcal{L}_{0}\wedge
R}$${\bigvee\limits_{j}\Sigma^{\infty+j}_{+}\mathcal{L}_{1}\wedge
R}$${\bigvee\limits_{j}\Sigma^{\infty+j}_{+}\mathcal{L}_{0}\wedge
R}$${\Sigma^{\infty}_{+}\mathcal{L}_{0}\wedge
R}$$\scriptstyle{[\mathcal{M}]\cdot}$$\scriptstyle{\Psi^{1}}$$\scriptstyle{[\mathcal{N}]\cdot}$$\scriptstyle{\Psi^{1}}$$\scriptstyle{\Psi^{1}}$$\scriptstyle{p}$$\scriptstyle{\Psi^{1}}$$\scriptstyle{\Psi^{1}_{*}[\mathcal{M}]\cdot}$$\scriptstyle{\Psi^{1}_{*}{[\mathcal{N}]}\cdot}$$\scriptstyle{p}$

By Lemma 5.2, the second vertical arrow is homotopic to the identity, and the
horizontal arrows along the top are all homotopic to the corresponding
horizontal arrows along the bottom. Furthermore Lemma 5.15 tells us that the
composition along the top (and similarly along the bottom) is an equivalence.
It follows that the map

$\Psi^{1}\mathrel{\mathop{\mathchar
58\relax}}\Sigma^{\infty}_{+}\mathcal{L}_{0}\wedge
R\rightarrow\Sigma^{\infty}_{+}\mathcal{L}_{0}\wedge R$

is homotopic to the identity. ∎

##  6 Families Over Other Bases

Our goal in this section will be to prove Theorem 1.21. In this section, we
only consider mod-2 singular homology. We will use the following purely
topological lemma:

######  Lemma 6.1 ([LM03, Lemma 4.3]).

Fix fibre bundles $A\hookrightarrow B\twoheadrightarrow C$ and
$F\hookrightarrow E\twoheadrightarrow B$. Assume that the fibre bundles

  1. 1.

$A\hookrightarrow B\twoheadrightarrow C$

  2. 2.

$E|_{A}\hookrightarrow E\twoheadrightarrow C$

  3. 3.

$F\hookrightarrow E|_{A}\twoheadrightarrow A$

all c-split. Then the fibre bundle $F\hookrightarrow E\twoheadrightarrow B$
c-splits.

###### Proof.

The inclusion of a fibre $F\hookrightarrow E$ is the composition
$F\hookrightarrow E|_{A}$ and $E|_{A}\hookrightarrow E$. The hypothesis of the
lemma is that both these maps induce injections on mod-2 singular homology. ∎

###### Proof of Theorem 1.21(i).

The case $i=0$ is trivial and the case $i=1$ follows from Theorem 1.4 along
with an application of the Mayer-Vietoris sequence. We proceed by induction,
and assume we know the result for $i-1$, for all $M$ and $L$ as in the
statement of Theorem 1.4.

We let $E=\gamma^{*}\mathcal{E}$. Then we have two fibre bundles

$L\hookrightarrow E\twoheadrightarrow(S^{1})^{i}$

and

$S^{1}\hookrightarrow(S^{1})^{i}\twoheadrightarrow(S^{1})^{i-1}$

where the inclusion map is inclusion to the first factor and the projection
map is projection to all the other factors.

The fibre bundle $E|_{S^{1}}\twoheadrightarrow S^{1}$ c-splits by Theorem 1.4.
So by Lemma 6.1, it suffices to show the fibre bundle given by the composition

$E\twoheadrightarrow(S^{1})^{i}\twoheadrightarrow(S^{1})^{i-1}$

c-splits.

For $\tau$ in $(S^{1})^{i-1}$, we let $\gamma_{\tau}$ be the map
$S^{1}\rightarrow Lag_{L}$ given by $\gamma_{\tau}(t)=\gamma(t,\tau)$.

######  Claim 6.2.

Possibly after perturbing $\gamma$, there exists a smooth family of smooth
maps $H_{\tau}\mathrel{\mathop{\mathchar 58\relax}}M\times
S^{1}\rightarrow\mathbb{R}$ for $\tau$ in $(S^{1})^{i-1}$ such that the
Hamiltonian flow of $H_{\tau}$ applied to $\gamma_{\tau}(0)$ is
$\gamma_{\tau}$.

###### Proof of Claim 6.2.

For each point $(t_{0},\tau)$ in $S^{1}\times(S^{1})^{i-1}$, let
$V_{t_{0},\tau}$ be the space of germs near $M\times\\{0\\}$ of compactly
supported smooth maps $M\times(-\varepsilon,\varepsilon)\rightarrow\mathbb{R}$
for some $\varepsilon>0$ whose Hamiltonian flow $\phi_{t}$ applied to
$\gamma(t_{0},\tau)$ agrees with $\gamma_{\tau}(t_{0}+t)$. Together these
define a fibre bundle over $(S^{1})^{i}$, and the family of $H_{\tau}$ we want
is a choice of smooth section of this bundle. The fibres of this bundle are
contractible so this exists. ∎

We define a map

$f\mathrel{\mathop{\mathchar 58\relax}}E\hookrightarrow M\times
T^{*}S^{1}\times(S^{1})^{i-1}$

by

$f(x)=(x,t,H(x),\tau)$

where $x$ lies in the fibre over $\tau$ in $(S^{1})^{i-1}$, and we identify
$T^{*}S^{1}$ with $S^{1}\times\mathbb{R}$ in the usual way. Note that this
lives over $(S^{1})^{i-1}$.

A direct computation shows that for each $\tau$, this gives a relatively exact
Lagrangian in $M\times T^{*}S^{1}$, therefore this realises
$E\twoheadrightarrow(S^{1})^{i-1}$ as a family of relatively exact Lagrangians
in $M\times T^{*}S^{1}$, and therefore by the induction hypothesis it
c-splits.

Therefore by Lemma 6.1, the fibre bundle $E\twoheadrightarrow(S^{1})^{i}$
c-splits. ∎

######  Lemma 6.3 ([LM03, Lemma 4.1(ii)]).

Let $E\twoheadrightarrow B$ be a fibre bundle, and
${f\mathrel{\mathop{\mathchar 58\relax}}B^{\prime}\rightarrow B}$ a map which
is surjective on mod-2 singular homology. Assume that the pullback bundle
$f^{*}E\twoheadrightarrow B^{\prime}$ c-splits. Then $E\twoheadrightarrow B$
c-splits.

###### Proof of Theorem 1.21(ii).

Follows from Lemma 6.3 by considering a degree 1 map $(S^{1})^{i}\rightarrow
S^{i}$. ∎

##  Appendix A de Silva’s Theorem

In the proof of Proposition 3.14, we quote a theorem of de Silva from [dS98].
We will recap a proof of this theorem since the original result was never
published.

Fix a finite based connected CW complex $X$ with basepoint $x_{0}$. A based
map

${\phi\mathrel{\mathop{\mathchar 58\relax}}\Sigma X\rightarrow U/O}$

determines a based virtual bundle pair $(E,F)$ over $(D^{2},\partial
D^{2})\times X$. Taking its index bundle gives a based virtual vector bundle
$\mathrm{Ind}(E,F)$ over $X$, classified by a based map $X\rightarrow BO$ up
to homotopy. This defines a map

$\mathrm{Ind}\mathrel{\mathop{\mathchar 58\relax}}[\Sigma
X,U/O]_{*}\rightarrow[X,BO]_{*}$

Let $H$ be the bundle pair over $(D^{2},\partial D^{2})\times\\{*\\}$ defined
as in Section 3.2. Then we see that for any based virtual vector bundle $E$
over $X$,

$\mathrm{Ind}((\mathbb{C},\mathbb{R})\otimes E)=E$

and

$\mathrm{Ind}(H\otimes E)=E\oplus E$

######  Theorem A.1 ([dS98, Theorem A]).

The map $\mathrm{Ind}$ is a bijection.

###### Proof.

We first prove surjectivity. We define a map

$\beta\mathrel{\mathop{\mathchar 58\relax}}[X,BO]_{*}\rightarrow[\Sigma
X,U/O]_{*}$

to send a based virtual vector bundle $E$ of rank $n$ to the based virtual
vector bundle

$H\otimes E-(\mathbb{C},\mathbb{R})\otimes E-H\otimes\mathbb{R}^{n}$

Then it is clear that $\beta$ is a section of $\mathrm{Ind}$ and so
$\mathrm{Ind}$ is surjective.

We now prove injectivity. We define

$KUO\left((D^{2},\partial D^{2})\times X\right)$

to be the set of virtual bundle pairs over $(D^{2},\partial D^{2})\times X$ up
to stable isomorphism, which is an abelian group under direct sums. There is a
natural forgetful map

$g\mathrel{\mathop{\mathchar 58\relax}}\left[\Sigma
X,U/O\right]_{*}\rightarrow KUO\left((D^{2},\partial D^{2})\times X\right)$

whose image lies in the set of virtual bundle pairs with vanishing Maslov
index. Note that by Bott periodicity, $U/O$ is a group-like monoid and so

$[\Sigma X,U/O]_{*}\cong[\Sigma X,U/O]$

From this, we see that $g$ is injective.

$g$ fits into a commutative square

${\left[\Sigma
X,U/O\right]_{*}}$${\left[X,BO\right]_{*}}$${KUO\left((D^{2},\partial
D^{2})\times
X\right)}$${\left[X,BO\times\mathbb{Z}\right]}$$\scriptstyle{\mathrm{Ind}}$$\scriptstyle{g}$$\scriptstyle{\mathrm{Ind}}$

Note that all the maps in this diagram are maps of $[X,BO]_{*}$\- modules,
where the module structure is induced by the tensor product.

By [Ati66, Theorem 2.1] (taking the line bundle $L$ in the statement of the
theorem to be trivial), we have that

$KUO\left((D^{2},\partial D^{2})\times
X\right)\cong[X,BO\times\mathbb{Z}][\theta]/(\theta^{2})$

as $[X,BO]$\- modules, where $\theta$ corresponds to
$H-(\mathbb{C},\mathbb{R})$, and the $\mathbb{Z}$ factor corresponds to the
Maslov index. Furthermore

$\mathrm{Ind}((\mathbb{C},\mathbb{R})\otimes E)=E$

and

$\mathrm{Ind}(\theta\otimes E)=E$

for all $E$.

Now suppose $\psi$ in $[\Sigma X,U/O]_{*}$ satisfies $\mathrm{Ind}\psi=0$.
Then $\mathrm{Ind}g(\psi)=0$ and so

$g(\psi)=(\theta-(\mathbb{C},\mathbb{R}))\otimes E$

for some real virtual vector bundle $E$ over $X$. Because this lies in the
image of $g$, by restricting to $\\{1\\}\times X$, we see that $E$ is trivial
of some rank. But the Maslov index of $g(\psi)$ is the rank of $E$, which must
therefore by 0. Therefore $g(\psi)$ and hence $\psi$ are trivial. ∎

##  Appendix B An Example in $K$-Theory

Our goal in this section is to prove the following.

######  Proposition B.1.

In all sufficiently high dimensions $n$, there exists a closed $n$-dimensional
manifold $V$ such that:

  1. 1.

$V$ is stably parallelisable (and hence spin).

  2. 2.

$V$ admits a self-diffeomorphism $\theta\mathrel{\mathop{\mathchar
58\relax}}V\rightarrow V$ which acts as the identity on integral cohomology
$H^{*}(V)$ but does not act as the identity on $K^{*}(V)$, where $K^{*}$
denotes complex $K$-theory.

  3. 3.

If $n$ is odd, we can take $V$ to be a rational homology sphere.

######  Corollary B.2.

If $L$ is diffeomorphic to $V$, then $\theta$ does not lie in
$\mathcal{G}_{L}$.

The rest of the section will be dedicated to a proof of Proposition B.1,
following a suggestion of Randal-Williams ([RW]). We will construct these in
sufficiently high odd dimensions and then observe that taking a product with
$S^{1}$ provides the desired examples in all sufficiently high even
dimensions.

Let $q$ be a positive integer, and $p$ an odd prime. Let $Y$ be the homotopy
mapping cone of the map $S^{2q}\rightarrow S^{2q}$ of degree $p$.

######  Lemma B.3.

We have the following:

  1. 1.

$\tilde{K}^{i}(Y)\cong\begin{cases}0&\,i\,\mathrm{even}\\\
\mathbb{Z}/p&\,i\,\mathrm{odd}\end{cases}$

  2. 2.

When $i$ is odd,

$K_{i}(Y)\cong 0$

  3. 3.

$\tilde{H}^{i}(Y)\cong\begin{cases}\mathbb{Z}/p&\,i=2q+1\\\
0&\,\mathrm{otherwise}\end{cases}$

  4. 4.

$\tilde{H}_{i}(Y)\cong\begin{cases}\mathbb{Z}/p&\,i=2q\\\
0&\,\mathrm{otherwise}\end{cases}$

  5. 5.

For all $i$,

$\tilde{H}^{i}(Y;\mathbb{Q})=0$

###### Proof.

Follows from the long exact sequence of a cone. (2) also uses the
decomposition

$K_{i}(Y)\cong\tilde{K}_{i}(Y)\oplus K_{i}(\mathrm{point})$

∎

By [Ada66, Theorem 1.7], for sufficiently large $q$, there is a map

$g\mathrel{\mathop{\mathchar 58\relax}}\Sigma^{2(p-1)}Y\rightarrow Y$

which induces an isomorphism on $\tilde{K}^{*}$, but must be 0 on
$\tilde{H}^{*}$ and $\tilde{H}_{*}$ for degree reasons. Furthermore we can
choose $q$ to be sufficiently large that $Y$ is simply connected.

Let $Z=\Sigma^{2(p-1)}Y\vee Y$, and we define $f\mathrel{\mathop{\mathchar
58\relax}}Z\rightarrow Z$ to be the composition

$Z=\Sigma^{2(p-1)}Y\vee Y\xrightarrow{\mathrm{Pinch}\vee
Id}\Sigma^{2(p-1)}Y\vee\Sigma^{2(p-1)}Y\vee Y\xrightarrow{Id\vee g\vee
Id}\Sigma^{2(p-1)}Y\vee Y=Z$

where the first map is the pinching map, which uses the fact that

$\Sigma^{2(p-1)}Y=\Sigma\left(\Sigma^{2(p-1)-1}Y\right)$

######  Lemma B.4.

Let $E$ be a generalised homology or cohomology theory. Let $A$ be a connected
finite CW complex, and $h\mathrel{\mathop{\mathchar 58\relax}}A\rightarrow A$
some map.

Then $h$ acts as the identity on $E(A)$ if and only if $h$ acts as the
identity on $\tilde{E}(A)$.

###### Proof.

There is a canonical splitting

$E(A)\cong\tilde{E}(A)\oplus E(\mathrm{point})$

$h$ acts diagonally with respect to this splitting and always acts as the
identity on $E(\mathrm{point})$. ∎

######  Lemma B.5.

Let $f\mathrel{\mathop{\mathchar 58\relax}}Z\rightarrow Z$ be the map
constructed above.

  1. 1.

$f$ does not act as the identity on $K^{*}(Z)$.

  2. 2.

$f$ acts as the identity on $H^{*}(Z)$ and $H_{*}(Z)$. In particular, since
$Z$ is simply connected, $f$ is a homotopy equivalence.

###### Proof.

With respect to the decomposition

$\tilde{K}^{*}(Z)\cong\tilde{K}^{*}\left(\Sigma^{2(p-1)}Y\right)\oplus\tilde{K}^{*}(Y)$

we see that $f^{*}$ looks like

$\begin{pmatrix}Id&0\\\ g^{*}&Id\end{pmatrix}$

and so $f$ does not act as the identity on $\tilde{K}^{*}(Z)$, and hence not
on $K^{*}(Z)$ either, by Lemma B.5. A similar argument shows the analogous
results for $H^{*}(Z)$ and $H_{*}(Z)$. ∎

Now if $Z$ were a closed manifold with the right properties and $f$ were a
diffeomorphism, we would be done, but unfortunately this is not the case.

Pick an embedding $Z\hookrightarrow\mathbb{R}^{N}$ for some large even
$N>4(p+q)$. Let $W$ be a regular neighbourhood of $Z$ with smooth boundary
such that $W$ deformation retracts to $Z$. Then $W$ is a compact manifold with
boundary, has one handle for each cell of $Z$, is simply connected, and for
sufficiently large $N$, $\partial W$ is also simply connected. The tangent
bundle $TW$ is trivial since $W$ is a codimension 0 submanifold of
$\mathbb{R}^{N}$, and so $T\partial W$ is also stably trivial.

Since $Z\simeq W$, $f$ induces a map $f\mathrel{\mathop{\mathchar
58\relax}}W\rightarrow W$, well-defined up to homotopy. If $N$ was chosen to
be sufficiently large, $f$ can be homotoped to an orientation-preserving
embedding $e\mathrel{\mathop{\mathchar 58\relax}}W\hookrightarrow W$, which we
can assume has image which does not touch $\partial W$.

######  Lemma B.6.

$e$ is homotopic to an orientation-preserving diffeomorphism
$\phi\mathrel{\mathop{\mathchar 58\relax}}W\rightarrow W$.

###### Proof.

The complement $C\mathrel{\mathop{\mathchar 58\relax}}=W\setminus
e\left(W\setminus\partial W\right)$ is a cobordism from $e(\partial W)$ to
$\partial W$. All three of $C$, $e(\partial W)$ and $\partial W$ are simply
connected, and since $e$ is a homotopy equivalence, by excision the inclusion
$e(\partial W)\hookrightarrow C$ is a homology equivalence.

By Poincaré-Lefschetz duality, the inclusion $\partial W\hookrightarrow C$ is
also a homology equivalence, and so by the $h$-cobordism theorem $C$ is a
trivial cobordism, and so the result follows. ∎

Now we let $V=\partial W$ and $\theta=\phi|_{\partial W}$.

######  Lemma B.7.

$V$ is a rational homology sphere.

###### Proof.

By Lemma B.3, $Y$ and hence $Z$ and $W$ are rationally homology equivalent to
a point. Then the result follows from the exact sequence of a pair (using
Poincaré duality):

$H_{i}(W;\mathbb{Q})\rightarrow H^{N-i}(W;\mathbb{Q})\rightarrow
H_{i-1}(V;\mathbb{Q})\rightarrow H_{i-1}(W;\mathbb{Q})\rightarrow
H^{N-i+1}(W;\mathbb{Q})$

∎

The following two lemmas complete the proof of Proposition B.1.

######  Lemma B.8.

$\theta$ acts as the identity on $H^{*}(V)$.

###### Proof.

From Lemma B.3, we see that

$\tilde{H}^{i}(W)\cong\begin{cases}\mathbb{Z}/p&\,i=2q+1\,\mathrm{or}\,2(p+q-1)+1\\\
0&\,\mathrm{otherwise}\end{cases}$

and since $W$ is orientable, by Poincaré duality

$H^{i}(W,\partial W)\cong H_{N-i}(W)\cong\begin{cases}\mathbb{Z}&\,i=N\\\
\mathbb{Z}/p&\,i=N-2q\,\mathrm{or}\,N-2(p+q-1)\\\
0&\,\mathrm{otherwise}\end{cases}$

Therefore by the long exact sequence of the pair $(W,\partial W)$, for all $i$
either the restriction map $\tilde{H}^{i}(W)\rightarrow\tilde{H}^{i}(\partial
W)$ is an isomorphism or the boundary map $\tilde{H}^{i}(\partial
W)\rightarrow H^{i+1}(W,\partial W)$ is an isomorphism. Both of these maps are
compatible with the actions of $\phi$ and $\theta$, so by Lemma B.5, the
result follows. ∎

######  Lemma B.9.

$\theta$ does not act as the identity on $K^{*}(V)$.

###### Proof.

We will show that the restriction map
$\tilde{K}^{i}(W)\rightarrow\tilde{K}^{i}(V)$ is injective for all $i$, then
the result follows from Lemmas B.5 and B.4.

By the long exact sequence of a pair, the kernel of the restriction map is the
image of

$K^{i}(W,\partial W)\rightarrow\tilde{K}^{i}(W)$

$TW$ is trivial and hence oriented with respect to $K^{*}$, so by Atiyah
duality

$K^{i}(W,\partial W)\cong K_{N-i}(W)$

Therefore using the decomposition

$\tilde{K}^{*}(W)\cong\tilde{K}^{*}\left(\Sigma^{2(p-1)}Y\right)\oplus\tilde{K}^{*}(Y)$

and Lemma B.3, we see that if $i$ is even then $\tilde{K}^{i}(W)=0$, and if
$i$ is odd then $K_{N-i}(W)=0$. In either case this implies that the
restriction map is injective in degree $i$. ∎

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