# An Introduction to Floer Homotopy 

The main reference for this note is [@AbouzBlum2021], however much of the content for this section comes from the earlier overview [@Cohen2019].

## What is a homotopy type?

Let us first start with a toy model of the type of question we are interested in:

:::{.question}
Given a bounded chain complex of $\ZZ\dash$modules
\[
\complex{C}: \qquad \cdots \to C_{n-1} \mapsvia{\bd_{n-1}} C_n \mapsvia{\bd_n} C_{n+1} \to \cdots
,\]
does there exist a topological space $X$ whose cellular chain complex $\complex{C}^{\cell}(X; \ZZ)$ is quasi-isomorphic to $\complex{C}$?
:::


:::{.remark}
For nice spaces $X$, we can immediately reduce this to a question of homotopy theory by noting that a homotopy equivalence of spaces $X \mapsvia{f} X'$ induces (at least!) a quasi-isomorphism on their cellular chain complexes
\[
f_*: \complex{C}^{\cell}(X; \ZZ) \iso \complex{C}^\cell(X'; \ZZ)
.\]
So we turn our attention to searching for a *homotopy type* $[X]$.
:::

:::{.remark}
Permit me, if you will, the following notation:

- $\cat{C}$ will generally denote a (small) category, and lowercase letters $x,y\in \cat{C}$ will denote objects in this category.
  $\Cat$ will denote the infinity category of small categories, and we'll abuse notation by writing $x\in \cat{C}$ when $x$ is an object of $\cat{C}$.
- $\Top$ denotes the category of (completely arbitrary) topological spaces whose morphisms are continuous maps.
- $\cat{C}(x, y)\da \Hom_{\cat C}(x, y)$ denotes the set of morphisms from $x$ to $y$.
  So for example, $\Top(x, y)$ denotes the set of continuous maps from the space $x$ to the space $y$.
- $\ho\Top$ denotes the associated (naive) homotopy category whose objects are precisely those in $\Top$, but whose morphisms are given by $\ho\Top(x,y) = \Top(x,y)/\sim$ where $f\sim g$ if $f$ is homotopic to $g$.
  Note that this is more properly thought of as $\Top\invert{W}$, the category obtained as a certain categorical localization of $\Top$ at a class $W$ of *weak equivalences*. 
- $\rmod$ will denote the category of modules over a ring $R$, and $\Ch(\rmod)$ will denote the category of chain complexes of \(R\dash\)modules.
  Its homotopy category will be written as $\ho\Ch(\rmod)$, and the bounded-below derived category as $\derivedcat{\Ch(\rmod)}$.
- $\SHC = \ho \Sp$ is the homotopy category of spectra, sometimes called the *stable homotopy category*.
:::

:::{.definition title="Homotopy equivalence and weak equivalence"}
Recall that two spaces $x, y\in \Top$ are **homotopy equivalent** precisely when there exist witnessing maps $f\in \Top(x, y)$ and $g\in \Top(y, x)$ whose compositions are homotopic to the identities on $x$ and $y$ respectively.
Note that as a consequence, there is an induced isomorphism of graded $\ZZ\dash$modules
\[
\pi(f): \pi_*(X) \iso \pi_*(Y)
.\]
The spaces $x$ and $y$ are **weakly equivalent** when there is a witnessing morphism $f\in \Top(x,y)$ which induces such an isomorphism on homotopy groups.
The difference is slightly subtle: we do not require the existence of a homotopy inverse for such a morphism, and indeed such an inverse may not exist -- there are weakly equivalent spaces that are *not* homotopy equivalent, which witnesses the fact that this is truly a weaker condition!
:::


:::{.warnings}
The relation of weak equivalence of spaces (or quasi-isomorphism of chain complexes) is not an equivalence relation!
Following the example discussed in [@MOQuasiEquiv], one can show that the relation on chain complexes given by $A\sim B \iff$ $A$ is quasi-isomorphic to $B$ is reflexive and transitive, but fails to by symmetric -- consider the following chain map of complexes over $\zmod$:

\begin{tikzcd}
	{\complex{A}\da } & \cdots & {} & 0 && \ZZ && \ZZ && 0 && \cdots \\
	\\
	{\complex{B}\da } & \cdots && 0 && 0 && {\ZZ/2\ZZ} && 0 & {} & \cdots
	\arrow[from=1-2, to=1-4]
	\arrow[from=1-4, to=1-6]
	\arrow[from=1-6, to=1-8]
	\arrow[from=1-8, to=1-10]
	\arrow[from=3-2, to=3-4]
	\arrow[from=3-6, to=3-8]
	\arrow[from=3-4, to=3-6]
	\arrow[from=3-8, to=3-10]
	\arrow[from=1-10, to=1-12]
	\arrow[from=3-10, to=3-12]
	\arrow["0", dashed, from=1-6, to=3-6]
	\arrow["{(\wait) \mod 2\ZZ}", dashed, from=1-8, to=3-8]
	\arrow[dashed, from=1-4, to=3-4]
	\arrow[dashed, from=1-10, to=3-10]
\end{tikzcd}

> [Link to Diagram](https://q.uiver.app/?q=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)

This induces an isomorphism in homology, but does not admit a nonzero inverse.

Presumably the same issue holds for weak equivalences of spaces and spectra, so we will implicitly take the symmetric closure of this relation whenever we discuss various homotopy types.
:::

:::{.remark}
In the setup of Floer homotopy, we'll want to associate not a space or a chain complex to some fixed starting set of Floer data, but rather a *spectrum*, which can be provisionally regarded as a "generalized space".

The general state of affairs is that one is usually interested in only the homotopy groups of a space, or only the homology of a chain complex.
We seek to enrich this information by remembering not only the results of such "homotopical extractions" (here taking homotopy groups or homology) but rather the object from whence it came.
Note that this strictly enriches the situation -- for example, with a space in hand, one can apply *other* homology theories to extract new and potentially interesting information.
However, we have some flexibility in the situation -- trading in a given space for another, homotopy-equivalent space recovers precisely the same homotopy groups.
Similarly trading a chain complex for a homotopy equivalent complex recovers precisely the same homology, which leads us to consider spaces or chain complexes up to homotopy or *homotopy types*.


Remarkably, the notion of a *homotopy type* is used in quite a few papers and books and seems to have a clear intended meaning but no precise definition.
I would love to be corrected on this, but in the meantime, let us proceed with the following provisional definition:
:::

:::{.definition title="Homotopy type (provisional)"}
A **(topological) homotopy type** is an object $x \in \ho\Top$, which can be concretely thought as a topological space up homotopy equivalence.
Note that one could instead only require *weak* equivalence, which is referred to as a **weak (topological) homotopy type**.

A **chain homotopy type** is an object in $x\in \ho\Ch(\rmod)$, regarded as a chain complex up chain homotopy.
Similarly, one could relax this slightly and only require $x\in \dchrmod$, so that $x$ is a complex up to quasi-isomorphism instead.

A **stable homotopy type** is an object $x\in \SHC$, regarded as a spectrum up to homotopy equivalence.
Again, one could instead require only weak equivalence to define a weak stable homotopy type.
:::

:::{.remark}
Since we'll need some infinity-categorical machinery eventually, it is perhaps worth introducing an equivalent definition of topological homotopy types:
there is a guiding principle, the *homotopy hypothesis*, which posits that there is an equivalence of infinity categories
\[
\Pi_*: \hoTop \iso \inftyGrpd \da (\infty, 1)\dash\Grpd
\]
where $\Pi_*$ is a certain *fundamental groupoid* construction.
This is a known theorem for source categories like Kan complexes, which are known to be closely related to $\ho\Top$ via simplicial sets.
Granted this hypothesis, we could equivalently define a homotopy type to just be an infinity-groupoid, although it is less clear (to me) what the corresponding definitions should be for chain homotopy types and stable homotopy types.
It is perhaps also worth noting that in the homotopy theory literature, the word "space" often means "topological homotopy type", and not necessarily an actual topological space up to homeomorphism.
:::


## An analogous question in Morse Theory 

:::{.remark}
To help motivate what exactly a "Floer homotopy type" should be, let's first recall the simpler analogous situation in classical Morse theory.
For $M$ a smooth finite-dimensional Riemannian manifold and $f\in C^\infty(M, \RR)$ a smooth real-valued functional, we can extra the Morse chain complex:
\[
\complex{C}(f): \qquad \cdots \to C_{n-1}(f) \mapsvia{\bd_{n-1}} C_n(f) \mapsvia{\bd_n} C_{n+1}(f) \to \cdots
.\]
Here the grading is by the indices $n$ of critical points of $f$, and $n_p(f) \da \ZZ\adjoint{\Crit_n(f)}$ is the free \(\ZZ\dash\)module on all critical points of index $n$.
The differential applied to a critical point $a$ is given by the formula
\[
\bd_n(a) \da \sum_{b\in \Crit_{n-1}(f)} \size \mcm(a, b)\cdot b
,\]
where $\mcm(a, b)$ is an appropriately normalized moduli space of gradient flow trajectories from $a$ to $b$.
Interestingly, Cohen notes that $\size \mcm(a, b) \in \ZZ \cong \pi_0 \MSO$, so that this count is somehow related to the homotopy groups of the Thom spectrum of the special orthogonal groups $\SO_n$.

The following fact is most relevant to our goal:
:::

:::{.fact}
If $M$ is closed, then there is a quasi-isomorphism of chain complexes
\[
\complex{C}(f) \homotopic \complex{C}^{\cell}(\mce_{M, f})
\]
where $\mce_{M, f}$ is a CW complex in the homotopy type of $M$.
:::

:::{.remark}
This seems to be a restatement (or perhaps a slight strengthening) of the standard theorems showing that $f$ induces a handle decomposition of $M$, and in fact a deformation retract of $M$ onto a CW skeleton, along with the fact that Morse homology recovers cellular (and thus singular) homology.
We are thus lead to formulate the following analogous question in Floer theory:
:::

## Floer homotopy types

:::{.question}
Let $(M, \omega, H)$ be a fixed symplectic manifold and Hamiltonian.
Is the Floer chain complex $\complex{\CF}(M, H)$ *realized* as $\complex{C}^{\cell}(\mce_{M, H})$ for some CW complex $\mce_{M, H}$?
:::

:::{.remark}
This question is somewhat subtle -- for starters, although one can actually produce the appropriate homotopy type in the Morse situation by taking a deformation retraction associated to the gradient flow, there is not an immediately analogous CW complex in the Floer situation. 
Moreover, *realization* has a precise definition which is somewhat more complicated than one might naively expect -- one first asks for not a CW complex but rather a CW *spectrum*, and Cohen defines a "realization" of a chain complex $\complex{C}$ which involves finding a filtered spectrum 
\[
X_1\injects X_2\injects \cdots \injects X_n = X
.\]
whose "associated graded" cofibers satisfy
\[
X_i/X_{i-1} \da \hocofib(X_{i-1}\injects X_{i}) \homotopic \Sigma^k \bigvee \SS
,\]
so that the associated graded pieces are suspensions of wedges of spheres whose integral homology (loosely speaking) recovers the chain complex, i.e. $H^*(X_i/X_{i-1}; \ZZ)\cong C_i$.
:::

:::{.remark}
Some recent interesting work along these lines in [@LurTan2018] -- they lay the groundwork for how this program might proceed for a Morse stable homotopy type.
The idea is that given starting Morse data which includes a Riemannian manifold $M$ with a generic metric $g$, a Morse function $f$, and auxiliary data including tangential structures and fundamental cycles on various moduli spaces of Morse trajectories $\mcm(a, b)$ between critical points $a$ and $b$, one should be able to produce a filtered $\Ainfty\dash$algebra object in spectra along with a Maurer-Cartan element whose associated graded pieces are precisely the Morse attaching spheres.
:::

:::{.remark}
Note that there is a notion of *geometric realization* for simplicial sets, and it is not so clear if Cohen's definition involving filtered spectra is related to this.
Rather, one cooks up a category $J$ with objects $[n]$ for each non-negative integer $n\in \ZZ_{\geq 0}$ whose morphisms are given by a particular combinatorial prescription.
One then defines "realization" as a functor
\[
R: \Fun(J, \SHC) \to \SHC
,\]
which takes a copresheaf of spectra on $J$ and associates a stable homotopy type.
With this in hand, we can begin to give a definition of our main object of study:
:::


:::{.definition title="Floer Homotopy Type"}
A **Floer homotopy type** associated to Floer data $(M, \omega, H,\cdots)$ is a stable homotopy type (i.e. a spectrum) obtained from a *smooth framed compact Morse-Smale category of finite type* which realizes the Floer chain complex $\complex{\CF}(M, H)$ as described above.
:::

:::{.remark}
There are many adjectives here that we have not defined!
My impression is that these are bespoke technical conditions -- examples of such categories include the *flow category* $\cat C$ associated to a pair $(M, f)$ as in the setup for Morse homology, where the objects are given by $\ts{ \crit_p(f)}_{p\in \ZZ_{\geq 0}}$, the sets of critical points of various indices, and the morphisms are given by $\cat{C}(a, b) \da \bar\mcm(a, b)$, the compactified moduli space of gradient flow lines from $a$ to $b$.
So we will restrict to just thinking about these particular flow categories of interest and elide these technical conditions; the curious reader can find them spelled out in [@Cohen2019, Definition 6].
:::