Formal aspects of Floer cohomology

From Talbot

Get PDF | Get log | Regenerate PDF | Export Source

---

\documentclass{article}
\usepackage{amsmath,amsfonts,amssymb,epsfig,color}
\newcommand{\C}{\mathbb{C}}
\newcommand{\Z}{\mathbb{Z}}
\newcommand{\R}{\mathbb{R}}
 
\newcommand{\iso}{\cong}
\parskip1em
\parindent0em
\begin{document}
 
\underline{Formal aspects of Floer cohomology}
 
The reason for this talk is to get a big-picture understanding of why
Floer cohomology only partially behaves like ordinary cohomology. For
instance, it can't be defined with $\Z$ coefficients in general, due to
sign ambiguities which are inherently not removable. This is based on a
formal analysis of infinite-dimensional Morse theories, similar to (but
not quite the same as) [Cohen-Jones-Segal, Floer's Infinite Dimensional
Morse Theory And Homotopy Theory]. The prerequisite is knowledge of
Atiyah-Singer index theory for families, as well as basic homotopy theory.
There is a lot of material here, and boiling it down to a presentation will
require some effort.
 
\underline{Plan}
 
{\it Morse theory.} A lightning review of Morse homology theory [Schwarz,
Morse homology]. Beyond that, we want to emphasize one property. Namely,
if $\mathcal M = \mathcal M(x,y)$ is a space of connecting trajectories of
a Morse function $f: M \rightarrow \R$, the canonical embedding $i:
\mathcal M \rightarrow M$ is stably framed, which means that the bundle
$i^*(TM) \oplus T\mathcal M^\vee$ has a canonical homotopy class of stable
trivializations. I think this should be obvious from the definition of
$\mathcal M$ as intersection of stable and unstable manifolds, which are
spheres and hence canonically framed. Informally speaking, this additional
information allows one to reconstruct the stable homotopy type of $M$ from
the Morse theory (no formal statement need to be given). The basic
reference is [Cohen-Jones-Segal, op. cit.].
 
%{\it Novikov homology.} Define Novikov homology for a closed one-form on
a manifold, introduce its coefficient field (the Novikov field in one
%variable; I like to allow all periods $t^a$, $a \in \R$, which yields the
universal Novikov ring that's independent of the particular one-form; see
%also [Fukaya-Oh-Ohta-Ono, the book manuscript]). An example would be
helpful. Again, give a lightning review of its description in terms of
%Morse theory. A possible reference is [Hutchings-Lee, Reidemeister
torsion in generalized Morse theory].
 
{\it Polarized manifolds.} A Hilbert manifold $\mathcal X$ is called
polarized if each tangent space $T\mathcal X_x$ comes with a bounded
selfadjoint operator $I_x$, $I_x^2 = 1 + {\it compacts}$, and where $I_x$
is itself determined up to compact operators only. This gives rise to an
approximate splitting of $T\mathcal X_x$ into $\pm$ parts. We want to
discuss the homotopy theory of this, I think using [Cohen-Jones-Segal, op.
cit.] as well as [Atiyah-Singer, Index theory for skew-adjoint Fredholm
operators]. There are two low-dimensional characteristic classes of a
polarization, one in $H^1(\mathcal X;\Z)$ and the second in $H^2(\mathcal
X;\Z/2)$. The first one can be viewed as spectral flow, and the second one
as a parametrized index. Alternatively, assume that $\mathcal X$ has a
covering such that over each open subset of the covering, the splittings
onto $\pm$ parts are made exact rather than approximate. Then these
classes arise naturally as Cech cocycles (the first is a $\Z$-valued Cech
cocycle, the second a Cech cocycle with values in the Picard groupoid of
real flat line bundles, through the determinant line construction).
 
{\it Morse theory on polarized manifolds.} This will have to be rather
metaphorical, but we're interested in how non-triviality of the
polarization affects the structure of Morse theory, for a Morse function
whose Hessian lies within the polarization class. In particular, why can't
the formal Morse homology be $\Z$-graded in general, and why are there
problems defining it over $\Z$? See also [Abbondandolo-Majer,
When the Morse index is infinite]. 
 
{\it Lagrangian Floer theory.} We now look at $\mathcal X$ being the space
of paths between two Lagrangian submanifolds, as in [Floer, Morse theory
for Lagrangian intersections]. In this case, the two characteristic
classes mentioned above are related to the Maslov class and second
Stiefel-Whitney class of the Lagrangian submanifolds under consideration.
For the Maslov class and index, see [Robbin-Salamon, the Maslov index for
paths] (and the sequely which relates this to index theory). For the
appearance of the second Stiefel-Whitney class, see [Fukaya-Oh-Ohta-Ono
the book manuscript] or [de Silva, index for a family of holomorphic
disks; unpublished but I may have a copy]. Alternatively, [Seidel, Fukaya
categories and Picard-Lefschetz theory] has a section about indices and
determinants of elliptic operators, which is relevant to the material
here.
 
\underline{Dependencies}
 
Towards the end, this increasingly depends on the talk about Lagrangian
submanifolds.
 
\end{document}