Homotopy types of stabilizers and orbits of Morse functions on surfaces
Sergey Maksymenko
备注：
49 pages, 8 figures. This version includes the proof of the fact that the orbits of a finite codimension of tame action of tame Lie group on tame Frechet manifold is a tame Frechet manifold itself
发表信息：
Annals of Global Analysis and Geometry 29 (2006) 241-285
Let $M$ be a smooth compact surface, orientable or not, with boundary or without it, $P$ either the real line $R^1$ or the circle $S^1$, and $Diff(M)$ the group of diffeomorphisms of $M$ acting on $C^{\infty}(M,P)$ by the rule $h\cdot f\mapsto f \circ h^{-1}$, where $h\in Diff(M)$ and $f \in C^{\infty}(M,P)$.
Let $f:M \to P$ be a Morse function and $O(f)$ be the orbit of $f$ under this action. We prove that $π_k O(f)=π_k M$ for $k\geq 3$, and $π_2 O(f)=0$ except for few cases. In particular, $O(f)$ is aspherical, provided so is $M$. Moreover, $π_1 O(f)$ is an extension of a finitely generated free abelian group with a (finite) subgroup of the group of automorphisms of the Reeb graph of $f$.
We also give a complete proof of the fact that the orbit $O(f)$ is tame Frechet submanifold of $C^{\infty}(M,P)$ of finite codimension, and that the projection $Diff(M) \to O(f)$ is a principal locally trivial $S(f)$-fibration.
We study Morse theory on noncompact manifolds equipped with exhaustions by compact pieces, defining the Morse homology of a pair which consists of the manifold and related geometric/homotopy data. We construct a collection of Morse data parametrized by cubes of arbitrary dimensions. From this collection, we obtain a family of linear maps subject to some coherency conditions, which can be packaged into a homotopy coherent diagram. We introduce a chain complex which is a colimit for the diagram and show that it computes the Morse homology.
By studying spaces of flow graphs in a closed oriented manifold, we construct operations on its cohomology, parametrized by the homology of the moduli spaces of compact Riemann surfaces with boundary marked points. We show that the operations satisfy the gluing axiom of an open homological conformal field theory. This complements previous constructions due to R. Cohen et al., K. Costello and M. Kontsevich and is also the Morse theoretic counterpart to a conjectural construction of operations on the Lagrangian Floer homology of the zero section of a cotangent bundle, obtained by studying uncomp…
We give a generalization of Fukaya's Morse homotopy theoretic approach for 2-loop Chern--Simons perturbation theory to 3-valent graphs with arbitrary number of loops at least 2. We construct a sequence of invariants of integral homology 3-spheres with values in a space of 3-valent graphs (Jacobi diagrams or Feynman diagrams) by counting graphs in an integral homology 3-sphere satisfying certain condition described by a set of ordinary differential equations.
Let $M$ be a smooth closed orientable surface. Let $F$ be the space of Morse functions on $M$ having fixed number of critical points of each index, moreover at least $χ(M)+1$ critical points are labeled by different labels (enumerated). A notion of a skew cylindric-polyhedral complex, which generalizes the notion of a polyhedral complex, is introduced. The skew cylindric-polyhedral complex $\mathbb{\widetilde K}$ (the "complex of framed Morse functions"), associated with the space $F$, is defined. In the case when $M=S^2$, the polyhedron $\mathbb{\widetilde K}$ is finite; its Euler characteris…
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Let $M$ be a smooth closed orientable surface. Let $F$ be the space of Morse functions on $M$, and $\mathbb{F}^1$ the space of framed Morse functions, both endowed with $C^\infty$-topology. The space $\mathbb{F}^0$ of special framed Morse functions is defined. We prove that the inclusion mapping $\mathbb{F}^0\hookrightarrow\mathbb{F}^1$ is a homotopy equivalence. In the case when at least $χ(M)+1$ critical points of each function of $F$ are labeled, homotopy equivalences $\mathbb{\widetilde K}\sim\widetilde{\cal M}$ and $F\sim\mathbb{F}^0\sim{\mathscr D}^0\times\mathbb{\widetilde K}$ are prove…
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Let A be an essential complex hyperplane arrangement in an n-dimensional complex vector space V. Let H denote the union of the hyperplanes, and M denote the complement to H in V. We develop the real-valued and circle-valued Morse theory for M and prove, in particular, that M has the homotopy type of a space obtained from a manifold fibered over a circle, by attaching cells of dimension n. We compute the Novikov homology of M for a large class of homomorphisms of the fundamental group of M to R.
A Morse function f on a manifold with corners M allows the characterization of the Morse data for a critical point by the Morse index. In fact, a modified gradient flow allows a proof of the Morse theorems in a manner similar to that of classical Morse theory. It follows that M is homotopy equivalent to a CW-complex with one cell of dimension λfor each essential critical point of index λ. The goal of this article is to determine the boundary maps of this CW-complex, in the case where M is compact and orientable. First, the boundary maps are defined in terms of the modified gradient flow. Then …
Let $M$ be a smooth closed orientable surface, and let $F$ be the space of Morse functions on $M$ such that at least $χ(M)+1$ critical points of each function of $F$ are labeled by different labels (enumerated). Endow the space $F$ with $C^\infty$-topology. We prove the homotopy equivalence $F\sim R\times{\widetilde{\cal M}}$ where $R$ is one of the manifolds ${\mathbb R}P^3$, $S^1\times S^1$ and the point in dependence on the sign of $χ(M)$, and ${\widetilde{\cal M}}$ is the universal moduli space of framed Morse functions, which is a smooth stratified manifold. Morse inequalities for the Bet…
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We define an algebraic/combinatorial object on the front projection $Σ$ of a Legendrian knot called a Morse complex sequence, abbreviated MCS. This object is motivated by the theory of generating families and provides new connections between generating families, normal rulings, and augmentations of the Chekanov-Eliashberg DGA. In particular, we place an equivalence relation on the set of MCSs on $Σ$ and construct a surjective map from the equivalence classes to the set of chain homotopy classes of augmentations of $L_Σ$, where $L_Σ$ is the Ng resolution of $Σ$. In the case of Legendrian knot c…
Let $B$ be a Möbius band and $f:B \to \mathbb{R}$ be a Morse map taking a constant value on $\partial B$, and $\mathcal{S}(f,\partial B)$ be the group of diffeomorphisms $h$ of $B$ fixed on $\partial B$ and preserving $f$ in the sense that $f\circ h = f$. Under certain assumptions on $f$ we compute the group $π_0\mathcal{S}(f,\partial B)$ of isotopy classes of such diffeomorphisms. In fact, those computations hold for functions $f:B\to\mathbb{R}$ whose germs at critical points are smoothly equivalent to homogeneous polynomials $\mathbb{R}^2\to\mathbb{R}$ without multiple factors. Together with…
Given two Morse functions $f, μ$ on a compact manifold $M$, we study the Morse homology for the Lagrange multiplier function on $M \times {\mathbb R}$ which sends $(x, η)$ to $f(x) + ημ(x)$. Take a product metric on $M \times {\mathbb R}$, and rescale its ${\mathbb R}$-component by a factor $λ^2$. We show that generically, for large $λ$, the Morse-Smale-Witten chain complex is isomorphic to the one for $f$ and the metric restricted to ${μ^{-1}(0)}$, with grading shifted by one. On the other hand, let $λ\to 0$, we obtain another chain complex, which is geometrically quite different but has the …
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Framed flow categories were introduced by Cohen-Jones-Segal as a way of encoding the flow data associated to a Floer functional. A framed flow category gives rise to a CW-complex with one cell for each object of the category. The idea is that the Floer invariant should take the form of the stable homotopy type of the resulting complex, recovering the Floer cohomology as its singular cohomology. Such a framed flow category was produced, for example, by Lipshitz-Sarkar from the input of a knot diagram, resulting in a stable homotopy type generalizing Khovanov cohomology. In this paper we give mo…
