Greg Friedman
We derive spectral sequences for the intersection homology of stratified fibrations and approximate tubular neighborhoods in manifold stratified spaces. These neighborhoods include regular neighborhoods in PL stratified spaces.
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In this paper, we develop Leray-Serre-type spectral sequences to compute the intersection homology of the regular neighborhood and deleted regular neighborhood of the bottom stratum of a stratified PL-pseudomanifold. The E^2 terms of the spectral sequences are given by the homology of the bottom stratum with a local coefficient system whose stalks consist of the intersection homology modules of the link of this stratum (or the cone on this link). In the course of this program, we establish the properties of stratified fibrations over unfiltered base spaces and of their mapping cylinders. We al…
This work forms a foundational study of factorization homology, or topological chiral homology, at the generality of stratified spaces with tangential structures. Examples of such factorization homology theories include intersection homology, compactly supported stratified mapping spaces, and Hochschild homology with coefficients. Our main theorem characterizes factorization homology theories by a generalization of the Eilenberg--Steenrod axioms; it can also be viewed as an analogue of the Baez--Dolan cobordism hypothesis formulated for the observables, rather than state spaces, of a topologic…
We show that intersection homology extends Poincare duality to manifold homotopically stratified spaces (satisfying mild restrictions). This includes showing that, on such spaces, the sheaf of singular intersection chains is quasi-isomorphic to the Deligne sheaf.
We compute the connected Heegaard Floer homology (defined by Hendricks, Hom, and Lidman) for a large class of 3-manifolds, including all linear combinations of Seifert fibered homology spheres. We show that for such manifolds, the connected Floer homology completely determines the local equivalence class of the associated $ι$-complex. Some identities relating the rank of the connected Floer homology to the Rokhlin invariant and the Neumann-Siebenmann invariant are also derived. Our computations are based on combinatorial models inspired by the work of Némethi on lattice homology.
In their previous work, Barraud and Cornea enriched the Lagrangian Floer complex by adding cubical chains in the based loop space of the Lagrangian, and recovered the Leray-Serre spectral sequence of the based path space fibration, assuming that the Lagrangian is weakly exact and simply connected. In the present article, we remove the simple connectivity assumption and adapt the construction to any Hurewicz fibration. To this end, we introduce a stronger notion of local systems than the classical one, as a topological functor from the free path space of a manifold to the category of topologica…
In this article, we generalize the results discussed in [arXiv:1004.3762] by introducing a genus to generic fibers of Lefschetz fibrations. That is, we give families of relations in the mapping class groups of genus-1 surfaces with boundaries that represent rational homology disk smoothings of weighted homogeneous surface singularities whose resolution graphs are $3$-legged with a bad central vertex.
Skeleta and other pure subsets of manifold stratified spaces are shown to have neighborhoods which are teardrops of stratified approximate fibrations (under dimension and compactness assumptions). In general, the stratified approximate fibrations cannot be replaced by bundles, and the teardrops cannot be replaced by mapping cylinder neighborhoods. Thus, this is the best possible topological tubular neighborhood theorem in the stratified setting.
Periodic Floer homology (PFH) is a Gromov-Floer type invariant for fibered three-manifolds with Hamiltonian structures. The cobordism maps on periodic Floer homology induced by symplectic cobordisms are currently only defined indirectly by using Seiberg-Witten theory. In this paper, we investigate the cobordism maps induced by a class of symplectic cobordisms constructed by P. Seidel, called elementary Lefschetz brations. We define the PFH cobordism maps induced by elementary Lefschetz fibrations in terms of holomorphic curves. Moreover, we compute these maps for some cases.
A Fibration Theorem and Chas-Sullivan product for Morse-Novikov Homology with differential graded coefficients
本文将具有微分 graded 系数的 Morse 同调及 Chas-Sullivan 积扩展至 Morse-Novikov 同调，构建了相应的 Morse-Novikov 复形，并在纤维化条件下证明了 Novikov 完备化同调上存在 Chas-Sullivan 类型的积。
We compute the $\mathrm{Pin}(2)$-equivariant Seiberg-Witten Floer homology of Seifert rational homology three-spheres in terms of their Heegaard Floer homology. As a result of this computation, we prove Manolescu's conjecture that $β=-\barμ$ for Seifert integral homology three-spheres. We show that the Manolescu invariants $α, β,$ and $γ$ give new obstructions to homology cobordisms between Seifert fiber spaces, and that many Seifert homology spheres $Σ(a_1,...,a_n)$ are not homology cobordant to any $-Σ(b_1,...,b_n)$. We then use the same invariants to give an example of an integral homology …
We extend Perutz's Lagrangian matching invariants to 3-manifolds which are not necessarily fibred using the technology of holomorphic quilts. We prove an isomorphism of these invariants with Ozsvath-Szabo's Heegaard Floer invariants for certain extremal spin^c structures. As applications, we give new calculations of Heegaard Floer homology of certain classes of 3-manifolds, and a characterization of Juhasz's sutured Floer homology.
We introduce a singular chain intersection homology theory which generalizes that of King and which agrees with the Deligne sheaf intersection homology of Goresky and MacPherson on any topological stratified pseudomanifold, compact or not, with constant or local coefficients, and with traditional perversities or superperversities (those satisfying p(2)>0). For the case p(2)=1, these latter perversitie were introduced by Cappell and Shaneson and play a key role in their superduality theorem for embeddings. We further describe the sheafification of this singular chain complex and its adaptabilit…
Lipshitz, Ozsváth and Thurston defined a bordered Heegaard Floer invariant CFDA for 3-manifolds with two boundary components, including mapping cylinders for surface diffeomorphisms. We define a related invariant for certain 4-dimensional cobordisms with corners, by associating a morphism F from CFDA(f) to CFDA(g) to each such cobordism between two mapping cylinders f and g. Like the Ozsváth-Szabó invariants of cobordisms between closed 3-manifolds, this morphism arises from counting holomorphic triangles on Heegaard triples. We demonstrate that the homotopy class of the morphism F only depend…
For an oriented irreducible 3-manifold M with non-empty toroidal boundary, we describe how sutured Floer homology ($SFH$) can be used to determine all fibered classes in $H^1(M)$. Furthermore, we show that the $SFH$ of a balanced sutured manifold $(M,γ)$ detects which classes in $H^1(M)$ admit a taut depth one foliation such that the only compact leaves are the components of $R(γ)$. The latter had been proved earlier by the first author under the extra assumption that $H_2(M)=0$. The main technical result is that we can obtain an extremal $\text{Spin}^c$-structure $\mathfrak{s}$ (i.e., one tha…
In this paper we compute the singular homology of the space of immersions of the circle into the $n$-sphere. Equipped with Chas-Sullivan's loop product these homology groups are graded commutative algebras, we also compute these algebras. We enrich Morse spectral sequences for fibrations of free loop spaces together with loop products, this offers some new computational tools for string topology.
We study bordism groups and bordism homology theories based on pseudomanifolds and stratified pseudomanifolds. The main seam of the paper demonstrates that when we uses classes of spaces determined by local link properties, the stratified and unstratified bordism theories are identical; this includes the known examples of pseudomanifold bordism theories, such as bordism of Witt spaces and IP spaces. Along the way, we relate the stratified and unstratified points of view for describing various (stratified) pseudomanifold classes.
We develop a theory of tubular neighborhoods for the lower strata in manifold stratified spaces with two strata. In these topologically stratified spaces, manifold approximate fibrations and teardrops play the role that fibre bundles and mapping cylinders play in smoothly stratified spaces. Applications include the classification of neighborhood germs, the construction of exotic stratifications, a multiparameter isotopy extension theorem and an h-coborsism extension theorem.
The characteristic varieties of a space are the jump loci for homology of rank 1 local systems. The way in which the geometry of these varieties may vary with the characteristic of the ground field is reflected in the homology of finite cyclic covers. We exploit this phenomenon to detect torsion in the homology of Milnor fibers of projective hypersurfaces. One tool we use is the interpretation of the degree 1 characteristic varieties of a hyperplane arrangement complement in terms of orbifold fibrations and multinets on the corresponding matroid. Another tool is a polarization construction, ba…
A chain-level model for Chas-Sullivan products in Morse homology with differential graded coefficients
本文提出了一种基于 Morse 理论的链级模型，用于研究自由循环空间的 Chas-Sullivan 乘积，并构建了微分格版本的 Künneth 公式和 Pontryagin-Thom 构造。
Homology manifolds via six functor formalisms
利用谱层六函子形式体系研究同调流形，刻画上同调光滑对象，证明ANR同调流形是Poincaré对偶复形，推广Wilder单调映射定理，引入同伦流形并证明不稳定版本的Wilder可定向猜想。
Our objective is to develop a stratified Morse theory with tangential conditions. We define a continuous strata-wise smooth Morse function on an abstract stratified space by using control conditions and radiality assumptions on the gradient vector field. For critical points of a Morse function one can show that the local unstable set is a manifold and the local stable set is itself an abstract stratified space. We also give a normal form for the gradient dynamics in the neighborhood of critical points. For a stratified Morse pair which satisfies the generic Morse-Smale condition we can build t…
We introduce hyperelliptic simplified (more generally, directed) broken Lefschetz fibrations, which is a generalization of hyperelliptic Lefschetz fibrations. We construct involutions on the total spaces of such fibrations of genus $g\geq 3$ and extend these involutions to the four-manifolds obtained by blowing up the total spaces. The extended involutions induce double branched coverings over blown up sphere bundles over the sphere. We also show that the regular fiber of such a fibration of genus $g\geq 3$ represents a non-trivial rational homology class of the total space.
This paper surveys the significant progress over the past couple of decades in the theory of stratified spaces through the application of controlled methods as well as through the application of intersection homology.
Topological invariance of torsion sensitive intersection homology
本文研究挠敏感相交同调的拓扑不变性，证明其定义层复形在拟同构意义下对分层选择的独立性，并推广经典不变性定理且获得经典情形下的新结果。
Morse homology with DG coefficients
本文构建带 DG 局部系统系数的 Morse 同调与上同调，证明其等价于 DG Tor/Ext，并建立整数扭曲上闭链、纤维化恢复及对偶等性质。
For fibred boundary and fibred cusp metrics, Hausel, Hunsicker, and Mazzeo identified the space of $L^2$ harmonic forms of fixed degree with the images of maps between intersection cohomology groups of an associated stratified space obtained by collapsing the fibres of the fibration at infinity onto its base. In the present paper, we obtain a generalization of this result to situations where, rather than a fibration at infinity, there is a Riemannian foliation with compact leaves admitting a resolution by a fibration. If the associated stratified space (obtained now by collapsing the leaves of…
This article is a survey of recent work of the author, together with Markus Banagl, Eric Leichtnam, Rafe Mazzeo, and Paolo Piazza, on the Hodge theory of stratified spaces. We discuss how to resolve a Thom-Mather stratified space to a manifold with corners with an iterated fibration structure and the generalization of a perversity in the sense of Goresky-MacPherson to a mezzoperversity. We define Cheeger spaces and their signatures and describe how to carry out the analytic proof of the Novikov conjecture on these spaces. Finally we review the reductive Borel-Serre compactification of a locall…
There are several topological spaces associated to a complex hyperplane arrangement: the complement and its boundary manifold, as well as the Milnor fiber and its own boundary. All these spaces are related in various ways, primarily by a set of interlocking fibrations. We use cohomology with coefficients in rank 1 local systems on the complement of the arrangement to gain information on the homology of the other three spaces, and on the monodromy operators of the various fibrations.
We develop a generalization to non-Witt spaces of the intersection homology theory of Goresky-MacPherson. The second author has described the self-dual sheaves compatible with intersection homology, and the other authors have described a generalization of Cheeger's L2 de Rham cohomology. In this paper we extend both of these cohomologies by describing all sheaf complexes in the derived category of constructible sheaves that are compatible with middle perversity intersection cohomology, though not necessarily self-dual. On Thom-Mather stratified spaces this refined intersection cohomology theor…
Witt spaces are pseudomanifolds for which the middle-perversity intersection homology with rational coefficients is self-dual. We give a new construction of the symmetric signature for Witt spaces which is similar in spirit to the construction given by Miscenko for manifolds. Our construction has all of the expected properties, including invariance under stratified homotopy equivalence.
We examine the action of the fundamental group $Γ$ of a Riemann surface with $m$ punctures on the middle dimensional homology of a regular fiber in a Lefschetz fibration, and describe to what extent this action can be recovered from the intersection numbers of vanishing cycles. Basis changes for the vanishing cycles result in a nonlinear action of the framed braid group $\widetilde{\mathcal B}$ on $m$ strings on a suitable space of $m\times m$ matrices. This action is determined by a family of cohomologous 1-cocycles ${\mathcal S}_c:\widetilde{\mathcal B}\to GL_m({\mathbb{Z}}[Γ])$ parametrized…
We survey the cohomology jumping loci and the Alexander-type invariants associated to a space, or to its fundamental group. Though most of the material is expository, we provide new examples and applications, which in turn raise several questions and conjectures. The jump loci of a space X come in two basic flavors: the characteristic varieties, or, the support loci for homology with coefficients in rank 1 local systems, and the resonance varieties, or, the support loci for the homology of the cochain complexes arising from multiplication by degree 1 classes in the cohomology ring of X. The ge…
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