Greg Friedman
We construct geometric examples of pseudomanifolds that satisfy the Witt condition for intersection homology Poincare duality with respect to certain fields but not others. We also compute the bordism theory of $K$-Witt spaces for an arbitrary field $K$, extending results of Siegel for $K=Q$.
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This note corrects an error in the char(K)=2 case of the author's computation of the bordism groups of K-Witt spaces for the field K.
The Bryant-Ferry-Mio-Weinberger surgery exact sequence for high-dimensional compact ANR homology manifolds is used to obtain transversality, splitting and bordism results for homology manifolds, generalizing previous work of Johnston.
This note provides a computation of the bordism groups of K-Witt spaces for fields K with characteristic 2. We provide a complete computation for the unoriented bordism groups. For the oriented bordism groups, a nearly complete computation is provided as well a discussion of the difficulty of resolving a remaining ambiguity in dimensions equivalent to 2 mod 4. This corrects an error in the char(K)=2 case of the author's prior computation of the bordism groups of K-Witt spaces for an arbitrary field K.
Using intersection theory in the context of Hilbert manifolds and geometric homology we show how to recover the main operations of string topology built by M. Chas and D. Sullivan. We also study and build an action of the homology of reduced Sullivan's chord diagrams on the singular homology of free loop spaces, extending previous results of R. Cohen and V. Godin and unifying part of the rich algebraic structure of string topology as an algebra over the prop of these reduced diagrams. Some of these operations are extended to spaces of maps from a sphere to a compact manifold.
We compute the equivariant bordism of free oriented $(\mathbb{Z}/p)^n$-manifolds as a module over $Ω_*^{SO}$, when $p$ is an odd prime. We show, among others, that this module is canonically isomorphic to a direct sum of suspensions of multiple tensor products of $Ω^{SO}_*(B \mathbb{Z}/p)$, and that it is generated by products of standard lens spaces. This considerably improves previous calculations by various authors. Our approach relies on the investigation of the submodule of the Brown-Peterson homology of $B (\mathbb{Z}/p)^n$ generated by elements coming from proper subgroups of $(\mathbb{…
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.
This is the last of five papers that construct an isomorphism between the Seiberg-Witten Floer homology and the Heegaard Floer homology of a given compact, oriented 3-manifold.
This is the first of five papers that construct an isomorphism between the embedded contact homology and Seiberg-Witten Floer cohomology of a compact 3-manifold with a given contact 1-form. This paper describes what is involved in the construction.
Mapping class groups of simply connected closed spin 5-manifolds with no 2- and 3-torsion in homology
该研究确定了同调中无2-挠和3-挠的单连通闭自旋5维流形的Torelli群与配边群的同构关系，明确了#^g(S²×S³)的映射类群及其相关代数性质，还研究了S³在S²×S³中的嵌入。
We construct Bott-type and equivariant Seiberg-Witten Floer homology and cohomology for 3-manifolds, in particular rational homology spheres, and prove their diffeomorphism invariance. This paper is a revised version of math.DG/9701010. Some typos are removed.
An action of the Witt algebra on Khovanov-Rozansky homology
本文构造了 Witt 代数正部分在 Khovanov-Rozansky gl_N 链环同调上的作用，并证明链环配边诱导同调的扭转之间的等变映射，同时将简单网的状态空间等同于多项式上的标准 Witt 代数表示，由此推导出与 Lee 同调和亏格界的简单关系。
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…
We construct equivariant and Bott-type Seiberg-Witten Floer homology and cohomology for 3-manifolds, in particular rational homology spheres, and prove their diffeomorphism invariance. We present several versions of the equivariant theory: the singular version, the de Rham version and the Cartan version, with the first playing the most important role. These versions are shown to be equivalent to each other. A few typos are removed.
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.
On a new geometric homology theory and an application in categorical Gromov-Witten theory
本文证明带角光滑orbifold可三角剖分，并据此提出几何同调理论GHT以克服奇异同调的组合刚性，同时展示其在范畴Gromov-Witten理论中的应用。
Morse complexity of homology classes
研究流形、配边和同调类的莫尔斯复杂度，通过手术理论证上界、指标理论证下界，得出允许离散序列表示的李群的局部对称空间莫尔斯复杂度随体积线性增长，且不允许开书分解的结论。
Families of diffeomorphisms, embeddings, and positive scalar curvature metrics via Seiberg-Witten theory
本文通过Seiberg-Witten理论构造4维流形微分同胚群的高阶同伦和同调中的无限秩直和项，并应用于嵌入空间和正标量曲率度量空间的无限生成性。
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