Eric Leichtnam, John Lott, Paolo Piazza
If M is a compact oriented manifold-with-boundary whose fundamental group is virtually nilpotent or Gromov-hyperbolic, we show that the higher signatures of M are oriented-homotopy invariants.
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Homotopy invariance of cohomology and signature of a riemannian foliation
本文证明 Riemannian 叶状结构的基本符号差与 Alvarez 类均为叶状同伦不变量，并说明叶状同伦映射诱导基本 Lichnerowicz 上同调间的同构。
We define the pull-back of a smooth principal fibre bundle, and show that it has a natural principal fibre bundle structure. Next, we analyse the relationship between pull-backs by homotopy equivalent maps. The main result of this article is to show that for a principal fibre bundle over a paracompact manifold, there is a principal fibre bundle isomorphism between pull-backs obtained from homotopic maps. This enables simple proofs of several results on the structure of principal fibre bundles. No new results are obtained---this is simply an accessible presentation of an important idea.
We prove the homotopy invariance of L^2 torsion for covering spaces, whenever the covering transformation group is either residually finite or amenable. In the case when the covering transformation group is residually finite and when the L^2 cohomology of the covering space vanishes, the homotopy invariance was established earlier by Lueck. We also give some applications of our results.
On the homotopy invariance of the twisted Lie algebroid cohomology
本文证明了扭曲李代数胚同调的同伦不变性，并利用Poincaré引理推广和Mayer-Vietoris定理导出了Künneth公式，为计算传递李代数胚的扭曲同调提供了系统方法。
We show that the homotopy type of the space of metrics of positive scalar curvature on a smooth manifold remains unchanged, after application of surgery in codimension at least three to the underlying manifold. This result is originally due to V. Chernysh, but remains unpublished.
We use K-area homology to summarize some results about the Novikov conjecture and the Hirzebruch L-class. In fact, we provide necessary and sufficient conditions for closed manifolds to have a homotopy invariant L-class. In order to obtain additional properties we also introduce the cohomology of infinite K--area which preserves the cup product of singular cohomology.
In this paper we study manifolds, $X_Σ$, with fibred singularities, more specifically, a relevant space ${\mathcal R}^{\rm psc}(X_Σ)$ of Riemannian metrics with positive scalar curvature. Our main goal is to prove that the space ${\mathcal R}^{\rm psc}(X_Σ)$ is homotopy invariant under certain surgeries on $X_Σ$.
In this paper we study the Roe index of the signature operator of manifolds of bounded geometry. Our main result is the proof of the uniform homotopy invariance of this index. In other words we show that, given an orientation-preserving uniform homotopy equivalence $f:(M,g) \longrightarrow (N,h)$ between two oriented manifolds of bounded geometry, we have that $f_\star(Ind_{Roe}D_M) = Ind_{Roe}(D_N)$. Moreover we also show that the same result holds considering a group $Γ$ acting on $M$ and $N$ by isometries and assuming that $f$ is $Γ$-equivariant. The only assumption on the action of $Γ$ is …
We show that Chen-Ruan cohomology is a homotopy invariant in certain cases. We introduce the notion of a T-representation homotopy, which is a stringent form of homotopy under which Chen-Ruan cohomology is invariant. We show that while hyperkahler quotients of the cotangent bundle to a complex vector space by a circle S^1 (here termed weighted hyperprojective spaces) are homotopy equivalent to weighted projective spaces, they are not S^1-representation homotopic. Indeed, we show that their Chen-Ruan cohomology rings (over the rationals) are distinct.
On the invariance of the string topology coproduct
研究弦拓扑余积在同伦等价下不变性失效问题，给出基于同伦数据及“伪对角线”构建阻碍类的奈夫公式变体，此阻碍类可衡量同伦等价有界性，且决定余积到高维环路推广的不变性失效。
Thin homotopy and the signature of piecewise linear surfaces
本文引入分段线性曲面的交叉模，定义曲面签名作为万有平移不变2-联络的曲面和乐，证明签名在平移和Thin同伦下唯一刻画曲面，推广Chen的结果并肯定回答Kapranov的问题，同时提供Thin同伦的等价定义及高效计算方法。
In this thesis we talk about lipschitz-homotopy invariants for manifolds of bounded geometry. There are three main results: the first one is the definition of a controvariant functor between the category of manifolds of bounded geometry with uniformly proper lipschitz maps and the category of complex vector space together linear maps. As consequence of this we obtain that the (un)-reduced $L^2$-cohomology is a lipschitz-homotopy invariant. The second result is the invariance of the Roe index of the signature operator under lipschitz-homotopy equivalence which preserves the orientations. Finall…
We prove that the higher harmonic signature of an even dimensional oriented Riemannian foliation of a compact Riemannian manifold with coefficients in a leafwise U(p,q)-flat complex bundle is a leafwise homotopy invariant. We also prove the leafwise homotopy invariance of the twisted higher Betti classes. Consequences for the Novikov conjecture for foliations and for groups are investigated. Replaces The Higher Harmonic Signature for Foliations I: The Untwisted Case, and contains significant improvements.
Building on the theory of elliptic operators, we give a unified treatment of the following topics: - the problem of homotopy invariance of Novikov's higher signatures on closed manifolds; - the problem of cut-and-paste invariance of Novikov's higher signatures on closed manifolds; - the problem of defining higher signatures on manifolds with boundary and proving their homotopy invariance.
For a closed, oriented, odd dimensional manifold $X$, we define the rho invariant $ρ(X,E,H)$ for the twisted odd signature operator valued in a flat hermitian vector bundle $E$, where $H = \sum i^{j+1} H_{2j+1}$ is an odd-degree closed differential form on $X$ and $H_{2j+1}$ is a real-valued differential form of degree ${2j+1}$. We show that the twisted rho invariant $ρ(X,E,H)$ is independent of the choice of metrics on $X$ and $E$ and of the representative $H$ in the cohomology class $[H]$. We establish some basic functorial properties of the twisted rho invariant. We express the twisted eta …
We give explicit formulas for the ranks of the third and fourth homotopy groups of all oriented closed simply-connected four manifolds in terms of their second Betti numbers. We also show that the rational homotopy type of these manifolds is classified by their rank and signature.
We study noncommutative eta- and rho-forms for homotopy equivalences. We prove a product formula for them and show that the rho-forms are well-defined on the structure set. We also define an index theoretic map from L-theory to C*-algebraic K-theory and show that it is compatible with the rho-forms. Our approach, which is based on methods of Hilsum-Skandalis and Piazza-Schick, also yields a unified analytic proof of the homotopy invariance of the higher signature class and of the L^2-signature for manifolds with boundary.
Dirac-harmonic maps with trivial index
本文针对平凡指标情形，通过定义并利用新量的同伦不变性，证明了 Dirac-harmonic 映射及 $α$-Dirac-harmonic 映射流的存在性，建立了相应的一般存在性理论。
This is a sequel to the paper "The signature package on Witt spaces, I. Index classes" by the same authors. In the first part we investigated, via a parametrix construction, the regularity properties of the signature operator on a stratified Witt pseudomanifold, proving, in particular, that one can define a K-homology signature class. We also established the existence of an analytic index class for the signature operator twisted by a C^*_rΓMischenko bundle and proved that the K-homology signature class is mapped to the signature index class by the assembly map. In this paper we continue our st…
In work of Higson-Roe the fundamental role of the signature as a homotopy and bordism invariant for oriented manifolds is made manifest in how it and related secondary invariants define a natural transformation between the (Browder-Novikov-Sullivan-Wall) surgery exact sequence and a long exact sequence of C*-algebra K-theory groups. In recent years the (higher) signature invariants have been extended from closed oriented manifolds to a class of stratified spaces known as L-spaces or Cheeger spaces. In this paper we show that secondary invariants, such as the rho-class, also extend from closed …
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