An Introduction to Pull-backs of Bundles and Homotopy Invariance

Remark on the Novikov conjecture and the homotopy invariance of the Hirzebruch L-class

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.

Homotopy invariance of the space of metrics with positive scalar curvature on manifolds with singularities

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 $X_Σ$.

Uniform homotopy invariance of Roe Index of the signature operator

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 …

Weighted hyperprojective spaces and homotopy invariance in orbifold cohomology

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.

论弦拓扑余积的不变性

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。

Lipschitz-homotopy invariants:$L^2$-cohomology, Roe index and $ρ$-class

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…

幂零分级 Lie 群上半经典微积分的几何不变性

Geometric invariance of the semi-classical calculus on nilpotent graded Lie groups

本文研究了幂零分级 Lie 群上半经典伪微分算子在保持过滤的 Pansu 可微微分同胚拉回下的几何不变性，并在过滤流形设定下解释了其几何意义。

Elliptic Operators and Higher Signatures

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.

Index type invariants for twisted signature complexes and 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 …

Almost flat bundles and homological invariance of infinite K-area

We extend the notion of an almost flat bundle over a closed Riemannian manifold to bundles over simplicial complexes, and prove that up to a constant factor, this notion is invariant under pullback via maps which induce isomorphisms on fundamental groups. As an application, we show that the property of having infinite K-area only depends on the image of the fundamental class under the classifying map of the universal cover.

Sigma-models and Homotopy Algebras

We review the relation between homotopy algebras of conformal field theory and geometric structures arising in sigma models. In particular we formulate conformal invariance conditions, which in the quasi-classical limit are Einstein equations with extra fields, as generalized Maurer-Cartan equations.

Fukaya's conjecture on Witten's twisted $A_\infty$-structures

Wedge product on deRham complex of a Riemannian manifold $M$ can be pulled back to $H^*(M)$ via explicit homotopy, constructed using Green's operator, to give higher product structures. We prove Fukaya's conjecture which suggests that Witten deformation of these higher product structures have semiclassical limits as operators defined by counting gradient flow trees with respect to Morse functions, which generalizes the remarkable Witten deformation of deRham differential from a statement concerning homology to one concerning real homotopy type of $M$. Various applications of this conjecture to…

An $\mathsf{A}_{\infty}$ version of the Poincaré lemma

We prove a categorified version of the Poincaré lemma. The natural setting for our result is that of $\infty$-local systems. More precisely, we show that any smooth homotopy between maps $f$ and $g$ induces an $\mathsf{A}_\infty$-natural transformation between the corresponding pullback functors. This transformation is explicitly defined in terms of Chen's iterated integrals. In particular, we show that a homotopy equivalence induces a quasi-equivalence on the DG categories of $\infty$-local system.