The Hausmann-Weinberger 4-manifold invariant of abelian groups

拓扑与光滑Hausmann-Weinberger不变量存在分歧

The topological and smooth Hausmann-Weinberger invariants disagree

本文证明了在拓扑流形与允许光滑结构的流形上，Hausmann-Weinberger不变量存在差异。

Thompson's group F is not SCY

In this note we prove that Thompson's group F cannot be the fundamental group of a symplectic 4-manifold with trivial canonical class by showing that its Hausmann-Weinberger invariant q(F) is strictly positive.

具有3-流形群的4-流形的最小欧拉示性数

Minimal Euler Characteristics of 4-manifolds with 3-manifold groups

本文针对紧致3-流形的基本群，在包括所有不含双侧 RP² 的情形下确定了4-流形的最小欧拉示性数 χ₄(π)，并厘清其与 Hausmann-Weinberger、Kotschick 和 Hillman 不变量 p、q* 的相等条件，回答了 Hillman 提出的问题。

Homology equivalences of manifolds and zero-in-the-spectrum examples

Working with group homomorphisms, a construction of manifolds is introduced to preserve homology groups. The construction gives as special cases Qullien's plus construction with handles obtained by Hausmann, the existence of one-sided $h$-cobordism of Guilbault and Tinsley, the existence of homology spheres and higher-dimensional knots proved by Karvaire. We also use it to get counter-examples to the zero-in-the-spectrum conjecture found by Farber-Weinberger, and by Higson-Roe-Schick.

Minimal Euler Characteristics for Even-Dimensional Manifolds with Finite Fundamental Group

We consider the Euler characteristics $χ(M)$ of closed orientable topological $2n$-manifolds with $(n-1)$-connected universal cover and a given fundamental group $G$ of type $F_n$. We define $q_{2n}(G)$, a generalized version of the Hausmann-Weinberger invariant for 4-manifolds, as the minimal value of $(-1)^nχ(M)$. For all $n\geq 2$, we establish a strengthened and extended version of their estimates, in terms of explicit cohomological invariants of $G$. As an application we obtain new restrictions for non-abelian finite groups arising as fundamental groups of rational homology 4-spheres.

The additivity of the $ρ$-invariant and periodicity in topological surgery

For a closed topological manifold M with dim (M) >= 5 the topological structure set S(M) admits an abelian group structure which may be identified with the algebraic structure group of M as defined by Ranicki. If dim (M) = 2d-1, M is oriented and M is equipped with a map to the classifying space of a finite group G, then the reduced rho-invariant defines a function, \wrho : S(M) \to \QQ R_{hat G}^{(-1)^d}, to a certain sub-quotient of the complex representation ring of G. We show that the function \wrho is a homomorphism when 2d-1 >= 5. Along the way we give a detailed proof that a geometrical…