A survey of Wall's finiteness obstruction

Wall's finiteness obstruction

The purpose of this note is to give a self contained description of Walls finiteness obstruction.

Infinitesimal finiteness obstructions

Does a space enjoying good finiteness properties admit an algebraic model with commensurable finiteness properties? In this note, we provide a rational homotopy obstruction for this to happen. As an application, we show that the maximal metabelian quotient of a very large, finitely generated group is not finitely presented. Using the theory of 1-minimal models, we also show that a finitely generated group $π$ admits a connected 1-model with finite-dimensional degree 1 piece if and only if the Malcev Lie algebra $\mathfrak{m}(π)$ is the lower central series completion of a finitely presented Li…

Finiteness obstructions and Euler characteristics of categories

We introduce notions of finiteness obstruction, Euler characteristic, L^2-Euler characteristic, and Möbius inversion for wide classes of categories. The finiteness obstruction of a category Gamma of type (FP) is a class in the projective class group K_0(RGamma); the functorial Euler characteristic and functorial L^2-Euler characteristic are respectively its RGamma-rank and L^2-rank. We also extend the second author's K-theoretic Möbius inversion from finite categories to quasi-finite categories. Our main example is the proper orbit category, for which these invariants are established notions i…

(G,n)-复形的消去律与Swan有限性障碍

Cancellation for $(G,n)$-complexes and the Swan finiteness obstruction

本文利用Swan有限性障碍与投射ZG-模，研究有限(G,n)-复形的消去律何时成立，并给出失败时同伦不同配对数量的下界，同时为Wall的D2问题提供了寻找反例的新思路。

On $π- π$ theorem for manifold pairs with boundaries

Surgery obstruction of a normal map to a simple Poincare pair $(X,Y)$ lies in the relative surgery obstruction group $L_*(π_1(Y)\toπ_1(X))$. A well known result of Wall, the so called $π$-$π$ theorem, states that in higher dimensions a normal map of a manifold with boundary to a simple Poincare pair with $π_1(X)\congπ_1(Y)$ is normally bordant to a simple homotopy equivalence of pairs. In order to study normal maps to a manifold with a submanifold, Wall introduced surgery obstruction group for manifold pairs $LP_*$ and splitting obstruction groups $LS_*$. In the present paper we formulate and …

Surgery obstructions on closed manifolds and the Inertia subgroup

The Wall surgery obstruction groups have two interesting geometrically defined subgroups, consisting of the surgery obstructions between closed manifolds, and the inertial elements. We show that the inertia group $I_{n+1}(π,w)$ and the closed manifold subgroup $C_{n+1}(π,w)$ are equal in dimensions $n+1\geq 6$, for any finitely-presented group $π$ and any orientation character $w\colon π\to \cy 2$.