The cell-dispensability obstruction for spaces and manifolds

Abstract

We compare two properties for a CW-space $X$ of finite type: (1) being homotopy equivalent to a CW-complex without $j$-cells for $k\le j\le \ell$ (($k,\ell$)-cellfree) and (2) $H^j(X;R)=0$ for any $\mathbb{Z}{\pi }_1(X)$-module $R$ when $k\le j\le \ell$ (cohomogy ($k,\ell$)-silent). Using the technique of Wall's finiteness obstruction, we show that a connected CW-space $X$ of finite type which is cohomogy ($k,\ell$)-silent determines a "cell-dispensability obstruction'' $w_k(X)\in {\tilde{K}}_0(\mathbb{Z}{\pi }_1(X))$ which vanishes if and only if $X$ is ($k,\ell$)-cellfree ($k\ge 4$). Any class in ${\tilde{K}}_0(\mathbb{Z}\pi )$ may occur as the cell-dispensability obstruction $w_k(X)$ for a CW-space $X$ with ${\pi }_1(X)$ identified with $\pi$. Using projective surgery, a similar theory is obtained for manifolds, replacing "cells" by "handles" (antisimple manifolds).

Minor corrections and a slight change on the definition of (k,l)-cellfreeness, required for the results to be correct. 42 pages