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# Homology, Homotopy and Applications

## Volume 12 (2010)

### Number 2

### Generalized Steenrod homology theories are strong shape invariant

Pages: 1 – 23

DOI: https://dx.doi.org/10.4310/HHA.2010.v12.n2.a1

#### Author

#### Abstract

It is shown that a reduced homology theory on the category of pointed compact metric spaces is strong shape invariant if and only if its homology functors $h_n$ satisfy the quotient exactness axiom, which means that for each pointed compact metric pair $(X, A, a_0)$ the natural sequence $h_n(A, a_0) \to h_n(X, a_0) \to h_n(X/A, *)$ is exact. As a consequence, all generalized Steenrod homology theories are strong shape invariant.

#### Keywords

Steenrod homology theory; pointed strong shape theory; strong excision axiom; cone collapsing axiom; quotient exactness axiom

#### 2010 Mathematics Subject Classification

55N20, 55N40, 55P55

Published 1 January 2010