
TL;DR
The paper introduces fine shape, a simplified shape theory for metrizable spaces that unifies and improves upon previous theories by ensuring invariance of both Čech cohomology and Steenrod-Sitnikov homology.
Contribution
It defines fine shape as a new, simpler shape theory that coincides with existing theories in special cases and proves its invariance properties.
Findings
Fine shape unifies strong shape and strong antishape for metrizable spaces.
Čech cohomology and Steenrod-Sitnikov homology are invariants of fine shape.
A (co)homology theory is fine shape invariant iff it satisfies the map excision axiom.
Abstract
We introduce and develop fine shape, which has a very simple definition and aims to supersede all previously known shape theories for metrizable spaces. The problem with known shape theories of metrizable spaces is illustrated by the following bizarre situation. \v{C}ech cohomology is an invariant of shape, and a fortiori of strong shape. But Steenrod-Sitnikov homology is not shape invariant (already for compacta) and has not been proved to be strong shape invariant (in more than 40 years). Worse yet, there is an ordinary homology theory that is strong shape invariant by design, but it cannot be computed in ZFC for simplest non-compact non-ANRs (such as the disjoint union of countably many copies of the one-point compactification of the countable discrete space). On the other hand, Steenrod-Sitnikov homology is an invariant of antishape (=compactly generated strong shape), and a…
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Taxonomy
TopicsHomotopy and Cohomology in Algebraic Topology · Advanced Topology and Set Theory
