Fine shape I

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 fortiori of strong antishape. However, \v{C}ech cohomology is not antishape invariant (already for ANRs) and has not been proved to be strong antishape invariant. And there is an ordinary cohomology theory that is strong antishape invariant by design, but it cannot be computed in ZFC for simplest non-compact non-ANRs. Even though strong shape and strong antishape differ from each other by exchanging direct and inverse limits, we show that their natural "corrections" (taking into account a topology on the indexing sets) coincide for all metrizable spaces. This common "correction", called fine shape, is much simpler than the original theories and has both \v{C}ech cohomology and Steenrod-Sitnikov homology as its invariants. For ANRs fine shape coincides with homotopy, for compacta with strong shape, and for locally compact separable metrizable spaces - with strong antishape. We prove that a (co)homology theory is fine shape invariant if and only if it satisfies the map excision axiom.