Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 1808.10228

Fine shape I

Sergey A. Melikhov
View original source ↗Read source document ↗Export metadata ↓

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.

Explore related subjects

math.ATmath.GT

Keep this discovery

BibTeXRIS

Sergey A. Melikhov. 2018-08-30. Fine shape I. https://arxiv.org/abs/1808.10228

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Homogeneous Milnor fibers and Kato--Matsumoto bounds via simplicial multiwedges

For every $n\geq 3$ and $s\geq 2$, we construct a homogeneous polynomial of degree $n(n+1)/2$ whose Milnor fiber is exactly $2s$-connected and whose rational cohomology contains a strictly defined nontrivial $n$-fold Massey product on classes of degree $2s+1$, implying that the Milnor fiber is non-formal, while attaining the Kato--Matsumoto connectivity bound. Our construction is based on the simplicial multiwedges of the nerve complexes of simple polytopes introduced by Limonchenko, combined with Suciu's realization of weighted homogeneous Milnor fibers. We thereby answer two problems posed by Suciu.

math.AT↗
arXiv2026

The homotopy types of directed path and trace spaces

We construct a saturated directed space with a Hausdorff $\Delta$-generated underlying space and two distinct points such that the trace space between them is homeomorphic to a square, whereas the directed path space has a nontrivial fundamental group. In particular, the canonical quotient map is not a weak homotopy equivalence. The same conclusion holds for regular directed paths modulo increasing homeomorphisms.

math.AT↗
arXiv2026

Moduli spaces of geometric functorial field theories

We develop tools to compute moduli spaces of geometric functorial field theories as mapping spaces of equivariant simplicial presheaves. Given a d-dimensional geometric structure F, presented as a presheaf on the site of smooth families of d-manifolds, we define its Cartesian realization, which is an O(d)-equivariant simplicial presheaf on the site of Cartesian spaces. We use Cartesian realizations to present the moduli space of functorial field theories with geometric structure F as a mapping space between O(d)-equivariant simplicial presheaves. In a companion paper, we use this result to compute the moduli space of smooth one-dimensional oriented Riemannian functorial field theories valued in an arbitrary smooth symmetric monoidal infinity-category.

math.AT↗