[Paper Review] Numerably Contractible Spaces

Eugenia Schwamberger, R. M. Vogt|arXiv (Cornell University)|Oct 30, 2008

Homotopy and Cohomology in Algebraic Topology12 references3 citations

TL;DR

This paper provides a systematic study of numerably contractible spaces—topological spaces admitting a numerable cover where each inclusion is nullhomotopic—demonstrating their central role in homotopy theory. It proves that the topological realization functor preserves fibration sequences when the base is path-connected and numerably contractible, implying the loop space functor commutes with realization up to homotopy, and establishes sufficient conditions for free algebras over topological operads to be numerably contractible.

ABSTRACT

Numerably contractible spaces play an important role in the theory of homotopy pushouts and pullbacks. The corresponding results imply that a number of well known weak homotopy equivalences are genuine ones if numerably contractible spaces are involved. In this paper we give a first systematic investigation of numerably contractible spaces. We list the elementary properties of the category of these spaces. We then study simplicial objects in this category. In particular, we show that the topological realization functor preserves fibration sequences if the base is path-connected and numerably contractible in each dimension. Consequently, the loop space functor commutes with realization up to homotopy. We give simple conditions which assure that free algebras over a topological operad are numerably contractible.

Motivation & Objective

• To establish a foundational theory of numerably contractible spaces, also known as Dold spaces, in algebraic topology.
• To investigate the behavior of simplicial objects and topological realization in the context of these spaces.
• To determine conditions under which free algebras over topological operads are numerably contractible.
• To strengthen classical homotopy equivalences by showing they become genuine equivalences when spaces involved are numerably contractible.
• To explore the limitations of closed classes in homotopy theory by constructing counterexamples involving Dold spaces.

Proposed method

• Define numerably contractible spaces via a numerable open cover where each inclusion is nullhomotopic.
• Use Dold’s theorem on partitions of unity in fibrations to derive strengthened results for homotopy pullbacks and pushouts.
• Apply the topological realization functor to simplicial objects in the category of numerably contractible spaces.
• Prove that realization preserves fibration sequences when the base space is path-connected and numerably contractible.
• Use the homotopy extension property and properties of path-connectedness to show that loop space and realization functors commute up to homotopy.
• Establish criteria for free algebras over topological operads (e.g., little n-cube operads) to be numerably contractible using path-connectedness and homotopy invariance.

Experimental results

Research questions

• RQ1 Under what conditions does a weak homotopy equivalence become a genuine homotopy equivalence in the presence of numerably contractible spaces?
• RQ2 Does the topological realization functor preserve fibration sequences when applied to simplicial objects over numerably contractible spaces?
• RQ3 When is the free algebra over a topological operad on a numerably contractible space itself numerably contractible?
• RQ4 Can the loop space functor commute with realization up to homotopy for numerably contractible base spaces?
• RQ5 Are there Dold spaces that are not of the homotopy type of a CW-complex, and what are the implications for homotopy classification?

Key findings

• The topological realization functor preserves fibration sequences if the base space is path-connected and numerably contractible, ensuring the loop space functor commutes with realization up to homotopy.
• Free algebras over the little n-cube operad on a path-connected, numerably contractible space are shown to be numerably contractible, generalizing results by May and Meiwes.
• The symmetric product $SP_n(X)$ is a Dold space when $X$ is path-connected and numerably contractible, as it is a retract of a free algebra over the commutative operad.
• There exist Dold spaces that are not of the homotopy type of a CW-complex, such as the unreduced suspension of $\{1/n\} \cup \{0\}$, which has non-finitely generated homology.
• Weak homotopy equivalences between Dold spaces need not be homotopy equivalences, as shown by a $CW$-approximation of the above suspension space.
• The loop space of a Dold space need not be a Dold space, as demonstrated by the rational numbers $\mathbb{Q}$, whose nerve realizes to a Dold space but whose loop space is not.