Fake Wedges

TL;DR

This paper investigates the homotopy types of the moduli space of fake wedges, proposing conjectures, verifying them after looping, and illustrating examples via manifold embeddings.

Contribution

It formulates two conjectures about the moduli space of fake wedges and verifies them after looping once, advancing understanding of their homotopy properties.

Findings

Conjectures hold after looping once.

Manifold embeddings yield examples of non-trivial fake wedges.

Connections to discussions with Arone and Thomason are established.

Abstract

A fake wedge is a diagram of spaces K <- A -> C whose double mapping cylinder is contractible. The terminology stems from the special case A = K v C with maps given by the projections. In this paper, we study the homotopy type of the moduli space D(K,C) of fake wedges on K and C. We formulate two conjectures concerning this moduli space and verify that these conjectures hold after looping once. We show how embeddings of manifolds in Euclidean space provide a wealth of examples of non-trivial fake wedges. In an appendix, we recall discussions that the first author had with Greg Arone and Bob Thomason in early 1995 and explain how these are related to our conjectures.