Extreme nonuniqueness of end-sum

Abstract

We give explicit examples of pairs of one-ended, open 4-manifolds whose end-sums yield uncountably many manifolds with distinct proper homotopy types. This answers strongly in the affirmative a conjecture of Siebenmann regarding nonuniqueness of end-sums. In addition to the construction of these examples, we provide a detailed discussion of the tools used to distinguish them; most importantly, the end-cohomology algebra. Key to our Main Theorem is an understanding of this algebra for an end-sum in terms of the algebras of summands together with ray-fundamental classes determined by the rays used to perform the end-sum. Differing ray-fundamental classes allow us to distinguish the various examples, but only through the subtle theory of infinitely generated abelian groups. An appendix is included which contains the necessary background from that area.

Publication Date

6-1-2022

Publication Title

Journal of Topography and Analysis

Department

Mathematics

Document Type

Article

DOI

https://dx.doi.org/10.1142/S179352532150014X

Notes

Dedicated to the memory of Andrew Ranicki.

Keywords

End-sum, Connected sum at infinity, End-cohomology, Proper homotopy, Direct limit, Infinitely generated abelian group

Language

English

Format

text

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