Connected sum at infinity and 4-manifolds

TL;DR

This paper investigates the connected sum at infinity operation on smooth, open manifolds, exploring conditions for independence from ray choices and constructing examples where different choices produce distinct manifolds, distinguished by cohomology at infinity.

Contribution

It introduces conditions for ray choice independence and constructs infinite families of manifolds with distinct connected sums at infinity, distinguished by cohomology algebras at infinity.

Findings

Connected sum at infinity can depend on ray choices in certain cases.

Constructed infinite families of manifolds with distinct connected sums at infinity.

Cohomology algebras at infinity can distinguish these manifolds.

Abstract

We study connected sum at infinity on smooth, open manifolds. This operation requires a choice of proper ray in each manifold summand. In favorable circumstances, the connected sum at infinity operation is independent of ray choices. For each m at least 3, we construct an infinite family of pairs of m-manifolds on which the connected sum at infinity operation yields distinct manifolds for certain ray choices. We use cohomology algebras at infinity to distinguish these manifolds.