Representing homology classes of 4-manifolds with topological 2-spheres

Gerard A. Venema

Calvin College / Michigan State University

Abstract:
In this talk I will address the problem of representing 2-dimensional homology classes of simply connected, piecewise linear (PL) 4-manifolds with topologically embedded 2-spheres. The first theorem states that each such class can be represented by a relatively simple codimension 0 submanifold.
Theorem 1. If $W$ is a compact, simply connected, PL 4-manifold, then each element of $H_2(W)$ can be represented by a compact PL submanifold $M\subset W$ such that $M$ consists of a Mazur-like contractible 4-manifold with a single 2-handle attached.
A compact contractible PL 4-manifold is Mazur-like if it has a handle decomposition in which there is one 0-handle, no handles of index greater than 2, and the attaching map for the $i$th 2-handle is homotopic to the loop represented by the $i$th 1-handle. The manifold $M$ represents a specified element of $H_2(W)$ in the sense that a generator of $H_2(M)\cong{\Bbb Z}$ is homologous in $W$ to the given element of $H_2(W)$.
Theorem 1 is the main ingredient in the proof of the following result.
Theorem 2. If $W$ is a compact, simply connected, PL submanifold of $S^4$, then each element of $H_2(W)$ can be represented by a locally flat topological embedding of $S^2$.
Both Theorems 1 and 2 are false without the hypothesis that $W$ is compact.