Non-simply connected copes

Abstract

Let ( W 4,∂ W 4) be a 4-manifold. Let f 1, f 2,…, f k :( D 2, ∂D 2)→ ( W 4, ∂W 4) be transverse immersions that have spherical duals α 1,α 2,…,α k:S 2→ W ̊ . Then there are open disjoint subsets V 1, V 2,…, V k of W, such that for each 1⩽ i⩽ k, ( a) ∂V i = V 1∩ ∂W and ∂V i is an open regular neighborhood of f i ( ∂D 2) in ∂ W, and (b) ( V i , ∂V i , f i ( ∂D 2)) is proper homotopy equivalent to ( M, ∂ M, d)—a standard object in which d bounds an embedded flat disk. If we could get a homeomorphism instead of a proper homotopy equivalence, then we would be able to prove a 5-dimensional s-cobordism theorem.