Embeddings of open manifolds

Abstract

Let T O P ( M ) TOP(M) be the simplicial group of homeomorphisms of M M . The following theorems are proved. Theorem A. Let M M be a topological manifold of dim ≥ \geq 5 with a finite number of tame ends ε i \varepsilon _{i} , 1 ≤ i ≤ k 1\leq i\leq k . Let T O P e p ( M ) TOP^{ep}(M) be the simplicial group of end preserving homeomorphisms of M M . Let W i W_{i} be a periodic neighborhood of each end in M M , and let p i : W i → R p_{i}: W_{i} \to \mathbb {R} be manifold approximate fibrations. Then there exists a map f : T O P e p ( M ) → ∏ i T O P e p ( W i ) f: TOP^{ep}(M) \to \prod _{i} TOP^{ep}(W_{i}) such that the homotopy fiber of f f is equivalent to T O P c s ( M ) TOP_{cs}(M) , the simplicial group of homeomorphisms of M M which have compact support. Theorem B. Let M M be a compact topological manifold of dim ≥ \geq 5, with connected boundary ∂ M \partial M , and denote the interior of M M by I n t M Int M . Let f : T O P ( M ) → T O P ( I n t M ) f: TOP(M)\to TOP(Int M) be the restriction map and let G \mathcal {G} be the homotopy fiber of f f over i d I n t M id_{Int M} . Then π i G \pi _{i} \mathcal {G} is isomorphic to π i C ( ∂ M ) \pi _{i} \mathcal {C} (\partial M) for i > 0 i > 0 , where C ( ∂ M ) \mathcal {C} (\partial M) is the concordance space of ∂ M \partial M . Theorem C. Let q 0 : W → R q_{0}: W \to \mathbb {R} be a manifold approximate fibration with dim W ≥ W \geq 5. Then there exist maps α : π i T O P e p ( W ) → π i T O P ( W ^ ) \alpha : \pi _{i} TOP^{ep}(W) \to \pi _{i} TOP(\hat W) and β : π i T O P ( W ^ ) → π i T O P e p ( W ) \beta : \pi _{i} TOP(\hat W) \to \pi _{i} TOP^{ep}(W) for i > 1 i >1 , such that β ∘ α ≃ i d \beta \circ \alpha \simeq id , where W ^ \hat W is a compact and connected manifold and W W is the infinite cyclic cover of W ^ \hat W .