Embeddings of open manifolds

Abstract:

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