Retractions of Closed Manifolds with Nonpositive Curvature

Relative hyperbolization and Pontrjagin classes

“…In this Note we point out that this result has a refined version as follows: Like the original version, Theorem 1.1 will also be proven via relative hyperbolization, a concept first appeared in [3]. The relative hyperbolization 'h' as described in [5] § §1-2 (see also [4] 2.2 and [2] 4a) is one of the rare relative hyperbolization that is universally good, and is the one that will be used here. There is another construction 'p' as described in [5] §3, which will also be used here.…”

“…Clarifying a point here: when applying relative hyperbolization to (P , K), you do not really need PL geometry on P if you are using the 'h' of [5], although you do if you use the 'h' of [4]. By the geometric retraction theorem ([5] 2.4), h(P , K) is a closed triangulated manifold with curvature 0 and K ⊂ h(P , K) is a retraction.…”

“…By the geometric retraction theorem ([5] 2.4), h(P , K) is a closed triangulated manifold with curvature 0 and K ⊂ h(P , K) is a retraction. Since K ⊂ h(P , K) is a retraction, the vector bundle E over K can be extended to one over h(P , K), still denoted as E. N = p(Q ∪ Q, h(P , K)), N ⊂ M is a codimension 0 submanifold, because M and N are of the same dimension as cell complexes, N ⊂ M is subcomplex, M is combinatorial manifold, while N is combinatorial manifold with boundary according to [5] K). Therefore T (M)| K stably = E. This proves Theorem 1.1.…”

“…The 'aspherical retraction theorem' of Davis says that any compact aspherical polyhedron is the retraction of a closed aspherical manifold, which was first proven using the reflection group trick, see, e.g., [1]. As discussed in [5], the construction 'h' applies equally well to aspherical polyhedra without geometry, and can yield a different proof of Davis's theorem. The reason is simply that, when it comes to the construction 'h', in dealing with piecewise Euclidean geometry, we only use Gromov's 'totally geodesic gluing lemma' without ever gotten involved in geometric links, and the gluing lemma has an easy aspherical counterpart, which I forgot to clarify in [5].…”

“…The relative hyperbolization 'h' as described in [5] § §1-2 (see also [4] 2.2 and [2] 4a) is one of the rare relative hyperbolization that is universally good, and is the one that will be used here. There is another construction 'p' as described in [5] §3, which will also be used here. Suppose X is a cell complex, K ⊂ X is subcomplex.…”