Special manifolds and shape fibrator properties

Abstract

Our main interest in this paper is further investigation of the concept of (PL) fibrators (introduced by Daverman [R.J. Daverman, PL maps with manifold fibers, J. London Math. Soc. (2) 45 (1992) 180-192]), in a slightly different PL setting. Namely, we are interested in manifolds that can detect approximate fibrations in the new setting. The main results state that every orientable, special (a new class of manifolds that we introduce) PL n-manifold with non-trivial first homology group is a fibrator in the new category, if it is a codimension-2 fibrator (Theorem 8.2) or has a non-cyclic fundamental group (Theorem 8.4). We show that all closed, orientable surface S with χ(S) < 0 are fibrators in the new category. Manifold homotopically determined by π 1 0. Introduction In 1977, Coram and Duvall [3,4] introduced the concept of approximate fibrations as a generalization of both Hurewicz fibrations and cell-like maps. Approximate fibrations are proper mappings that satisfy an approximate version of the homotopy lifting property-the defining property for fibrations. Why study approximate fibrations? Coram and Duvall showed that approximate fibrations have shape theoretic properties analogous to the homotopy theoretic properties of Hurewicz fibrations. For each approximate fibration p : M → B, where M, B are locally compact ANRs, they showed that all fibers of p are fundamental absolute neighborhood retracts [1, p. 266], and moreover that if B is path connected, then any two fibers have the same shape. The most useful property is the existence of an exact sequence involving the homotopy groups of domain, target and shape-theoretical homotopy groups of any point inverse of p. When working with the PL approximate fibration these properties of an approximate fibration reduce to the usual properties of Hurewicz fibration because the fibers are ANRs, so the ith shape homotopy groups are isomorphic to ith homotopy groups. A question that arises is this: how can we detect approximate fibration so we can use these nice properties? Sometimes a proper map defined on an arbitrary manifold of a specific dimension can be recognized as an approximate fibration due to having point inverses all of a certain homotopy type (or shape). So the above question that has been addressed for more than two decades can be restated in a different form: * Tel.: +(

Coperfectly Hopfian groups and shape fibrator's properties

“…The next result (that we use later) and its proof is the analog to the Fundamental Theorem [34,Theorem 5.5] and its proof. Proof.…”

“…A group G is hyper-Hopfian [8] if every homomorphism ϕ : G → G with ϕ(G) ⊳ G and G/ϕ(G) cyclic is necessarily an automorphism. A group G is ultra-Hopfian [34] if every nontrivial homomorphism ϕ : G → G with ϕ(G) ✂ G is an automorphism.…”

“…Furthermore, D 2 n+1 = x, y | x 2 = y 2 n+1 = 1, x −1 yx = y −1 are 2-groups, so coperfectly Hopfian by Corollary 3.4. Note that D 2 n+1 are not ultra-Hopfian (see [34,Section 2]). The quaternionic group Q = c, d | c 2 = (cd) 2 = d 2 , of order 8, is a hyper-Hopfian group (see [8,Section 4]) and a coperfectly Hopfain group by Corollary 3.4, which is not ultra-Hopfian (see [34,Section 2]).…”

“…The group of rational numbers, Q, is a coperfectly Hopfian group since it is Abelian and ultra-Hopfian (see [34,Section 2]).…”

“…All closed orientable surfaces S with genus g > 1 are shape m simpl o-fibrators.Proof. S is homotopically determined by π 1 , has coperfectly Hopfian fundamental group by Theorem 3.9, and is a codimension-2 shape m simpl o-fibrator by[34, Corollary 6.3]. So by Theorem 5.4 it follows that S is a shape m simpl ofibrator.In this section we discuss the shape m simpl o-fibrator's properties of direct product of Hopfian manifolds.…”