Geometric branched covers between generalized manifolds

Abstract

We develop a theory of geometrically controlled branched covering maps between metric spaces that are generalized cohomology manifolds. Our notion extends that of maps of bounded length distortion, or BLD-maps, from Euclidean spaces. We give a construction that generalizes an extension theorem for branched covers by I. Berstein and A. Edmonds. We apply the theory and the construction to show that certain reasonable metric spaces that were shown by S. Semmes not to admit bi-Lipschitz parametrizations by a Euclidean space nevertheless admit BLD-maps into Euclidean space of same dimension. Theorem 0.3 was proved by Berstein and Edmonds [BE, Th. 6.2] for p = 2 and d 0 = 3, and without the restriction d = 3i. The restriction d = 3i on the degree is needed for our proof, and we do not know if it is really necessary. Similarly, in our construction the degree lower bound d 0 depends on W . It could well be that one can choose d 0 = d 0 ( p, m), but we do not know this. In our proof we use the techniques of [Ri1] to reduce the general case to the case considered by Berstein and Edmonds. This reduction is presented here in a self-contained manner; no reference to [Ri1] is needed.A special case of Theorem 0.3 was proved in [HR], where we showed the existence of a BLD-map f : S 3 → S 3 whose branch set contains a wild Cantor set. The general construction of the present paper simplifies that of [HR], except the use of the Berstein-Edmonds theorem. (In [HR] we did not use [BE] but relied instead on S. Rickman's paper [Ri1] in a more substantial way.)By the aid of Theorem 0.3 we can construct BLD-maps from some interesting compact generalized 3-manifolds onto S 3 . Semmes [S3] showed that the classical (compact) decomposition spaces arising from the Whitehead continuum, Bing's dogbone space and Bing's double, for example, all admit metrics that are smooth Riemannian outside a totally disconnected closed singular set and that are indistinguishable from the standard metric on S 3 by means of classical analysis. We show that all these spaces, although bi-Lipschitz inequivalent to the standard or PL 3-sphere, can be mapped onto standard S 3 by a BLD-map. As alluded to above, the claims made by Semmes, as well as our constructions, work for each space arising in a self-similar manner from an "excellent package," as defined in [S3]. All this is made more precise in Section 8.At present, the method of constructing interesting branched covers in the spirit of [Ri1] and [HR] is limited to dimension n = 3. When n = 2, every generalized manifold is actually a manifold, and the existence and nature of branched covers is well understood. In higher dimensions, every closed orientable n-pseudomanifold can be mapped onto S n by a BLD-branched cover. For smooth manifolds this was proved by J. Alexander [A] in 1920, and his method works for orientable pseudomanifolds, too. Recall that a closed n-pseudomanifold (cf. [ST, Sec. 24]) is a pure n-dimensional finite complex where every (n −1)-simplex is a face of precisely two n-simplexes and where ...

Open and discrete maps with piecewise linear branch set images are piecewise linear maps

“…We remark that the regularity of being a CW‐complex is not enough for our main results. The quasiregular mappings constructed by Heinonen and Rickman in [13] and [12] also have CW‐complex branch sets but otherwise behave pathologically. In particular, the boundaries of normal domains are not manifolds in those examples.…”

“…The set is not simply connected. So, as a Cantor set with a topologically nontrivial complement, the set cannot be contained in a codimension 2 simplicial complex (see [13] and [12]). Here a wild Cantor set refers to any Cantor set in such that there is no homeomorphism for which .…”

Cartan--Whitney Presentation, Non-smooth Analysis and Smoothability of Manifolds: On a theorem of Kondo--Tanaka

“…Roughly speaking, ϑ is a local coframe with weak regularity and an essentially nondegenerate volume density, and the integration of ϑ along segments gives rise to a branched covering map F ϑ . Heinonen-Rickman [14] and Heinonen-Sullivan [15] proved that the local smoothability of a Lipschitz manifold is equivalent to that the local degree of F ϑ = 1; furthermore, Heinonen-Keith [13] established its equivalence with the Sobolev regularity condition ϑ ∈ W 1,2 loc . In a recent paper [17], a brand-new perspective has been adopted by Kondo-Tanaka to approach the smoothability problem.…”

“…In this note we present a simple, new proof of Theorem 1.2, which also establishes its converse at the same strike. Our proof is based on the geometric measure theoretic studies on the smoothability problem (see [27,28,29,14,15,13]). In particular, we make crucial use of the results due to Heinonen-Keith [13].…”

Mappings of finite distortion: Size of the branch set

“…These sets have very rich and complex topological and geometrical structures both of which have been studied in a variety of different contexts, see e.g. [1,2,12,13,14,25,35] and references there.…”