Simplicial Structures and Transverse Cellularity

Abstract

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On the 𝐾-theory of crystallographic groups

“…The lower ^-groups K-¡(Zii) were first defined by Bass in [1] and later by Ranicki as transfer invariant elements of Wh(7r x Z!+1) (see [21]). The Whitehead group Wh(A') of a CW complex X is defined geometrically in various places (see [6]). A typical element in Wh(A") is an equivalence class of pairs (Y, f), where F is a CW complex and /: Y -* X is a strong deformation retraction.…”

“…Moreover, as in the finite case, Wh(X) « Wh(nx(X)). This description of Wh(Z) is suitable for carrying over the construction given in [6] for Wh(X)c ; i.e. the definition of Wh(J)c, for an arbitrary CW complex X, is completely analogous with the finite case.…”

Cone complexes and PL transversality

“…(We will say that A is a subcomplex of C, when no confusion can occur.) A subdivision B of C is a cone complex with I B I = I CI such that aE C implies ctyTBCone complexes were introduced by Akin [2], following ideas of Cohen [7]. He also introduced the concept of "structuring" a cone complex to study its geometry.…”

“…So suppose height C = «, and the lemma is true for complexes of height < n. Thus we assume 8 I IC"_ xl has been defined, so we just need to define ¡517 for each principal cone 7 of C, given 8137-Let K = (r_©(8|37))'> a triangulation of 7 (depending on the given structure of 7). Let the cones of 817 be the following sets (not the cones of 7 © (81 oy)):…”

“…Our main tool is a canonical geometric duality for "structured" cone complexes. Cone complexes have also been used by M. Cohen [7] and E. Akin [2] to study simplicial maps. The rigid geometry provided by a structured cone complex on a polyhedron is analogous to a Riemannian metric on a smooth manifold (as manifested by its relation to duality and transversality).…”