Finite ${\mathbb Z}/2{\mathbb Z}$-CW complexes which are not homotopically stratified by orbit type

Abstract

For k > 2, we construct finite Z/2Z-CW complexes with one 2/22 cell in dimensions 0, 1 and k + 1. Using a theorem of Bruce Hughes, we show that these complexes are not homotopically stratified by orbit type in the sense of Quinn. Homotopically stratified sets were introduced by Quinn in [5] as a means of studying purely topological stratified phenomena. Quinn showed, under suitable conditions, that the orbit space of a finite group acting on a manifold, with the orbit type partition, is a homotopically stratified set ([5, Corollary 1.6]). In this paper we construct examples of 2/22-CW complexes having few 2/22-cells whose orbit spaces, with the orbit type partition, are not homotopically stratified. A closed subspace Y of a space X is forward tame in X if there exists a neigh borhood U of Y in X and a homotopy H: U x I -* X such that Ho is in clusion U -* X, HtIy is inclusion Y )-* X for every t E I, H1(U) = Y and H((U Y) x [0, 1)) C X Y. The homotopy link of Y in X is holink(X, Y) = {w c XI w (t) E Y if and only if t = 0}. A stratification of a space X consists of an indexed locally finite partition {Xi I i C I} of X by locally closed subspaces. We refer to X together with its stratification as a stratified space. Given a space X with an action of a group G, the orbit type corresponding to a subgroup H C G is the set of all points in X whose isotropy group is conjugate to H. The orbit type partition of X consists of the connected components of the orbit types of X. The orbits of these components give a partition of the orbit space G\X. A stratified space X is said to satisfy the frontier condition if for every i, j C 1, Xi n closure(Xj) $& 0 implies that Xi C closure(Xj). This induces a relation < on X, defined by i < j if and only if i 74 j and Xi C closure(Xj). The orbit type stratification of a finite G-CW complex need not satisfy the frontier condition. For example, let X = Si V S1 V S' be the wedge of three circles along a basepoint *. Express X * as the union of three disjoint 1-cells el, el, e3 whose closures are the corresponding S1 factors and give X the 2/22-CW complex structure: one 2/22-0 cell, the basepoint * (isotropy 2/22), and two 2/22-1-cells: el U el on which 2/22 acts by interchanging el and el (trivial isotropy) and el (isotropy 2/22). The orbit Received by the editors December 14, 2007. 2000 Mathematics Subject Classification. Primary 57N80, 57S17, 57N40.