A formula for Stiefel-Whitney homology classes

The purpose of this paper is to define for mod 2 Euler spaces a formula which enables one to compute the Stiefel-Whitney homology classes in the original triangulation without passing to the first barycentric subdivision. The purpose of this paper is to define for mod 2 Euler spaces a formula which enables one to compute the Stiefel-Whitney homology classes in the original triangulation without passing to the first barycentric subdivision. The formula has a somewhat tenuous connection to the Steenrod reduced squares. In the case when we are dealing with a smooth triangulation, the Wu formulae [7] and the Whitney theorem [4] establish such a connection. The authors would like to thank S. Halperin and D. Toledo for a copy of their preprint [5]; the use of their map 4> (see §2) simplifies an earlier proof of the main theorem. The homology theory used is that based on infinite chains. 1. Statement of the theorem. Let A" be a finite-dimensional, locally finite simplicial complex. K is said to be a mod 2 Euler space if the link of every simplex in A" has even Euler characteristic [9]. The pth Stiefel-Whitney class of K, denoted io (K), is the p-dimensional mod 2 homology class which has a representative, the p-dimensional chain consisting of all p-simplexes in the first barycentric subdivision of A"-this chain is a cycle for each p iff A" is a mod 2 Euler space. From now on we assume that A is given an ordering of its vertices and any representation of a simplex in A is written with its vertices in increasing order. We now recall a definition due to Steenrod [8]. Let s be a p-simplex in K, say s = (v0,vx,... ,Vp). Let t be another simplex which has s as a face; i.e., s E t (s may be equal to t). Let B_x = set of vertices of t less than v0, B0 = set of vertices of t strictly between v0 and vx, Bm = set of vertices of t strictly between vm and i>m+1 B = set of vertices of / greater than vp. We say that s is regular in t, if # (Bm) = 0 for every odd m. Let ap(t) denote the mod 2 chain which consists of all p-dimensional simplexes s in t so that 5 Presented to the Society, April 11, 1975; received by the editors April 28, 1975 and, in revised form, November 10, 1975. AMS (MOS) subject classifications (1970). Primary 57C99; Secondary 57D20.