A combinatorial formula for normal Stiefel-Whitney classes

Using an ordering of the vertices of a combinatorial n-manifold K, we give an explicit description of a simplicial mod 2 cycle c„_¡(K) which represents the dual of the z'th normal Stiefel-Whitney class of K. 1. The formula. A local ordering of a simplical complex K is a partial ordering of the vertices of K which restricts to a total order on the closed star of each vertex. If K is a locally ordered «-dimensional complex, and (a, r) is an ordered pair of n-simplices of K such that a n r ¥= 0, define u(o, t) G Z/2Z as follows. Since o n r J= 0, there is a vertex v such that a and t are in star(u). Let vx,. .., vs be the vertices of a and r in the given total order of star(u). (If a vertex belongs to both a and r, it should be listed only once.) Let ¡i(a, r) = 1 if vx, v3, v5, . . . are in o and v2, v4, v6, . . . are in r, and u(a, t) = 0 otherwise. For i = 1, 2,. . ., n, let cn_i(K) be the simplicial mod 2 (n — z')-cnain 2 [i(o, r)a n r, summed over all ordered pairs (a, r) of n-simplices such that dim(a n r) = n — i. Theorem 1. Let K be a finite locally ordered combinatorial n-manifold without boundary. The chain c„_¡(K) is a mod 2 cycle, and its homology class is Poincaré dual to w¡(K), the ith normal StiefelWhitney class of K. A theorem of Levitt and Rourke [6] asserts that some combinatorial formula using a local ordering exists for every characteristic class of combinatorial manifolds. The prototype is the combinatorial formula for the tangential Stiefel-Whitney classes w¡ (cf. [4]). A generalization of this formula for w¡ has been given by Goldstein and Turner [3]. The observation that the Goldstein-Turner formula can be proved using our geometric definition of Stiefel-Whitney classes ([2], [7]) led us to the formula for w¡ presented here. Its form was simplified by a remark of Lee Rudolph. 2. An example. Let K be the triangulation of the projective plane obtained by identifying opposite points of the icosahedron. Order the vertices of K as in Figure 1. Presented to the Society, January 27, 1977; received by the editors April 13, 1978. AMS (MOS) subject classifications (1970). Primary 57D20, 55G05; Secondary 57C35.