Yang index of the deleted product

Abstract

For any s > 1 a ,i-dimensional polyhedron Y,, is constructed such that the Yang index of its deleted product Y,K equals 2,i. This answers a question of lzydorek and Jaworowski (1995). For any ,. > 1 a 2,i-dimensional closed manifold M with involution is constructed such that index M = 2,i, but M can be mapped into a K,dimensional polyhedron without antipodal coincidence. The deleted product of Y is the space y* = y2 z where A is the diagonal of y2. There is a natural free involution T(x, y) = (y, x) acting in Y*. Our goal is to compute the Yang index of the deleted product of some polyhedra (with respect to the involution T). In particular, we answer the question in [3] of whether there exists a n-dimensional polyhedron Y,, with indexY, = 2s. It is shown that the space Y,, = [A2,+2]1 has indexY,Z = 2,. In fact, we shall find in Y, a closed manifold M21C with index M2, = 2s. Then the projection p(x, y) = x is a map p: M2,1 -Y,, without antipodal coincidence. Other examples of such manifolds (or even polyhedra) are not known to us. Let us note the theorem of Schepin [4], which asserts that every map f: S2, -+ of the 2ii-sphere into a ,-dimensional polyhedron has an antipodal coincidence. First some notation. A'n is a standard n-simplex in R n with center in the origin 0. Let P be a simplicial complex. For x E P, [x] denotes the carrier of x, i.e. the (closed) simplex containing x in its interior. If a E P is a vertex, its star St(a) is the union of all open simplexes with vertex a. [P]' denotes the ,-dimensional skeleton of P. All maps are assumed to be continuous. Now we shall list some properties of the Yang index that we shall make use of, and refer the reader to [7] for the definition and the whole index theory. Let X be a compact metric space with a free involution T: X -X. Then its Yang index is defined inductively by means of the equivariant homology groups with coefficients in 22. We denote it here by indexX. An important property of Received by the editors December 18, 1995 and, in revised form, September 5, 1996. 2000 Mathematics Subject Classification. Primary 55M20.