R. James Milgram

Immersing Projective Spaces

THEOREM 2. (a) HPn immerses in R8 n-Ea(n)-3J. (b) For n even, CPn immerses in R4ln-a(n)-1]. (c) For n odd, CPn immerses in R4n-a(n). Here a(n) is the number of ones in the dyadic expansion of n, and k(n) is a non-negative function depending only on the mod (8) residue class of n with k(1) = 0, k(3) = k(5) = 1 and k(7) = 4. As a consequence, for every j>O there is some n so that RPn immerses in R2f-. From the techniques used to prove these theorems, we also obtain some results on the existence of non-singular bilinear maps.

THE HOMOLOGY OF SYMMETRIC PRODUCTS

In this paper we compute the homology groups for the various symmetric products of any space X of finite type. Thus we complete the calculations begun by M. Morse, Smith and Richardson in the 1930s and carried dramatically forward by N. Nakaoka in a series of papers dating from 1955. Our methods are essentially geometric in nature and are based on a close examination of the geometry of the topological bar construction introduced in [10]. Indeed it was the study of the symmetric products which led to [10], but the exposition given here is selfcontained. (1) The w-fold symmetric product SPm(X) is the set of all unordered «j-tuples of points in X. Equivalently, SPm(X) is the orbit space of the Cartesian product Xm under the action of ?^m, the symmetric group on m letters. It has the quotient topology. Let a base point *elbe given, then there is an inclusion j:SPm(X)<= SPm + 1(X)

Semicharacteristics, bordism, and free group actions

In this paper we give characteristic class formulae for all semicharac- teristic classes of all compact, closed manifolds with finite fundamental groups. These invariants are identified with elements in certain odd L-groups, and ex- actly which elements occur is specified. An appendix calculates the cohomology of the model groups needed. A second appendix determines the structure of the L-groups needed. - The Euler characteristic of an odd-dimensional manifold is zero; a natural substitute is a semicharacteristic—an alternating sum of the homology up to the middle dimension. Study of semicharacteristics was initiated by Kervaire (K) who examined their role in differential topology and geometry. The first semicharacteristic bordism invariant was introduced by DeRham. It turns out k Xx/2(M2k+X;K) = ?(-l)diffl(ff?(*f;A)j i=i