Peter S. Landweber

Circle actions on spin manifolds and characteristic numbers

II+ 1970 Atiyah and Hirzebruch [3] proved that the existence of a non-trivial smooth action of the circle group S’ on a closed connected Spin manifold M implies that the A-genus of M vanishes. It is our purpose to explore the vanishing of further Pontrjagin numbers of Spin manifolds admitting smooth S’ actions of odd type; assuming that MS’ # 4, and viewing Z, = ( k 1) c S’, these are the actions for which all components of the fixed set MzL have codimensions congruent to 2 (mod 4) (see [2,3]). We have one general result for such actions; the rest of our results require that the action also be semifree, i.e. that it be free on the complement of the fixed set MS’. The following result was proved during conversations with S. Weinberger.

Symmetric topological complexity of projective and lens spaces

For real projective spaces, (a) the Euclidean immersion dimension, (b) the existence of axial maps, and (c) the topological complexity are known to be three facets of the same problem. But when it comes to embedding dimension, the classical work of Berrick, Feder and Gitler leaves a small indeterminacy when trying to identify the existence of Euclidean embeddings of these manifolds with the existence of symmetric axial maps. As an alternative we show that the symmetrized version of (c) captures, in a sharp way, the embedding problem. Extensions to the case of even torsion lens spaces and complex projective spaces are discussed.

The integral cohomology of configuration spaces of pairs of points in real projective spaces

Abstract. We compute the integral cohomology ring of configuration spaces of two points on a given real projective space. Apart from an integral class, the resulting ring is a quotient of the known integral cohomology of the dihedral group of order 8 (in the case of unordered configurations, thus has only 2- and 4-torsion) or of the elementary abelian 2-group of rank 2 (in the case of ordered configurations, thus has only 2-torsion). As an application, we complete the computation of the symmetric topological complexity of real projective spaces with and .

A bilinear form for Spin manifolds

On etudie la forme bilineaire sur H j (M;Z 2 ) definie par [x,y]=xSq 2 y[M] quand M est une variete spin fermee de dimension 2j+2

A note on the thick subcategory theorem

In this paper We will discuss an algebraic version (Theorem 1.6) of the thick subcategory theorem of Hopkins-Smith [HS] (Theorem 1.4). The former is stated as Theorem 3.4.2 in [Rav92], hut the proof given there is incorrect. (A list of errata for [Rav92] can be obtained by e-mail from the third author.)

A survey of bordism and cobordism

In the paper ‘Bordism and Cobordism’, which appeared in vol. 57 (1961) of this journal [5], Michael Atiyah introduced and began the study of bordism and cobordism theory. The present article will trace developments in this area since this beginning.