Donald M. Davis

Network security via private-key certificates

We present some practical security protocols that use private-key encryption in the public-key style. Our system combines a new notion of private-key certificates, a simple key-translation protocol, and key-distribution. These certificates can be administered and used much as public-key certificates are, so that users can communicate securely while sharing neither an encryption key nor a network connection.

The geometric dimension of some vector bundles over projective spaces

We prove that in many cases the geometric dimension of the p-fold Whitney sum p1Hk of the Hopf bundle Hk over quaternionic projective space QPk is the smallest n such that for all i < k the reduction of the ith symplectic Pontryagin class of pHk to coefficients 74i-1((RP /RRPn1 ) A bo) is zero, where bo is the spectrum for connective KO-theory localized at 2. We immediately obtain new immersions of real projective space RP4k+3 in Euclidean space if the number of Is in the binary expansion of k is between 5 and 8.

₁-periodic homotopy groups of exceptional Lie groups: torsion-free cases

The v 1 -periodic homotopy groups v 1 #751 π. (X; p) are computed explicitly for all pairs (X, p), where X is an exceptional Lie group whose integral homology has no p-torsion. This yields new lower bounds for p-exponents of actual homotopy groups of these spaces. Delicate calculations with the unstable Novikov spectral sequence are required in the proof

The spectrum ( P ⋀ bo ) −∞

There are spectra P −k constructed from stunted real projective spaces as in [1] such that H * ( P −k ) is the span in ℤ/2[x, x −1 ] of those x i with i ≥ −k. (All cohomology groups have ℤ/2-coefficients unless specified otherwise.) Using collapsing maps, these form an inverse system which is similar to those of Lin ([15], p. 451). It is a corollary of Lins work that there is an equivalence of spectra where holim is the homotopy inverse limit ([3], ch. 5) and Ŝ –1 the 2-adic completion of a sphere spectrum. One may denote by this holim ( P –κ ), although one must constantly keep in mind that , but rather

Divisibility by 2 of Stirling-like numbers

We give a characterization of functions of the form f(n) = v(n E), where v(-) denotes the exponent of 2, and E is a 2-adic integer. We show that it applies to the restriction to even or odd integers of the function f (n) = v(a * 5n + b * 3n + c), with mild restrictions on a, b, and c. This function is closely related to divisibility of certain Stirling numbers of the second kind. Let v (m) denote the exponent of 2 in m, and let N denote the set of nonnegative integers. Our first result is a characterization of a certain class of functions. We think of a 2-adic integer as a possibly infinite sum of distinct 2-powers. Theorem 1. Let f be a function N -k N U {oo}. Then the following are equivalent: (i) There is a 2-adic integer E such that f(n) = v(n E) for all n; (ii) For all n and d, = =d if d f(n) if d = f(n) =f(n) if d > f(n); (iii) f satisfies (a) for all n, f (n + 2f (n)) > f(n), and (b) if d 1 2e, with e1 f(n). Received by the editors September 29, 1989 and, in revised form, January 11, 1990. 1980 Mathematics Subject Classification (1985 Revision). Primary 11 B73.

The complex bordism of groups with periodic cohomology

Is is proved that if BG is the classifying space of a group G with periodic cohomology, then the complex bordism groups MU*(BG) are obtained from the connective K-theory groups ku*(BG) by just tensoring up with the generators of MU. as a polynomial algebra over ku* . The explicit abelian group structure is also given. The bulk of the work is the verification when G is a generalized quaternionic group. 1. STATEMENT OF RESULTS It is well known [CE] that a finite group has periodic cohomology if and only if its Sylow subgroups are all cyclic or generalized quaternionic. Another characterization [Sw] is that these are precisely the finite groups which can act freely on a finite simplicial homotopy sphere. In [Wo], it was shown exactly which of these (the spherical space-form groups) admit a free orthogonal action on a standard sphere. Let MU.( ) denote (reduced) complex bordism and bu*( ) connective Ktheory homology. It is well known [CF1] that if BG denotes the classifying space of a finite group G, then MUn(BG+) is isomorphic to the group of bordism classes of stably almost complex n-manifolds with free G-action. Here and elsewhere X+ is the space obtained from X by adjoining a disjoint basepoint. The coefficient rings are MU. --MU*(S?) = Z[X2 : i > 1] and bu* --bu*(S0) = Z[x2] where x21 is a generator of degree 2i in a polynomial algebra. Our main result proves an extension of a conjecture of Gilkey [G, BD]. Theorem 1.1. If G is any finite group with periodic cohomology, then there is an isomorphism of graded abelian groups MU*(BG) bu*(BG) 0 Z[x2i: i > 2]. Received by the editors May 1, 1988. 1980 Mathematics Subject Classification (1985 Revision). Primary 57R85; Secondary 55R35, 55N22.

Chapter 20 - Computing v 1 -periodic Homotopy Groups of Spheres and some Compact Lie Groups

This chapter presents an account of the principal methods that are used to compute the v 1 -periodic homotopy groups of spheres and many compact simple Lie groups. The two main tools have been J-homology and the unstable Novikov spectral sequence (UNSS). The v 1 -periodic homotopy groups are important because for spaces such as spheres and compact simple Lie groups they give a significant portion of the actual homotopy groups and yet are often completely calculable. The goal of this chapter is to explain how those calculations can be made.

Projective product spaces

Let nbar=(n_1,...,n_r). The quotient space P_nbar:=(S^{n_1} x...x S^{n_r})/(x ~ -x)is what we call a projective product space. We determine the integral cohomology ring and the action of the Steenrod algebra. We give a splitting of Sigma P_nbar in terms of stunted real projective spaces, and determine when S^{n_i} is a product factor. We relate the immersion dimension and span of P_nbar to the much-studied sectioning question for multiples of the Hopf bundle over real projective spaces. We show that the immersion dimension of P_nbar depends only on min(n_i), sum n_i, and r, and determine its precise value unless all n_i exceed 9. We also determine exactly when P_nbar is parallelizable.

2-primary v 1-periodic homotopy groups of SU(n) revisited

Abstract In 1991, Bendersky and Davis used the BP-based unstable Novikov spectral sequence to study the 2-primary v 1-periodic homotopy groups of SU(n). Here we use a K-theoretic approach to add more detail to those results. In particular, whereas only the order of the groups was determined in the 1991 paper, here we determine the number of summands in these groups and much information about the orders of those summands. In addition, we give explicit conditions for certain differentials and extensions in a spectral sequence, which affect the homotopy groups. Finally, we give complete results for for n ≤ 13.

The image of the stable J-homomorphism

IN THIS PAPER we give detailed proofs of results about the image of the stable J-homomorphism similar to some announced in [24, 25, 26, and 271. The methods are those introduced in those papers; we just take a bit more care here to clarify certain aspects of the proof. The paper could be viewed as a response to Frank Adams’ challenge (Cl]): “for the classical J-homomorphism, we have no published proof independent of the ideas I am now discussing (the Adams conjecture); homotopy theorists know in principle where they should look for an independent proof, but nobody has yet been willing to undertake the heavy task of working it out in detail and writing it down properly.” The J-homomorphism, as introduced by G. W. Whitehead in [41], is the homomorphism X:”