David Copeland Johnson

PROJECTIVE DIMENSION AND BROWN-PETERSON HOMOLOGY

Abstract BP ∗ (X) is the reduced Brown-Peterson homology of a finite complex X for a fixed prime p. We study a sequence of homology theories BP ∗ (X)→⋯→BP〈n+1〉 ∗ (X) → p(n,n + 1) BP〈n〉 ∗ (X) → p(n - 1,n) BP〈n–1〉 ∗ (X) →⋯→BP〈0〉 ∗ (X) → p( - 1,0) BP〈–1〉 ∗ (X) . Our main result states that there is an n such that ⋯, ϱ(n,n + 1), ϱ(n– 1, n) are all epimorphisms; each of the remaining homomorphism fails to be onto; and n is the projective dimension of BP ∗ (X) as a module over the coefficient ring BP ∗ . The first two sections are introductory in nature while the third contains the core results. The reader should be pleased to know that §4–6 are independent of each other.

Brown-Peterson homology of elementary p-groups II

Abstract We compute the complete Abelian group structure of the Brown-Peterson homology of BV, the classifying space for V=( Z p ) n , the elementary Abelian p-group of rank n.

On the [p]-series in Brown-Peterson homology

Abstract The coefficients of the [ p ]-series for Brown-Peterson homology lie in products of prime invariant ideals of BP ∗ . The particular product of ideals containing a s , the dimension-2 s coefficient, depends directly on the p -adic expansion of s + 1.

The projective dimension of the complex bordism of Eilenberg-MacLane spaces

Let MU*X be the complex bordism of the space X and let MU* be the complex bordism coefficient ring [7]. There is a standard conjecture that the projective dimension of MU*K(Z/(p),n) (horn. dim. MUjfMU*K(ZI(p),ri), [4]) should be n. The conjecture was motivated by its truth for w— 0,1 and by the early establishment of the lower bound horn. dim. MU^MU*K(Z/(p)yn)^n[2, 3, 4]. The purpose of this note is to disprove the conjecture in the strongest possible way. Let/) be a prime and let Z/(p) denote the integers modulo p*.

On a theorem of Ossa

If V is an elementary abelian 2-group, Ossa proved that the connective K-theory of BV splits into copies of Z/2 and of the connective Ktheory of the infinite real projective space. We give a brief proof of Ossas theorem.

On the relation of complex cobordism to connective K-theory

The objective of this note is to complete in one essential way the study undertaken in (2) of the relation between complex bordism and the connective k-homology theory. Specifically, let us denote by MU * ( ) the generalized homology theory associated to the Thorn spectrum MU(6), and by k * ( ) the generalized homology theory associated to the connective bu spectrum (2, 4). Recall that

The Brown-Peterson [2k]-series revisited

Abstract Each coefficient a k , s of the [2 k ]-series for Brown-Peterson homology has a distinguished shortest monomial which is determined by the dyadic expansion of s + 1.