The Category of H-modules over a Spectrum

Introduction.

Let H^( ;M) denote the generalized homology theory associated with the

spectrum M. (See [9].) Let MU denote the Thorn spectrum associated with

weakly complex bordism. TT^(MU) , the coefficient ring of H^( ;MU) , is a

graded polynomial algebra over the integers Z with generators {x,x ,•••},

x_ in dimension 2n. In the effort to obtain results about EL(X;MU) ,

2n *

related homology theories whose coefficient rings are polynomial algebras

over Z with generators {x_,x„,•••,x_ } have been considered. In [1],

z 4 2n

Baas proved the existence of such homology theories using manifolds with

singularities, an idea due to Sullivan [8]. A theorem of Brown asserts the

existence of representing spectra for these homology theories. This paper

arose from an attcrpt to construct such spectra from MU using only the

properties of I1U.

One of the most important properties of MU is that it is both a ring

spectrum and a module over itself. Among ether things this means there is

a pairing w: (MU.MU) - MU whose maps w : MU A MU -^MU satisfy

^ ^ r r,q r q r+q

certain homotopy commutativity properties. Given x e T T (MU) , represent x

P

by f: S - M . One can construct a spectrum MU(f) by letting MU(f)

J

v q+v

be the mapping cone of the composition

w

p+v fAl v.q

s^

A M

,— y M A M '-*—+ M

q V q V+q

For any CW complex X there is an exact triangle

H^(X;MU)

-X—

H^(X;MU)

H^(X;MU(f))

where i is induced by an inclusion, and 3 is of degree -(p+1). If one

Received by the editors May 31, 1973.

1