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
Previous Page Next Page