The Category of H-modules over a Spectrum 7

Further H-pairings are obtained by induction and the usual limit procedure.

Let M be one of the Thorn spectra MO, MSO,MU, MSU, or MS . In

P

section 8 we show that M is a convergent H-module over itself with an

H-pairing P: (M,M)-» • M. Consider M = MO. Let £ be a universal

q

0(q)-bundle with fibre R over a locally finite CW complex BO(q); let

M = T(£ ),the Thorn space of £ ; and let £ = £ x ••• x £ . we define

q q * q R r±

srk

M(R,q) to be the subspace of Map(£ x£ ,£

v

) consisting of all

R q q+LR

0 (q+ZR)-bundle maps. We define e: M(R,q) - Map^(M AM ,M „ ) by

0 R q q+LR

e(F) = T(F), the map of the Thorn spaces induced by the 0(q+lR)-bundle map

F. The maps c, a', and W are defined using appropriate compositions

r,s

of bundle maps. The other Thorn spectra are treated similarly.

Section 9 shows that N(f ,»-«,f ) is H-equivalent to

N(f ,_. ,"-,f , .) , where a e S (n), and then extends this result, in a weaker

a(l) a(n)

form, to the limit H-module N(f°°).

The previous results are applied in section 10 to MU to obtain a tower

of homology theories and natural transformations:

MU^( )-••••-- MUn+l^( ) • MUn^( )-•••- MU0^( ) = H^( ;Z) .

These homology theories have the nice properties of the homology theories

constructed by Baas [1] using ideas of Sullivan. In particular,

Mn (S°) = Z[x_,--*,x_ ].

* 2 2n

This paper is a revision of the author's doctoral dissertation written

at the University of Virginia under the direction of Professor E. E. Floyd.

The author is grateful to Professor Floyd for his most helpful advice.