The Category of H-modules over a Spectrum 5

defined. For each (R,q), there is a space F{R,q) all of whose homotopy

groups vanish and a map e: F(R,q) - Map (M A A ,B ) . There are maps

U R q q+LR

c, a', and W similar to those above. The category of H-modules over

r, s

M has the H-modules over M as objects and equivalence classes of words

[F I •••IF,]: A - B as morphisms, where F.: C. , - C. is an H-map, C^ = A,

n1 ' 1 1 l-l

I

^' 0

and C = B. We prove

2.12. Proposition. There is a functor from the category of H-modules

over M to the category of TT^(M)-modules and TT^(M)-homomorphisms which

assigns TT^(A) to A and f ^ : TT^(A) - TT^(B) to f: A - B.

In section 3 we construct a mapping cylinder and a mapping cone for a

morphism f: A - B. The constructions are far from unique. We define an

equivalence relation, H-equivalence, and prove that the mapping cylinder and

mapping cone are unique up to H-equivalence.

Let x e T T (M) be represented by f: S - M . For an H-module N,

p v

p+v p+v

there are an H-module S A N , an H-map F: S A N - N, and hence a

mapping cone which we denote by N(f). In particular, N(f) is the

mapping cone of the composition

p+v fAl v,q

si^

A N y

M A N '-^ * N

q v q q+v

where N e N(v,q). This kind of mapping cone is especially interesting

v,q

since the process can be repeated. Furthermore, if N is a convergent

H-module, TT^(N) and ir^(N(f)) are related. N is a convergent H-module if

N(R,q) contains only one point whenever q _ 0 or r. _ 0 for some i,

and if both M and N are convergent spectra. We prove that if N is a

convergent H-module over M, and if p _ 0 and v _ 0, then N(f) is a