The operation v∕c is bilinear, and it is easy to verify that

Assume now that v is an equivariant cochain; for ow 𝜖π we have αc = αΣn_Je_f = Σ(αn_j)(αe_j), then

Thus, in this case,

(7.3) v∕αc = v∕c and v∕(αc − c) = 0.

Consequently, the definition of v∕c extends to the case of v, an equi- variant cochain, and c an element of [C_i(W; Z_m^⟨q))] _π ≈ C_i(Z_m^(q) ⊗ _πW);the relation (7.2) holds for this extended operation.

Now take v = ∅^#uⁿ and c𝜖C _i(Z_m^1q) ⊗ _πW), then

is defined as the reduction by c of the n^th power of u. Suppose that u is a cocycle, then ϕ#uⁿ is an equivariant cocycle, and if c is a cycle, it follows from (7.2) that ϕ#uⁿ∕c is a cocycle. Moreover, if the cycle c is varied by a boundary, then (7.2) implies that ϕ#uⁿ∕c varies by a co- boundary. If u is varied by a coboundary ϕ#uⁿ∕c also varies by a coboundary. We only remark here that the proof of this last fact requires a special argument and is not, as in the preceding case, an immediate consequence of (7.2). Thus the class {ϕ#uⁿ∕c} is a function of the classes {u},{c}, and it is independent of the particular ϕ_#, since by (3.1) any two choices of ϕ_# are equivariantly homotopic. Then Steenrod defines {u}ⁿ∕{c}, the reduction by {c} of the n^th power of {u}, by

This gives the Steenrod reduced power operations; they are operations defined for u𝜖H^q(K; Z _m) and c𝜖H_i(π; Z_m^⟨q)), and the value is

In general, the reduced powers uⁿ∕c are linear operations in c, but may not be linear in u. We will list some of their proφ rties. Unless otherwise stated, we assume u and c as above.

First, we have

(7.4) uⁿ∕c = 0 if i > nq − q.

Let f : K → L be a map and f^∗ : H^q(L; Z _m) → H^q(K; Z _m), the induced homomorphism; then

This result implies topological invariance for reduced powers

OCR based on this tiff scan