test ocr
The operation
is bilinear, and it is easy to verify that
Assume now that
is an equivariant cochain; for ow
we have
, then
Thus, in this case,
(7.3)
and
.
Consequently, the definition of
extends to the case of
, an equi- variant cochain, and
an element of
;the relation (7.2) holds for this extended operation.
Now take
and
, then
is defined as the reduction by
of the
power
of
. Suppose that
is a cocycle, then
is an equivariant cocycle, and if
is a cycle, it follows from (7.2) that
is a cocycle. Moreover, if the cycle
is varied by a boundary, then (7.2) implies that
varies by a co- boundary. If
is varied by a coboundary
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
is a function of the classes
, and it is independent of the particular
, since by (3.1) any two choices of
are equivariantly homotopic. Then Steenrod defines
, the reduction by
of the
power of
, by
This gives the Steenrod reduced power operations; they are operations defined for
and
, and the value is
In general, the reduced powers
are linear operations in
, but may not be linear in
. We will list some of their
rties. Unless otherwise stated, we assume
and
as above.
First, we have
(7.4)
if
.
Let
be a map and
, the induced homomorphism; then
This result implies topological invariance for reduced powers
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