test ocr

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

$\displaystyle (7.2)$ $\displaystyle \quad \delta v/c=v/\partial c+(-1)^{\mathfrak{i}}\delta(v/c).
$

    Assume now that $ v$ is an equivariant cochain; for ow $ \epsilon\pi$ we have $ \alpha c=\alpha\Sigma n_{J}e_{f}=\Sigma(\alpha n_{j})(\alpha e_{j})$ , then

$\displaystyle (v/\alpha c)\cdot\sigma=\Sigma(\alpha n_{j})v\cdot(\alpha e_{f})\otimes\sigma=\Sigma(\alpha n_{j})\alpha v\cdot(e_{j}\otimes\sigma)
$

$\displaystyle =\alpha^{2}\Sigma n_{j}v\cdot(e_{f}\otimes\sigma)=(v/c)\cdot\sigma.
$

Thus, in this case,

(7.3) $ v/\alpha c=v/c$ and $ v/(\alpha c-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}^{\langle q)})]_{\pi}\approx C_{i}(Z_{m}^{(q)}\otimes_{\pi}W)$ ;the relation (7.2) holds for this extended operation.

    Now take $ v=\emptyset^{\char93 }u^{n}$ and $ c\epsilon C_{i}(Z_{m}^{1q)}\otimes_{\pi}W)$ , then

$\displaystyle \phi\char93  u^{n}fc\epsilon C^{nq-i}(K;Z_{m})
$

is defined as the reduction by $ c$ of the $ n^{\mathrm{t}\mathrm{h}}$ power of $ u$ . Suppose that $ u$ is a cocycle, then $ \phi\char93  u^{n}$ is an equivariant cocycle, and if $ c$ is a cycle, it follows from (7.2) that $ \phi\char93  u^{n}/c$ is a cocycle. Moreover, if the cycle $ c$ is varied by a boundary, then (7.2) implies that $ \phi\char93  u^{n}/c$ varies by a co- boundary. If $ u$ is varied by a coboundary $ \phi\char93  u^{n}/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 $ \{\phi\char93  u^{n}/c\}$ is a function of the classes $ \{u\}, \{c\}$ , and it is independent of the particular $ \phi_{\char93 }$ , since by (3.1) any two choices of $ \phi_{\char93 }$ are equivariantly homotopic. Then Steenrod defines $ \{u\}^{n}/\{c\}$ , the reduction by $ \{c\}$ of the $ n^{\mathrm{t}\mathrm{h}}$ power of $ \{u\}$ , by

$\displaystyle \{u\}^{n}/\{c\}=\{\phi\char93  u^{n}/c\}.
$

This gives the Steenrod reduced power operations; they are operations defined for $ u\epsilon H^{q}(K;Z_{m})$ and $ c\epsilon H_{i}(\pi;Z_{m}^{\langle q)})$ , and the value is

$\displaystyle u^{n}/c\epsilon H^{nq-i}(K;Z_{m}).
$

    In general, the reduced powers $ u^{n}/c$ are linear operations in $ c$ , but may not be linear in $ u$ . We will list some of their $ \mathrm{p}\mathrm{r}\mathrm{o}\varphi$ rties. Unless otherwise stated, we assume $ u$ and $ c$ as above.

    First, we have

(7.4) $ u^{n}/c=0$ if $ i>nq-q$ .

    Let $ f:K\rightarrow L$ be a map and $ f^{*}: H^{q}(L;Z_{m})\rightarrow H^{q}(K;Z_{m})$ , the induced homomorphism; then

$\displaystyle (7.5)$ $\displaystyle \quad f^{*}(u^{n}/c)=(f^{*}u)^{n}/c.
$

This result implies topological invariance for reduced powers

OCR based on this tiff scan



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