PlanetPhysics/Test OCR

The operation $$v/c$$ is bilinear, and it is easy to verify that $$ (7.2)= = \quad \delta v/c=v/\partial c+(-1)^{\mathfrak{i}}\delta(v/c). $$ \quad 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 $$ (v/\alpha c)\cdot\sigma=\Sigma(\alpha n_{j})v\cdot(\alpha e_{f})\otimes\sigma=\Sigma(\alpha n_{j})\alpha v\cdot(e_{j}\otimes\sigma) $$ $$ =\alpha^{2}\Sigma n_{j}v\cdot(e_{f}\otimes\sigma)=(v/c)\cdot\sigma. $$ Thus, in this case,

\noindent (7.3) $$v/\alpha c=v/c$$ and $$v/(\alpha c-c)=0$$.

\noindent 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.

\quad Now take $$v=\emptyset^{\#}u^{n}$$ and $$c\epsilon C_{i}(Z_{m}^{1q)}\otimes_{\pi}W)$$, then $$ \phi\# u^{n}fc\epsilon C^{nq-i}(K;Z_{m}) $$ is defined as the reduction by $$c$$ of the $$n^{\mathrm{t}\mathrm{h}}$$ power of $$u$$. Suppose that $$u$$ is a cocycle, then $$\phi\# u^{n}$$ is an equivariant cocycle, and if $$c$$ is a cycle, it follows from (7.2) that $$\phi\# u^{n}/c$$ is a cocycle. Moreover, if the cycle $$c$$ is varied by a boundary, then (7.2) implies that $$\phi\# u^{n}/c$$ varies by a co- boundary. If $$u$$ is varied by a coboundary $$\phi\# u^{n}/c$$ 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 $$\{\phi\# u^{n}/c\}$$ is a function of the classes $$\{u\}, \{c\}$$, and it is independent of the particular $$\phi_{\#}$$, since by (3.1) any two choices of $$\phi_{\#}$$ are equivariantly homotopic. Then Steenrod defines $$\{u\}^{n}/\{c\}$$, the reduction by $$\{c\}$$ of the $$n^{\mathrm{t}\mathrm{h}}$$ power of $$\{u\}$$, by $$ \{u\}^{n}/\{c\}=\{\phi\# u^{n}/c\}. $$ 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 $$ u^{n}/c\epsilon H^{nq-i}(K;Z_{m}). $$ \quad 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.

\quad First, we have

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

\quad 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 $$ (7.5)= = \quad f^{*}(u^{n}/c)=(f^{*}u)^{n}/c. $$ This result implies topological invariance for reduced powers

OCR based on this tiff scan