PlanetPhysics/Test OCR

The [[../Cod/|operation]] $v/c$ is bilinear, and it is easy to verify that $(7.2)==\delta v/c=v/\partial c+(-1{)}^{𝔦}\delta (v/c).$ \quad Assume now that $v$ is an equivariant cochain; for ow $ϵ\pi$ we have $\alpha c=\alpha \mathrm{\Sigma }{n}_J{e}_f=\mathrm{\Sigma }(\alpha n_j)(\alpha e_j)$, then $(v/\alpha c)\cdot \sigma =\mathrm{\Sigma }(\alpha n_j)v\cdot (\alpha e_f)\otimes \sigma =\mathrm{\Sigma }(\alpha n_j)\alpha v\cdot (e_j\otimes \sigma )$ $={\alpha }^2\mathrm{\Sigma }{n}_{j}v\cdot (e_f\otimes \sigma )=(v/c)\cdot \sigma .$ 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 $[C_i(W;Z_m^{⟨q)}){]}_{\pi }\approx C_i(Z_m^{(q)}{\otimes }_{\pi }W)$;the [[../Bijective/|relation]] (7.2) holds for this extended operation.

\quad Now take $v={\mathrm{\varnothing }}^{\mathrm{\#}}{u}^n$ and $cϵC_i(Z_m^{1q)}{\otimes }_{\pi }W)$, then $\varphi \mathrm{\#}{u}^{n}fcϵC^{nq-i}(K;Z_m)$ is defined as the reduction by $c$ of the $n^{\mathrm{t}\mathrm{h}}$ [[../Power/|power]] of $u$. Suppose that $u$ is a cocycle, then $\varphi \mathrm{\#}{u}^n$ is an equivariant cocycle, and if $c$ is a cycle, it follows from (7.2) that $\varphi \mathrm{\#}{u}^n/c$ is a cocycle. Moreover, if the cycle $c$ is varied by a [[../GenericityInOpenSystems/|boundary]], then (7.2) implies that $\varphi \mathrm{\#}{u}^n/c$ varies by a co- boundary. If $u$ is varied by a coboundary $\varphi \mathrm{\#}{u}^n/c$ 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 $\{\varphi \mathrm{\#}{u}^n/c\}$ is a [[../Bijective/|function]] of the classes $\{u\},\{c\}$, and it is independent of the particular ${\varphi }_{\mathrm{\#}}$, since by (3.1) any two choices of ${\varphi }_{\mathrm{\#}}$ are equivariantly homotopic. Then Steenrod defines $\{u{\}}^n/\{c\}$, the reduction by $\{c\}$ of the $n^{\mathrm{t}\mathrm{h}}$ power of $\{u\}$, by $\{u{\}}^n/\{c\}=\{\varphi \mathrm{\#}{u}^n/c\}.$ This gives the Steenrod reduced power operations; they are operations defined for $uϵH^q(K;Z_m)$ and $cϵH_i(\pi ;Z_m^{⟨q)})$, and the value is $u^n/cϵH^{nq-i}(K;Z_m).$ \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 $\mathrm{p}\mathrm{r}\mathrm{o}\phi$ 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 $f:K\to L$ be a map and $f^{*}:H^q(L;Z_m)\to H^q(K;Z_m)$, the induced [[../TrivialGroupoid/|homomorphism]]; then $(7.5)==f^{*}(u^n/c)=(f^{*}u{)}^n/c.$ This result implies [[../CoIntersections/|topological]] invariance for reduced powers

OCR based on this tiff scan