Talk:PlanetPhysics/Test OCR

Original TeX Content from PlanetPhysics Archive

%%% This file is part of PlanetPhysics snapshot of 2011-09-01 %%% Primary Title: test ocr %%% Primary Category Code: 00. %%% Filename: TestOcr.tex %%% Version: 2 %%% Owner: bloftin %%% Author(s): bloftin %%% PlanetPhysics is released under the GNU Free Documentation License. %%% You should have received a file called fdl.txt along with this file. %%% If not, please write to gnu@gnu.org. \documentclass[12pt]{article} \pagestyle{empty} \setlength{\paperwidth}{8.5in} \setlength{\paperheight}{11in}

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\begin{document}

The \htmladdnormallink{operation}{http://planetphysics.us/encyclopedia/Cod.html} $v/c$ is bilinear, and it is easy to verify that

$$ (7.2)\text{ }\quad \delta v/c=v/\partial c+(-1)^{\mathfrak{i}}\delta(v/c). $$ \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 \htmladdnormallink{relation}{http://planetphysics.us/encyclopedia/Bijective.html} (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 $n^{\mathrm{t}\mathrm{h}}$ \htmladdnormallink{power}{http://planetphysics.us/encyclopedia/Power.html} 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 \htmladdnormallink{boundary}{http://planetphysics.us/encyclopedia/GenericityInOpenSystems.html}, 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 \htmladdnormallink{function}{http://planetphysics.us/encyclopedia/Bijective.html} 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 \htmladdnormallink{homomorphism}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html}; then $$ (7.5)\text{ }\quad f^{*}(u^{n}/c)=(f^{*}u)^{n}/c. $$ This result implies \htmladdnormallink{topological}{http://planetphysics.us/encyclopedia/CoIntersections.html} invariance for reduced powers

\htmladdnormallink{OCR based on this tiff scan}{http://aux.planetphysics.us/files/objects/502/algebraicgeometr032092mbp_0216.tif}

\end{document}