Question:Let $G$ be a finite abelian group and $X$ be a G-set. $K$ be a subgroup of $G$. let $i$ be a group homomorphism from $K$ to $G$ . I am looking for the map $$ i^{*} : H^{\alpha}_{G}(X,M) \rightarrow H^{i^{*}\alpha}_{K}(i^*X,i^*M).$$How can I compute this map $i^{*}$ for a constant Mackey functor $M$? Where $\alpha$ belongs to $RO(G)$ and $i^{*}X = X$ with $K$ - action induced by $G$. Note: For constant $G$- Mackecy fuctor $M$, $i^{*}M$ is $M$ as constant Mackey functor of $K$. Can anyone give some hint atleast for $X= point$.
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$\begingroup$ I don't think we even know $H_G^*(*;M)$ for all finite abelian groups, in $RO(G)$ grading. However, the calculations I've seen are Mackey-functor valued, so include all the restriction maps and transfers. $\endgroup$– Steve CostenobleCommented Mar 19, 2015 at 19:15
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$\begingroup$ @SteveCostenoble: Thank you very much for your comment.I am interested in $ G = Z/6 $, $ M = Z/6$-constant Mackey functor and $\alpha = \xi^2 - \xi$, where $\xi$- is nontrivial irreducible representation of $Z/6$ with out fixed point.Is it possible to find $i^*$ in this setting? $\endgroup$– Surojit GhoshCommented Mar 20, 2015 at 13:37
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$\begingroup$ Unfortunately, the only computations I know of are for $G = {\mathbb Z}/p$ for $p$ prime. (Look for papers by L. Gaunce Lewis, Jeff Caruso, and William Kronholm.) I think I saw a paper also looking at prime power order, but can't seem to find it. $\endgroup$– Steve CostenobleCommented Mar 20, 2015 at 16:59
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$\begingroup$ @SteveCostenoble : Thank you for your valuable comment.I had read most of the papers of Gaunce Lewis and Jeff Caruso(just One paper).But I never saw any paper on the calculation $H^*_{G}(*,M)$ for $|G| = p^n$.It will be nice if you can find it for me.I have another question that'll help me for the above calculation.Question:What is $Z/6$-complex structure of $S(\xi + \xi^2 + 2\sigma)$ ? Where $\xi$ ,$\xi^2$ are defined in above comment and $\sigma$ is 1 real dimensional sign representation of $Z/6$. $\endgroup$– Surojit GhoshCommented Mar 22, 2015 at 1:23
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$\begingroup$ I must be misremembering, as I can't find any calculations now for groups other than ${\mathbb Z}/p$. I don't know the structure on $S(\xi+\xi^2+2\sigma)$ offhand, but if I wanted to work it out I would first put structures on $D(\xi)$, $D(\xi^2)$, and $D(2\sigma)$, then take their product and look at what the structure on the boundary is. $\endgroup$– Steve CostenobleCommented Mar 23, 2015 at 15:55
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