Transfer homomorphisms with coefficients - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-21T21:08:26Z http://mathoverflow.net/feeds/question/45670 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/45670/transfer-homomorphisms-with-coefficients Transfer homomorphisms with coefficients Oscar Randal-Williams 2010-11-11T10:00:43Z 2010-12-24T04:06:01Z <p>In group cohomology, for $H$ a finite-index subgroup of $G$ and $M$ a $G$-module, there is a transfer (or corestriction) map $Cor : H^* (H;M) \to H^*(G;M)$.</p> <p>In homotopy theory, there is a transfer map for finite covering spaces $\bar{X} \to X$, and it exists for all coefficient systems on $X$. It is given on homology (say) by sending a small simplex in $X$ to its finitely-many lifts to $\bar{X}$. The group transfer is obtained from the topological one by the finite covering space $BH \to BG$.</p> <p>There is a fancier transfer, due to Becker and Gottlieb, for any map $E \to B$ whose homotopy fibre is stably equivalent to a finite complex. Can this be extended to give a transfer map for coefficient systems on $B$?</p> http://mathoverflow.net/questions/45670/transfer-homomorphisms-with-coefficients/45724#45724 Answer by Charles Rezk for Transfer homomorphisms with coefficients Charles Rezk 2010-11-11T16:46:36Z 2010-11-11T16:46:36Z <p>I believe the answer is yes. The idea should be to generalize the construction of the group-theoretic transfer. I'll describe how I think this ought to go. I'm pretty sure what I'm describing is a known construction, but I haven't been able to dig up a reference which describes it.</p> <p>Let $f:X\to Y$ be a map of spaces (CW-complexes), and assume (for simplicity) that both are path connected. Then there is an inclusion of topological groups $\phi: G\to H$, such that $B\phi: BG\to BH$ is equivalent to $f:X\to Y$.</p> <p>A <em>local system</em> on $X$, in the most general possible context, is a spectrum $M$ equipped with a $G$-action. (I'm not doing anything fancy here; in particular, I'm not doing equivariant stable homotopy theory. Just spectra with a $G$-action; a $G$ map $f:M\to N$ of such things is a weak equivalence if it is a weak equivalence of the underlying spectra.) </p> <p>Let $S_G$ be the category of $G$-spectra. It'll be important to note that this is a closed monoidal category: the smash product $X\wedge Y$ of two objects in $S_G$ is defined in $S_G$ (smash the spectra, and use the diagonal $G$-action), as well as a function object $\mathrm{Hom}(X,Y)$ (take the spectrum function object, with diagonal $G$-action).</p> <p>The Becker Gottlieb transfer $f_!$ of $f$ associated to an $M\in S_H$ should be a map $M_{hH}\to M_{hG}$ (where "$hG$" is "homotopy orbits"). So, if $M=S^0$ with trivial $H$-action, you get a map $S^0_{hH}\to S^0_{hG}$, which is a map $\Sigma^\infty Y_+\to \Sigma^\infty X_+$. You should interpret the map $\pi_* f_!$ as being a map <code>$$H_*(Y,M) \to H_*(X,f^*M),$$</code> where these are "homology with coefficients in the local systems $M$ and $f^*M$". There should be a similar way to get a cohomolgy transfer. </p> <p>Let $F=\Sigma^\infty(H/G)_+$, as a spectrum with $H$-action. Note that the space $H/G$ is just the fiber of $f:X\to Y$. </p> <p>Here is a sequence of maps in $S_H$: <code>$$M\to \mathrm{Hom}(F,F\wedge M) \leftarrow \mathrm{Hom}(F,M)\wedge F \to \mathrm{Hom}(F,M)\wedge F\wedge F \to M\wedge F \approx M\wedge_G H.$$</code> The first is smashing with the identity map of $F$; the second comes from smashing a map $F\to M$ with a map $S^0\to F$; the third comes from the diagonal map $H/G\to H/G\times H/G$; the fourth is "evaluation".</p> <p>If $F$ has the homotopy type of a <em>finite</em> CW-spectra (i.e., if $H/G$ is finitely dominated), then the backwards map in this sequence is a weak equivalence. In this case, we get a map $M\to M\wedge_G H$ of $H$-spectra, and taking homotopy orbits gives $M_{hH}\to M_{hG}$.</p> <p>If $Y$ is contractible, this amounts to a map <code>$$S^0 \to \mathrm{Hom}(F,F) \leftarrow \mathrm{Hom}(F, S^0)\wedge F \to \mathrm{Hom}(F, S^0)\wedge F\wedge F \to S^0\wedge F,$$</code> which is to say, a map $S^0\to \Sigma^\infty X_+$, and in cohomology this will send $1\in H^0(X)$ to $\chi(X)\in H^0(\mathrm{point})$. So apparently we recover the Becker-Gottlieb transfer.</p> <p>If the spaces aren't connected, you can still do all this, but you need to allow $H$ and $G$ to be groupoids.</p> <p>(I learned this way of thinking from some paper of John Klein about the "dualizing spectrum" of a toplogical group; I can't find one where he addresses the BG transfer this way.)</p> http://mathoverflow.net/questions/45670/transfer-homomorphisms-with-coefficients/50269#50269 Answer by John Klein for Transfer homomorphisms with coefficients John Klein 2010-12-24T04:06:01Z 2010-12-24T04:06:01Z <p>I think what Charles has written up above is closely related to what Becker and Gottlieb did in</p> <p>Becker, J. C.; Gottlieb, D. H. Transfer maps for fibrations and duality. Compositio Math. 33 (1976), no. 2, 107–133. </p> <p>They work fiberwise, but that is equivalent to working equivariantly in the sense that a $G$-space $X$ gives rise to a fibration $X \times_G EG \to BG$ (Borel construction; this induces a Quillen equivalence). </p>