Kappa classes (Ex)

Latest revision as of 01:17, 26 August 2013

Let $<wikitex>; Let $i \colon SG \to G/TOP$ be the canonical map and let $\kappa_{4k-2} \in H^{4k-2}(G/PL; \Zz/2)$ be defined as in \cite{Madsen&Milgram1979|Theorem 4.9}. In addition let $\varphi_K \colon S^{4k-2} \to G/PL$ be the normal invariant of a degree one normal map $(f, b) \colon M_K \to S^{4k-2}$ from the Kervaire manifold to the $(4k-2)$-sphere. {{beginthm|Exercise}} # Show that $\phi_K^*(\kappa_{4k-2}) \in H^{4k-2}(S^{4k-2}; \Zz/2)$ is the generator. # Show that there is a framed manifold of Kervaire invariant one in dimension $(4k-2)$ if and only if there is a map $\varphi \colon S^{4k-2} \to SG$ such that $\phi^*(\kappa_{4k-2}) \in H^{4k-2}(M; \Zz/2)$ is a generator. # Show that there is a framed manifold of Kervaire invariant one in dimension $(4k-2)$ if and only if there is a closed framed $(4k-2)$-manifold $M$ and a map $\varphi \colon M \to SG$ such that $\phi^*(\kappa_{4k-2}) \in H^{4k-2}(M; \Zz/2)$ is a generator. {{endthm}} </wikitex> == References == {{#RefList:}} [[Category:Exercises]] [[Category:Exercises without solution]]i \colon SG \to G/TOP$ be the canonical map and let $\kappa_{4k-2} \in H^{4k-2}(G/PL; \Zz/2)$ be defined as in [Madsen&Milgram1979, Theorem 4.9]. In addition let $\varphi_K \colon S^{4k-2} \to G/PL$ be the normal invariant of a degree one normal map $(f, b) \colon M_K \to S^{4k-2}$ from the Kervaire manifold to the $(4k-2)$-sphere.

[edit] References

• [Madsen&Milgram1979] I. Madsen and R. J. Milgram, The classifying spaces for surgery and cobordism of manifolds, Princeton University Press, Princeton, N.J., 1979. MR548575 (81b:57014) Zbl 0446.57002