Kappa classes (Ex)
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# 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. | # 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. | ||
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== References == | == References == |
Latest revision as of 01:17, 26 August 2013
Let be the canonical map and let be defined as in [Madsen&Milgram1979, Theorem 4.9]. In addition let be the normal invariant of a degree one normal map from the Kervaire manifold to the -sphere.
Exercise 0.1.
- Show that is the generator.
- Show that there is a framed manifold of Kervaire invariant one in dimension if and only if there is a map such that is a generator.
- Show that there is a framed manifold of Kervaire invariant one in dimension if and only if there is a closed framed -manifold and a map such that is a generator.
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