zbMATH — the first resource for mathematics

Transchromatic twisted character maps. (English) Zbl 1318.55011
In earlier work [N. Stapleton, Algebr. Geom. Topol. 13, No. 1, 171–203 (2013; Zbl 1300.55011)], the author constructed a transchromatic generalized character map for the Morava \(E\)-theories \(E_n\). It takes values in a height \(t\) cohomology theory where \(0 \leq t \leq n\). For \(t=0\), his construction recovers the generalized character map introduced in influential work by Hopkins, Kuhn, and Ravenel [M. J. Hopkins et al., J. Am. Math. Soc. 13, No. 3, 553–594 (2000; Zbl 1007.55004)].
In the paper under review, the author constructs a twisted transchromatic generalized character map \[ E_n^*(EG \times_G X) \to B_t^*\otimes_{L_{K(t)} E_n^*(B \mathbb Q_p/\mathbb Z_p^{n-t})}L_{K(t)}E_n^*({\text{Twist}}_{n-t}(X)) \] that refines (a completed version of) the transchromatic generalized character map from his earlier work. Here \(L_{K(t)}\) is the localization with respect to height \(t\) Morava \(K\)-theory and \(B_t\) is a certain universal \(L_{K(t)} E_n^0\)-algebra over which the formal group associated to \(E_n\) is a non-trivial extension of a height \(t\) formal group by a height \(n-t\) constant étale \(p\)-divisible group. The space \({\text{Twist}}_{n-t}(X)\) is a variant of the target space for the earlier generalized character maps that takes torus actions into account. The main theorem states that for a finite group \(G\), the twisted transchromatic generalized character map induces an isomorphism when tensored up with \(B_t\).

55P42 Stable homotopy theory, spectra
55N91 Equivariant homology and cohomology in algebraic topology
55P48 Loop space machines and operads in algebraic topology
Full Text: DOI arXiv
[1] Ando, M., Morava, J.: A renormalized Riemann-Roch formula and the Thom isomorphism for the free loop space. In: Topology, geometry, and algebra: interactions and new directions (Stanford, CA, 1999). Contemp. Math., vol. 279, pp. 11-36. Amer. Math. Soc., Providence (2001) · Zbl 1009.57042
[2] Carchedi, D, Compactly generated stacks: a Cartesian closed theory of topological stacks, Adv. Math., 229, 3339-3397, (2012) · Zbl 1254.14003
[3] Carlsson, G; Douglas, CL; Dundas, BI, Higher topological cyclic homology and the Segal conjecture for tori, Adv. Math., 226, 1823-1874, (2011) · Zbl 1223.55004
[4] Ganter, N.: Stringy power operations in tate, k-theory (2013, submitted) · Zbl 1277.19003
[5] Hopkins, MJ; Kuhn, NJ; Ravenel, DC, Generalized group characters and complex oriented cohomology theories, J. Am. Math. Soc., 13, 553-594, (2000) · Zbl 1007.55004
[6] Rezk, C.: Fibrations and homotopy colimits of simplicial sheaves. arXiv:math/9811038 [math.AT] · Zbl 1254.14003
[7] Segal, G, Categories and cohomology theories, Topology, 13, 293-312, (1974) · Zbl 0284.55016
[8] Stapleton, N.J.: Transchromatic generalized character maps. Algebr. Geom. Topol. (2013, to appear) · Zbl 1300.55011
[9] Tate, J.T.: p-divisible groups. In Proc. Conf. Local Fields (Driebergen, 1966), pp. 158-183. Springer, Berlin (1967) · Zbl 0157.27601
[10] Witten, E, Elliptic genera and quantum field theory, Commun. Math. Phys., 109, 525-536, (1987) · Zbl 0625.57008
[11] Witten, E.: The index of the Dirac operator in loop space. In Elliptic curves and modular forms in algebraic topology (Princeton, NJ, 1986). Lecture Notes in Math., vol. 1326, pp. 161-181. Springer, Berlin (1988) · Zbl 1223.55004
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.