Algebraic & Geometric Topology Volume 21, issue 5 (2021)

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Twisted Morava $K$–theory and connective covers of Lie groups

Hisham Sati and Aliaksandra Yarosh

Algebraic & Geometric Topology 21 (2021) 2223–2255

DOI: 10.2140/agt.2021.21.2223

Bibliography

1	J F Adams, Stable homotopy and generalised homology, Univ. Chicago Press (1974) MR0402720
2	D W Anderson, L Hodgkin, The K–theory of Eilenberg–MacLane complexes, Topology 7 (1968) 317 MR231369
3	M Ando, A J Blumberg, D Gepner, Twists of K–theory and TMF, from: "Superstrings, geometry, topology, and C^∗–algebras" (editors R S Doran, G Friedman, J Rosenberg), Proc. Sympos. Pure Math. 81, Amer. Math. Soc. (2010) 27 MR2681757
4	M Ando, A J Blumberg, D Gepner, M J Hopkins, C Rezk, Units of ring spectra, orientations and Thom spectra via rigid infinite loop space theory, J. Topol. 7 (2014) 1077 MR3286898
5	N Andruskiewitsch, J Devoto, Extensions of Hopf algebras, Algebra i Analiz 7 (1995) 22 MR1334152
6	M F Atiyah, F Hirzebruch, Vector bundles and homogeneous spaces, from: "Differential geometry" (editor C B Allendoerfer), Proc. Sympos. Pure Math. 3, Amer. Math. Soc. (1961) 7 MR0139181
7	M Atiyah, G Segal, Twisted K–theory, Ukr. Mat. Visn. 1 (2004) 287 MR2172633
8	M Atiyah, G Segal, Twisted K–theory and cohomology, from: "Inspired by S S Chern" (editor P A Griffiths), Nankai Tracts Math. 11, World Sci. (2006) 5 MR2307274
9	P Baum, A Carey, B L Wang, K–cycles for twisted K–homology, J. K–Theory 12 (2013) 69 MR3126635
10	A K Bousfield, K–localizations and K–equivalences of infinite loop spaces, Proc. Lond. Math. Soc. 44 (1982) 291 MR647434
11	A K Bousfield, On K(n)–equivalences of spaces, from: "Homotopy invariant algebraic structures" (editors J P Meyer, J Morava, W S Wilson), Contemp. Math. 239, Amer. Math. Soc. (1999) 85 MR1718077
12	P Bouwknegt, A L Carey, V Mathai, M K Murray, D Stevenson, Twisted K–theory and K–theory of bundle gerbes, Comm. Math. Phys. 228 (2002) 17 MR1911247
13	P Bouwknegt, J Evslin, B Jurčo, V Mathai, H Sati, Flux compactifications on projective spaces and the S–duality puzzle, Adv. Theor. Math. Phys. 10 (2006) 345 MR2250274
14	V Braun, Twisted K–theory of Lie groups, J. High Energy Phys. 2004 (2004) MR2061550
15	P Donovan, M Karoubi, Graded Brauer groups and K–theory with local coefficients, Inst. Hautes Études Sci. Publ. Math. 38 (1970) 5 MR282363
16	C L Douglas, On the twisted K–homology of simple Lie groups, Topology 45 (2006) 955 MR2263220
17	A D Elmendorf, I Kriz, M A Mandell, J P May, Rings, modules, and algebras in stable homotopy theory, 47, Amer. Math. Soc. (1997) MR1417719
18	M R Gaberdiel, T Gannon, D–brane charges on non-simply connected groups, J. High Energy Phys. 2004 (2004) MR2080884
19	K Gomi, Clifford modules, finite-dimensional approximation and twisted K–theory, J. Math. Sci. Univ. Tokyo 19 (2012) 587 MR3086748
20	D Grady, H Sati, Spectral sequences in smooth generalized cohomology, Algebr. Geom. Topol. 17 (2017) 2357 MR3686400
21	M J Hopkins, N J Kuhn, D C Ravenel, Morava K–theories of classifying spaces and generalized characters for finite groups, from: "Algebraic topology" (editors J Aguadé, M Castellet, F R Cohen), Lecture Notes in Math. 1509, Springer (1992) 186 MR1185970
22	M Hopkins, J Lurie, Ambidexterity in K(n)–local stable homotopy theory, preprint (2013)
23	M J Hopkins, D C Ravenel, W S Wilson, Morava Hopf algebras and spaces K(n) equivalent to finite Postnikov systems, from: "Stable and unstable homotopy" (editors W G Dwyer, S Halperin, R Kane, S O Kochman, M E Mahowald, P S Selick), Fields Inst. Commun. 19, Amer. Math. Soc. (1998) 137 MR1622344
24	D C Johnson, W S Wilson, BP operations and Morava’s extraordinary K–theories, Math. Z. 144 (1975) 55 MR377856
25	D C Johnson, W S Wilson, The Brown–Peterson homology of elementary p–groups, Amer. J. Math. 107 (1985) 427 MR784291
26	M Khorami, A universal coefficient theorem for twisted K–theory, J. Topol. 4 (2011) 535 MR2832567
27	N Kitchloo, G Laures, W S Wilson, The Morava K–theory of spaces related to BO, Adv. Math. 189 (2004) 192 MR2093483
28	N Kitchloo, G Laures, W S Wilson, Splittings of bicommutative Hopf algebras, J. Pure Appl. Algebra 194 (2004) 159 MR2086079
29	I Kriz, Morava K–theory of classifying spaces : some calculations, Topology 36 (1997) 1247 MR1452850
30	I Kriz, H Sati, M–theory, type IIA superstrings, and elliptic cohomology, Adv. Theor. Math. Phys. 8 (2004) 345 MR2086925
31	J Maldacena, G Moore, N Seiberg, D–brane instantons and K–theory charges, J. High Energy Phys. 2001 (2001) MR1877986
32	V Mathai, J Rosenberg, Group dualities, T–dualities, and twisted K–theory, J. Lond. Math. Soc. 97 (2018) 1 MR3764065
33	E Meinrenken, Twisted K–homology and group-valued moment maps, Int. Math. Res. Not. 2012 (2012) 4563 MR2989614
34	J W Milnor, J C Moore, On the structure of Hopf algebras, Ann. of Math. 81 (1965) 211 MR174052
35	T Nishimoto, Higher torsion in the Morava K–theory of SO(m) and Spin(m), J. Math. Soc. Japan 53 (2001) 383 MR1815140
36	V K Rao, On the Morava K–theories of SO(2n + 1), Proc. Amer. Math. Soc. 108 (1990) 1031 MR994784
37	V K Rao, Towards the algebra structure of the Morava K–theory of the orthogonal groups, Manuscripta Math. 94 (1997) 287 MR1485437
38	D C Ravenel, Complex cobordism and stable homotopy groups of spheres, 121, Academic (1986) MR860042
39	D C Ravenel, W S Wilson, The Morava K–theories of Eilenberg–MacLane spaces and the Conner–Floyd conjecture, Amer. J. Math. 102 (1980) 691 MR584466
40	D C Ravenel, W S Wilson, N Yagita, Brown–Peterson cohomology from Morava K–theory, K–Theory 15 (1998) 147 MR1648284
41	J Rosenberg, Homological invariants of extensions of C^∗–algebras, from: "Operator algebras and applications, I" (editor R V Kadison), Proc. Sympos. Pure Math. 38, Amer. Math. Soc. (1982) 35 MR679694
42	J Rosenberg, Continuous-trace algebras from the bundle theoretic point of view, J. Austral. Math. Soc. Ser. A 47 (1989) 368 MR1018964
43	Y B Rudyak, On Thom spectra, orientability, and cobordism, Springer (1998) MR1627486
44	H Sati, A higher twist in string theory, J. Geom. Phys. 59 (2009) 369 MR2501747
45	H Sati, Ninebrane structures, Int. J. Geom. Methods Mod. Phys. 12 (2015) MR3332238
46	H Sati, U Schreiber, J Stasheff, Fivebrane structures, Rev. Math. Phys. 21 (2009) 1197 MR2588823
47	H Sati, U Schreiber, J Stasheff, Twisted differential string and fivebrane structures, Comm. Math. Phys. 315 (2012) 169 MR2966944
48	H Sati, H b Shim, String structures associated to indefinite Lie groups, J. Geom. Phys. 140 (2019) 246 MR3924563
49	H Sati, C Westerland, Twisted Morava K–theory and E–theory, J. Topol. 8 (2015) 887 MR3431663
50	H Sati, M Wheeler, Variations of rational higher tangential structures, J. Geom. Phys. 130 (2018) 229 MR3807127
51	S Stolz, A conjecture concerning positive Ricci curvature and the Witten genus, Math. Ann. 304 (1996) 785 MR1380455
52	R E Stong, Determination of H^∗(BO(k,…,∞),Z₂) and H^∗(BU(k,…,∞),Z₂), Trans. Amer. Math. Soc. 107 (1963) 526 MR151963
53	M E Sweedler, Hopf algebras, Benjamin (1969) MR0252485
54	R G Underwood, An introduction to Hopf algebras, Springer (2011) MR2986663
55	B L Wang, Geometric cycles, index theory and twisted K–homology, J. Noncommut. Geom. 2 (2008) 497 MR2438341
56	W S Wilson, Brown–Peterson homology : an introduction and sampler, 48, Amer. Math. Sci. (1982) MR655040
57	U Würgler, Morava K–theories : a survey, from: "Algebraic topology" (editors S Jackowski, B Oliver, K Pawałowski), Lecture Notes in Math. 1474, Springer (1991) 111 MR1133896
58	A Yarosh, Computations in twisted Morava K–theory, PhD thesis, University of Pittsburgh (2017)