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Spectral sequences in smooth generalized cohomology. (English) Zbl 1375.55002
Let \(\text{Sh}_\infty({\mathcal C}\text{art}{\mathcal S}\text{p})_+\) be the category of pointed smooth stacks over \({\mathcal C}\text{art}{\mathcal S}\text{p}\), the category of Cartesian spaces, and let \({\mathcal A}\text{b}_{\text{gr}}\) be the category of graded abelian groups. A smooth cohomology theory is a functor \[ {\mathcal E}^\ast : \text{Sh}_\infty({\mathcal C}\text{art}{\mathcal S}\text{p})_+^{\text{op}} \to {\mathcal A}\text{b}_{\text{gr}} \] satisfying four axioms: invariance, additivity, Mayer-Vietoris and suspension. Herein the homotopy axiom is dropped since it does not necessarily yield an equivalence of stacks, while on the other hand an equivalence of stacks induces a weak homotopy equivalence of geometric realizations. Let \({\mathcal E}\) denote a smooth spectrum representing \({\mathcal E}^\ast\) which consists of pointed stacks. Then we can define \({\mathcal E}^\ast(M)\) for any manifold \(M\) by considering the category of manifolds as an \(\infty\)-subcategory of \(\text{Sh}_\infty({\mathcal C}\text{art}{\mathcal S}\text{p})\). The authors prove (Theorem 25) that there is a spectral sequence with \[ E_2^{p, q}=H^p(M, {\mathcal E}^q) \Rightarrow {\mathcal E}^{p+q}(M) \] where \(H^p\) denotes the \(p\)th Čech cohomology with coefficients in the presheaf \({\mathcal E}^q\). The proof can be done in the usual way using a filtration by the Čech resolution with respect to an open covering of \(M\). In this paper the authors mainly consider the spectral sequence for the differential refinement \({\hat{\mathcal E}}^\ast\) of \({\mathcal E}^\ast\) with the intention of comparing with that for \({\mathcal E}^\ast\). The differential function spectrum representing \({\hat{\mathcal E}}^\ast\) is written here \(\text{diff}({\mathcal E}, \text{ch})\). Then we have a natural map \(I : \text{diff}({\mathcal E}, \text{ch}) \to \underline{\mathcal E}\) where \(\underline{\mathcal E}\) denotes the presheaf of spectra \({\mathcal E}\) which is equivalent to the underlying theory of \({\mathcal E}\) as smooth spectra. It is shown that this map \(I\) induces a morphism between the corresponding spectral sequences. The use of this morphism enables us to compare the differentials in those two spectral sequences; however, as a result, it turns out that this does not work well. For this reason the authors try to look for a different map and obtain (Theorem 34) that under certain conditions there is a map between the lower quadrants of the two spectral sequences corresponding to \(\text{diff}({\mathcal E}, \text{ch})\) and \({\mathcal E}\). This morphism allows us to carry out what we want to do above. The final section (Section 4) is devoted to examining the spectral sequences for the following three differential cohomology theories: smooth Deligne cohomology, differential complex \(K\)-theory and differential Morava \(K\)-theory where the last one is a smooth extension of integral Morava \(K\)-theory which has been introduced by the authors.

MSC:
55N15 Topological \(K\)-theory
55T10 Serre spectral sequences
55T25 Generalized cohomology and spectral sequences in algebraic topology
53C05 Connections (general theory)
55S05 Primary cohomology operations in algebraic topology
55S35 Obstruction theory in algebraic topology
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References:
[1] 10.1007/BF00966117 · Zbl 0768.55012
[2] ; Atiyah, Differential Geometry. Proc. Sympos. Pure Math., 3, 7, (1961)
[3] 10.1016/0040-9383(62)90094-0 · Zbl 0108.36401
[4] 10.1017/CBO9780511526398.005
[5] ; Björner, Handbook of combinatorics, II, 1819, (1995)
[6] 10.1007/978-1-4757-3951-0
[7] 10.1016/0040-9383(79)90018-1 · Zbl 0417.55007
[8] 10.1007/s00220-007-0396-y · Zbl 1162.58008
[9] 10.2307/1996573
[10] 10.1007/978-0-8176-4731-5
[11] 10.5802/ambp.276 · Zbl 1200.55007
[12] 10.1007/s40062-014-0092-5 · Zbl 1341.57020
[13] ; Bunke, 328, 45, (2009)
[14] 10.1112/jtopol/jtq002 · Zbl 1252.55002
[15] 10.1016/j.geomphys.2004.02.008 · Zbl 1092.81055
[16] 10.1007/BFb0075216
[17] 10.4310/ATMP.2002.v6.n6.a2
[18] 10.1006/aima.2001.2015 · Zbl 1001.18001
[19] 10.7146/math.scand.a-14961 · Zbl 1101.14024
[20] 10.1007/978-3-0348-8600-0
[21] 10.4310/ATMP.2012.v16.n1.a5 · Zbl 1420.57074
[22] 10.4310/SDG.2002.v7.n1.a6
[23] 10.2140/gt.2010.14.903 · Zbl 1197.58007
[24] 10.4310/AJM.1999.v3.n4.a6 · Zbl 1028.81052
[25] 10.1007/s002220050118 · Zbl 0873.14025
[26] 10.1007/s40062-017-0178-y · Zbl 1404.55011
[27] 10.1007/978-1-4614-8468-4 · Zbl 1281.55002
[28] 10.1016/j.difgeo.2015.02.001 · Zbl 1317.19013
[29] ; Hilton, General cohomology theory and K-theory. London Math. Soc. Lecture Note Ser., 1, (1971) · Zbl 1386.55001
[30] 10.1007/s10455-012-9325-1 · Zbl 1257.19005
[31] 10.5802/ambp.337 · Zbl 1329.19010
[32] 10.4310/jdg/1143642908 · Zbl 1116.58018
[33] 10.1007/978-3-540-74956-1
[34] 10.4153/CJM-1987-035-8 · Zbl 0645.18006
[35] 10.1007/978-1-4939-2300-7 · Zbl 1320.18001
[36] ; Karoubi, Homologie cyclique et K-théorie. Astérisque, 149, (1987)
[37] 10.4310/ATMP.2004.v8.n2.a3 · Zbl 1082.81070
[38] 10.4310/CAG.1994.v2.n2.a6 · Zbl 0840.58044
[39] 10.1007/978-1-4612-1314-7_10
[40] 10.1017/S0305004100037245
[41] 10.1090/surv/132
[42] ; McCleary, A user’s guide to spectral sequences. Cambridge Studies in Advanced Mathematics, 58, (2001) · Zbl 0959.55001
[43] 10.1016/0022-4049(81)90064-5 · Zbl 0459.55012
[44] 10.1088/1126-6708/1997/11/002
[45] 10.1090/gsm/074
[46] 10.2307/2374308 · Zbl 0586.55003
[47] 10.1007/978-3-540-77751-9 · Zbl 0906.55001
[48] 10.1090/pspum/081/2681765
[49] 10.1007/s00220-012-1510-3 · Zbl 1252.81110
[50] 10.1112/jtopol/jtv020 · Zbl 1330.55007
[51] 10.1353/ajm.2007.0014 · Zbl 1120.55007
[52] 10.1112/jtopol/jtm006 · Zbl 1163.57020
[53] ; Switzer, Algebraic topology — homotopy and homology. Grundl. Math. Wissen., 212, (1975) · Zbl 0305.55001
[54] 10.1016/S0022-4049(97)00177-1 · Zbl 0927.55013
[55] 10.1017/is013002018jkt218 · Zbl 1327.19014
[56] ; Tradler, New York J. Math., 22, 527, (2016)
[57] 10.2140/agt.2015.15.65 · Zbl 1312.55004
[58] 10.1112/jlms/s2-22.3.423 · Zbl 0453.55013
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