×

zbMATH — the first resource for mathematics

A parametrized index theorem for the algebraic \(K\)-theory Euler class. (English) Zbl 1077.19002
Summary: Riemann-Roch theorems assert that certain algebraically defined wrong way maps (transfers) in algebraic \(K\)-theory agree with topologically defined ones [see P. Baum, W. Fulton and R. MacPherson, Acta Math. 143, 155–192 (1979; Zbl 0474.14004).
J.-M. Bismut and J. Lott [J. Am. Math. Soc. 8, 291–363 (1995; Zbl 0837.58028)] proved such a Riemann-Roch theorem where the wrong way maps are induced by the projection of a smooth fiber bundle, and the topologically defined transfer map is the Becker-Gottlieb transfer. We generalize and refine their theorem, and prove a converse stating that the Riemann-Roch condition is equivalent to the existence of a fiberwise smooth structure. In the process, we prove a family index theorem where the \(K\)-theory used is algebraic \(K\)-theory, and the fiber bundles have topological (not necessarily smooth) manifolds as fibers.

MSC:
19D10 Algebraic \(K\)-theory of spaces
57R10 Smoothing in differential topology
58J22 Exotic index theories on manifolds
58J52 Determinants and determinant bundles, analytic torsion
14C40 Riemann-Roch theorems
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Adams, J. F.,Stable Homotopy and Generalised Homology. Chicago Lectures in Mathematics. University of Chicago Press, Chicago, IL, 1974. · Zbl 0309.55016
[2] –,Infinite Loop Spaces. Ann. of Math. Stud., 90. Princeton Univ. Press. Princeton, NJ, 1978. · Zbl 0398.55008
[3] Anderson, D. R., Connolly, F. X., Ferry, S. C. &Pedersen, E. K., AlgebraicK-theory with continuous control at infinity.J. Pure Appl. Algebra, 94 (1994), 25–47. · Zbl 0854.55003
[4] Bass, H.,Algebraic K-Theory. Benjamin, New York-Amsterdam, 1968.
[5] Baum, P. &Douglas, R.,K homology and index theory, inOperator Algebras and Applications, Part I (Kingston, ON, 1980), pp. 117–173. Proc. Sympos. Pure Math., 38. Amer. Math. Soc., Providence, RI, 1982.
[6] –, Index theory, bordism, andK-homology, inOperator Algebras and K-theory (San Francisco, CA, 1981), pp. 1–31. Contemp. Math., 10. Amer. Math. Soc., Providence, RI, 1982.
[7] Becker, J. C., Extensions of cohomology theories.Illinois J. Math., 14 (1970), 551–584. · Zbl 0211.55103
[8] Becker, J. C. &Gottlieb, D. H., The transfer map and fiber bundles.Topology, 14 (1975), 1–12. · Zbl 0306.55017
[9] –, Transfer maps for fibrations and duality.Compositio Math., 33 (1976), 107–133. · Zbl 0354.55009
[10] Becker, J. &Schultz, R., The real semicharacteristic of a fibered manifold.Quart. J. Math. Oxford Ser., (2), 33 (1982), 385–403. · Zbl 0512.55015
[11] Bousfield, A. K. &Kan, D. M.,Homotopy Limits, Completions and Localizations. Lecture Notes in Math., 304. Springer-Verlag, Berlin-New York, 1972. · Zbl 0259.55004
[12] Bismut, J.-M. &Lott, J., Flat vector bundles, direct images and higher real analytic torsion.J. Amer. Math. Soc., 8 (1995), 291–363. · Zbl 0837.58028
[13] Bödigheimer, C.-F., Stable splittings of mapping spaces, inAlgebraic Topology (Seattle, WA, 1985), pp. 174–187. Lecture Notes in Math., 1286. Springer-Verlag, Berlin, 1987.
[14] Brown, M., A proof of the generalized Schoenflies theorem.Bull. Amer. Math. Soc., 66 (1960), 74–76. · Zbl 0132.20002
[15] Burghelea, D. &Lashof, R., The homotopy type of diffeomorphisms, I; II.Trans. Amer. Math. Soc., 196 (1974), 1–36; 37–50. · Zbl 0296.58003
[16] –, Geometric transfer and the homotopy type of the automorphism groups of a manifold.Trans. Amer. Math. Soc., 269 (1982), 1–38. · Zbl 0489.57008
[17] Burghelea, D., Lashof, R. &Rothenberg, M.,Groups of Automorphisms of Manifolds. Lecture Notes in Math. 473. Springer-Verlag, Berlin-New York, 1975. · Zbl 0307.57013
[18] Casson, A. &Gottlieb, D. H., Fibrations with compact fibres.Amer. J. Math., 99 (1977), 159–189. · Zbl 0375.55015
[19] Cerf, J., La stratification naturelle des espaces de fonctions différentiables réelles et le théorème de la pseudo-isotopie.Inst. Hautes Études Sci. Publ. Math., 39 (1970), 5–173. · Zbl 0213.25202
[20] Chapman, T.,Lectures on Hilbert Cube Manifolds. CBMS Regional Conf. Ser. in Math., 28. Amer. Math. Soc., Providence, RI, 1976.
[21] Chapman, T. A., Piecewise linear fibrations.Pacific J. Math., 128 (1987), 223–250. · Zbl 0654.57011
[22] Cheeger, J., Analytic torsion and the heat equation.Ann. of Math., (2) 109 (1979), 259–322. · Zbl 0412.58026
[23] Clapp, M., Duality and transfer for parametrized spectra.Arch. Math. (Basel), 37 (1981), 462–472. · Zbl 0482.55014
[24] Carlsson, G. &Pedersen, E. K., Controlled algebra and the Novikov conjectures forK- andL-theory.Topology, 34 (1995), 731–758. · Zbl 0838.55004
[25] Carlsson, G., Pedersen, E. K. &Vogell, W., Continuously controlled algebraicK-theory of spaces and the Novikov conjecture.Math. Ann., 310 (1998), 169–182. · Zbl 0897.55003
[26] Dold, A.,Lectures on Algebraic Topology. Grundlehren Math. Wiss., 200. Springer-Verlag, New York-Berlin, 1972. · Zbl 0234.55001
[27] –, The fixed point transfer of fibre-preserving maps.Math. Z., 148 (1976), 215–244. · Zbl 0329.55007
[28] Dold, A. &Puppe, D., Duality, trace and transfer, inGeometric Topology (Warsaw, 1978), pp. 81–102. PWN, Warsaw, 1980.
[29] Dundas, B. I., RelativeK-theory and topological cyclic homology.Acta Math., 179 (1997), 223–242. · Zbl 0913.19002
[30] Dundas, B. I. &McCarthy, R., StableK-theory and topological Hochschild homology.Ann. of Math. (2), 140 (1994), 685–701; Erratum.Ibid., 142 (1995), 425–426. · Zbl 0833.55007
[31] Dwyer, W., The centralizer decomposition ofBG, inAlgebraic Topology: New Trends in Localization and Periodicity (Sant Feliu de Guíxols, 1994), pp. 167–184. Progr. Math., 136. Birkhäuser, Basel, 1996.
[32] Fulton, W. &MacPherson, R.,Categorical Framework for the Study of Singular Spaces. Mem. Amer. Math. Soc., 31 (243). Amer. Math. Soc., Providence, RI, 1981. · Zbl 0467.55005
[33] Gauld, D., Mersions of topological manifolds.Trans. Amer. Math. Soc., 149 (1970), 539–560. · Zbl 0197.20403
[34] Gillet, H., Riemann-Roch theorems for higher algebraicK-theory.Adv. in Math., 40 (1981), 203–289. · Zbl 0478.14010
[35] Goodwillie T. G., Calculus I: The first derivative of pseudoisotopy theory.K-Theory, 4 (1990), 1–27. · Zbl 0741.57021
[36] –, Calculus II: Analytic functors.K-Theory, 5 (1991/92), 295–332. · Zbl 0776.55008
[37] Goodwillie, T., Klein, J. & Weiss, M., A Haefliger style description of the embedding calculus tower. To appear inTopology. · Zbl 1034.57027
[38] Hatcher, A. &Wagoner, J.,Pseudo-Isotopies of Compact Manifolds. Astérisque, 6. Soc. Math. France, Paris, 1973. · Zbl 0274.57010
[39] Hilton, P., Mislin, G. &Roitberg J.,Localization of Nilpotent Groups and Spaces. North-Holland Math. Stud. 15. North-Holland, Amsterdam, 1975. · Zbl 0323.55016
[40] Hu, S.-T.,Theory of Retracts. Wayne State Univ. Press, Detroit 1965. · Zbl 0145.43003
[41] Igusa, K., What happens to Hatcher and Wagoner’s formula for {\(\pi\)}0 C(M) when the first Postnikov invariant ofM is nontrivial?, inAlgebraic K-Theory, Number Theory Gometry and Analysis (Bielefeld, 1982), pp. 104–172. Lecture Notes in Math., 1046. Springer-Verlag, Berlin, 1984.
[42] –, The stability theorem for smooth pseudoisotopies.K-Theory, 2 (1988), 1–355. · Zbl 0691.57011
[43] –,Higher Franz-Reidemeister Torsion. AMS/IP Stud. Adv. Math., 31. Amer. Math. Soc., Providence, RI; International Press, Somerville, MA, 2002.
[44] Igusa, K. &Klein, J., The Borel regulator map on pictures, II. An example from Morse theoryK-Theory, 7 (1993), 225–267. · Zbl 0793.19002
[45] James, I. M.,Fibrewise Topology. Cambridge Tracts in Math., 91. Cambridge Univ. Press, Cambridge, 1989.
[46] Kan, D. M., On c.s.s. complexes.Amer. J. Math., 79 (1957), 449–476. · Zbl 0078.36901
[47] Karoubi, M.,Homologie cyclique et K-théorie. Astérisque, 149. Soc. Math.. France, Paris, 1987. · Zbl 0648.18008
[48] Kister, J. M., Microbundles are fiber bundles.Ann. of Math. (2) 80 (1964), 190–199. · Zbl 0131.20602
[49] Kirby, R. C. &Siebenmann, L. C.,Foundational Essays on Topological Manifolds, Smoothings, and Triangulations. Ann. of Math. Stud., 88. Princeton Univ. Press, Princeton, NJ, 1977. · Zbl 0361.57004
[50] Lacher, R. C., Cell-like mappings, I.Pacific J. Math., 30 (1969), 717–731. · Zbl 0182.57601
[51] –, Cell-like mappings, II,Pacific J. Math., 35 (1970), 649–660. · Zbl 0212.55702
[52] –, Cell-like mappings and their generalizations.Bull. Amer. Math. Soc., 83 (1977), 495–552. · Zbl 0364.54009
[53] Lott, J.,Diffeomorphisms and Noncommutative Analytic Torsion. Mem. Amer. Math. Soc., 141 (673) Amer. Math. Soc., Providence, RI, 1999. · Zbl 0942.58002
[54] MacLane, S.,Categories for the Working Mathematician. Graduate Texts in Math., 5. Springer-Verlag, New York-Berlin, 1971. · Zbl 0705.18001
[55] Mather, J. N., The vanishing of the homology of certain groups of homeomorphisms.Topology, 10 (1971), 297–298. · Zbl 0221.57021
[56] Mazur, B., Differential topology from the point of view of simple homotopy theory.Inst. Hautes Études Sci. Publ. Math., 15 (1963), 1–93. · Zbl 0173.51203
[57] McDuff, D., Configuration spaces of positive and negative particles.Topology, 14 (1975), 91–107. · Zbl 0296.57001
[58] –, The homology of some groups of diffeomorphisms.Comment. Math. Helv., 55 (1980), 97–129. · Zbl 0448.57015
[59] Mishchenko, A. S. &Fomenko, A. T., The index of elliptic operators overC *-algebras.Izv. Akad. Nauk SSSR Ser. Mat., 43 (1979), 831–859, 967 (Russian). · Zbl 0416.46052
[60] Morlet, C., Lissage des homéomorphismes.C. R. Acad. Sci. Paris Sér. A-B, 268 (1969), A1323-A1326. · Zbl 0198.28802
[61] Müller, W., Analytic torsion andR-torsion for unimodular representations.J. Amer. Math. Soc., 6 (1993), 721–753. · Zbl 0789.58071
[62] Pedersen, E. K. &Weibel, C. A.,K-theory homology of spaces, inAlgebraic Topology (Arcata, CA, 1986), pp. 346–361. Lecture Notes in Math., 1370. Springer-Verlag, Berlin, 1989.
[63] Quillen, D., Higher algebraicK-theory, I, inAlgebraic K-theory, I:Higher K-theories (Seattle, WA, 1972), pp. 85–147. Lecture Notes in Math., 341. Springer-Verlag, Berlin, 1973.
[64] Quinn, F., Thesis, Princeton University, 1969.
[65] –, A geometric formulation of surgery, inTopology of Manifolds (Athens, GA, 1969), pp. 500–511. Markham, Chicago, IL, 1970.
[66] Ranicki, A. &Yamasaki, M., ControlledK-theory.Topology Appl., 61 (1995), 1–59. · Zbl 0835.57013
[67] Rosenberg, J.,C *-algebras, positive scalar curvature and the Novikov conjecture, II, inGeometric Methods in Operator Algebras (Kyoto, 1983), pp. 341–374. Pitman Res. Notes Math. Ser., 123. Longman Sci. Tech., Harlow, 1986.
[68] –,C *-algebras, positive scalar curvature and the Novikov conjecture, III.Topology, 25 (1986), 319–336. · Zbl 0605.53020
[69] –, TheKO-assembly map and positive scalar curvature, inAlgebraic Topology (Poznan, 1989), pp. 170–182. Lecture Notes in Math., 1474. Springer-Verlag, Berlin, 1991.
[70] Rourke, C. P. &Sanderson, B. J., {\(\Delta\)}-sets, I. Homotopy theory.Quart. J. Math. Oxford Ser. (2), 22 (1971), 321–338. · Zbl 0226.55019
[71] –, {\(\Delta\)}-sets, II. Block bundles and block fibrations.Quart. J. Math. Oxford Ser. (2), 22 (1971), 465–485. · Zbl 0226.55020
[72] Segal, G., Categories and cohomology theories.Topology, 13 (1974), 293–312. · Zbl 0284.55016
[73] –, Classifying spaces related to foliations.Topology, 17 (1978), 367–382. · Zbl 0398.57018
[74] –, The topology of spaces of rational functions.Acta Math., 143 (1979), 39–72. · Zbl 0427.55006
[75] Smale, S., On the structure of manifolds.Amer. J. Math., 84 (1962), 387–399. · Zbl 0109.41103
[76] Steinberger, M., The classification of PL fibrations,Michigan Math. J., 33 (1986), 11–26. · Zbl 0606.55012
[77] Strøm, A., The homotopy category is a homotopy category.Arch. Math. (Basel), 23 (1972), 435–441. · Zbl 0261.18015
[78] Thomason, R. W., Homotopy colimits in the category of small categories.Math. Proc. Cambridge Philos. Soc., 85 (1979), 91–109. · Zbl 0392.18001
[79] Thomason, R. W. &Trobaugh, T. F., Higher algebraicK-theory of schemes and of derived categories, inThe Grothendieck Festschrift, Vol. III, pp. 247–435. Progr. Math., 88. Birkhäuser Boston, Boston, MA, 1990.
[80] Thurston, W., Foliations and groups of diffeomorphisms.Bull. Amer. Math. Soc., 80 (1974), 304–307. · Zbl 0295.57014
[81] Vogell, W., AlgebraicK-theory of spaces, with bounded control.Acta Math., 165 (1990), 161–187. · Zbl 0722.19002
[82] Waldhausen, F., AlgebraicK-theory of spaces, a manifold approach, inCurrent Trends in Algebraic Topology, Part 1 (London, ON, 1981), pp. 141–186. CMS Conf. Proc., 2, Amer. Math. Soc., Providence, RI, 1982.
[83] –, AlgebraicK-theory of spaces, inAlgebraic and Geometric Topology (New Brunswick, NJ, 1983), pp. 318–419. Lecture Notes in Math., 1126. Springer-Verlag, Berlin, 1985.
[84] –, AlgebraicK-theory of spaces, concordance, and stable homotopy theory, inAlgebraic topology and Algebraic K-Theory (Princeton, NJ, 1983), pp. 392–417. Ann. of Math. Stud., 113. Princeton Univ. Press, Princeton, NJ, 1987.
[85] –, An outline of how manifolds relate to algebraicK-theory, inHomotopy Theory (Durham, 1985), pp. 239–247. London Math. Soc. Lecture Note Ser., 117. Cambridge Univ. Press, Cambridge, 1987.
[86] Wall, C. T. C., Finiteness conditions for CW-complexes.Ann. of Math., (2), 81 (1965), 56–69. · Zbl 0152.21902
[87] Weiss, M., Excision and restriction in controlledK-theory.Forum Math., 14 (2002), 85–119. · Zbl 0995.19003
[88] Williams, B., Bivariant Riemann-Roch theorems inGeometry and Topology (Aarhus, 1998), pp. 377–393. Contemp. Math., 258. Amer. Math. Soc., Providence, RI, 2000.
[89] Waldhausen, F. & Vogell, W., Spaces of PL manifolds and categories of simple maps (the non-manifold part). Preprint, Bielefeld University, 2000.
[90] Waldhausen, F. & Vogell, W., Spaces of PL manifolds and categories of simple maps (the manifold part). Preprint, Bielefeld University, 2000.
[91] Weiss, M. &Williams, B., Assembly, inNovikov Conjectures, Index Theorems and Rigidity, Vol. 2 (Oberwolfach, 1993), pp. 332–352. London Math. Soc. Lecture Note Ser., 227. Cambridge Univ. Press, Cambridge, 1995.
[92] – Pro-excisive functors, inNovikov Conjectures, Index Theorems and Rigidity, Vol. 2 (Oberwolfach, 1993), pp. 353–364. London Math. Soc. Lecture Note Ser., 227. Cambridge Univ. Press, Cambridge, 1995.
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.