Torsion invariants for families

Goette, Sebastian

From probability to geometry (II) - Volume in honor of the 60th birthday of Jean-Michel Bismut, Astérisque, no. 328 (2009), pp. 161-206.

MR Zbl

@incollection{AST_2009__328__161_0,
     author = {Goette, Sebastian},
     title = {Torsion invariants for families},
     booktitle = {From probability to geometry (II) - Volume in honor of the 60th birthday of Jean-Michel Bismut},
     editor = {Dai Xianzhe and L\'eandre R\'emi and Xiaonan Ma and Zhang Weiping},
     series = {Ast\'erisque},
     pages = {161--206},
     publisher = {Soci\'et\'e math\'ematique de France},
     number = {328},
     year = {2009},
     mrnumber = {2674876},
     zbl = {1247.58019},
     language = {en},
     url = {http://www.numdam.org/item/AST_2009__328__161_0/}
}

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Goette, Sebastian. Torsion invariants for families, dans From probability to geometry (II) - Volume in honor of the 60th birthday of Jean-Michel Bismut, Astérisque, no. 328 (2009), pp. 161-206. http://www.numdam.org/item/AST_2009__328__161_0/

Bibliographie
Cité par

[1] B. Badzioch & W. Dorabiala - "Additivity for the parametrized topological Euler characteristic and Reidemeister torsion", $K$ -Theory 38 (2007), p. 1-22. | DOI | MR | Zbl

[2] B. Badzioch, W. Dorabiala, J. Klein & B. Williams - "Equivalence of higher torsion invariants", preprint arXiv:0904.4684, 2009. | MR | Zbl

[3] B. Badzioch, W. Dorabiala & B. Williams - "Smooth parametrized torsion: a manifold approach", Adv. Math. 221 (2009), p. 660-680. | DOI | MR | Zbl

[4] J. C. Becker & D. H. Gottlieb - "The transfer map and fiber bundles", Topology 14 (1975), p. 1-12. | DOI | MR | Zbl

[5] N. Berline, E. Getzler & M. Vergne - Heat kernels and Dirac operators, Grund. Math. Wiss., vol. 298, Springer, 1992. | MR | Zbl

[6] J.-M. Bismut - "The Atiyah-Singer index theorem for families of Dirac operators: two heat equation proofs", Invent. Math. 83 (1985), p. 91-151. | DOI | EuDML | MR | Zbl

[7] J.-M. Bismut, "Eta invariants, differential characters and flat vector bundles", Chinese Ann. Math. Ser. B 26 (2005), p. 15-44. | DOI | MR | Zbl

[8] J.-M. Bismut, "The hypoelliptic Laplacian on the cotangent bundle", J. Amer. Math. Soc. 18 (2005), p. 379-476. | DOI | MR | Zbl

[9] J.-M. Bismut & J. Cheeger - " $η$ -invariants and their adiabatic limits", J. Amer. Math. Soc. 2 (1989), p. 33-70. | MR | Zbl

[10] J.-M. Bismut & J. Cheeger, "Families index for manifolds with boundary, superconnections and cones. II. The Chern character", J. Funct. Anal. 90 (1990), p. 306-354. | DOI | MR | Zbl

[11] J.-M. Bismut & S. Goette - "Holomorphic equivariant analytic torsions", Geom. Funct. Anal. 10 (2000), p. 1289-1422. | DOI | MR | Zbl

[12] J.-M. Bismut & S. Goette, "Families torsion and Morse functions", Astérisque 275 (2001). | Numdam | MR | Zbl

[13] J.-M. Bismut & S. Goette, "Equivariant de Rham torsions", Ann. of Math. 159 (2004), p. 53-216. | DOI | MR | Zbl

[14] J.-M. Bismut & K. Köhler - "Higher analytic torsion forms for direct images and anomaly formulas", J. Algebraic Geom. 1 (1992), p. 647-684. | MR | Zbl

[15] J.-M. Bismut & G. Lebeau - The hypoelliptic laplacian and Ray-singer metrics, Annals of Math. Studies, vol. 167, Princeton Univ. Press, 2008. | MR | Zbl

[16] J.-M. Bismut & J. Lott - "Flat vector bundles, direct images and higher real analytic torsion", J. Amer. Math. Soc. 8 (1995), p. 291-363. | DOI | MR | Zbl

[17] J.-M. Bismut & W. Zhang - "An extension of a theorem by Cheeger and Müller", Astérisque 205 (1992). | Numdam | MR | Zbl

[18] J.-M. Bismut & W. Zhang, "Milnor and Ray-Singer metrics on the equivariant determinant of a flat vector bundle", Geom. Funct Anal. 4 (1994), p. 136-212. | DOI | EuDML | MR | Zbl

[19] M. Bökstedt - "The rational homotopy type of $Ω W h^{Diff} (*)$ ", in Algebraic Topology, Aarhus 1982 (Aarhus, 1982), Lecture Notes in Math., vol. 1051, Springer, 1984, p. 25-37. | MR | Zbl

[20] A. Borel - "Stable real cohomology of arithmetic groups", Ann. Sci. Ecole Norm. Sup. 7 (1974), p. 235-272. | DOI | EuDML | Numdam | MR | Zbl

[21] M. Braverman & T. Kappeler - "A refinement of the Ray-Singer torsion", C. R. Math. Acad. Sci. Paris 341 (2005), p. 497-502. | DOI | MR | Zbl

[22] U. Bunke - "Equivariant torsion and $G - C W$ -complexes", Geom. Funct. Anal. 9 (1999), p. 67-89. | DOI | MR | Zbl

[23] U. Bunke, "Equivariant higher analytic torsion and equivariant Euler characteristic", Amer. J. Math. 122 (2000), p. 377-401. | DOI | MR | Zbl

[24] U. Bunke, "On the functoriality of Lott's secondary analytic index", $K$ -Theory 25 (2002), p. 51-58. | DOI | MR | Zbl

[25] U. Bunke & X. Ma - "Index and secondary index theory for flat bundles with duality", in Aspects of Boundary problems in analysis and geometry, Oper. Theory Adv. Appl., vol. 151, Birkhäuser, 2004, p. 265-341. | DOI | MR | Zbl

[26] J. Cheeger - "Analytic torsion and the heat equation", Ann. of Math. 109 (1979), p. 259-322. | DOI | MR | Zbl

[27] J. Cheeger & J. Simons - "Differential characters and geometric invariants", in Geometry and Topology (College Park, Md., 1983/84-), Lecture Notes in Math., vol. 1167, Springer, 1985, p. 50-80. | MR | Zbl

[28] X. Dai - "Adiabatic limits, nonmultiplicativity of signature, and Leray spectral sequence", J. Amer. Math. Soc. 4 (1991), p. 265-321. | DOI | MR | Zbl

[29] X. Dai & W. Zhang - "Adiabatic limit, Bismut-Freed connection, and the real analytic torsion form", preprint arXiv:0807.3782. | MR | Zbl

[30] W. Dwyer, M. Weiss & B. Williams - "A parametrized index theorem for the algebraic $K$ -theory Euler class", Acta Math. 190 (2003), p. 1-104. | DOI | MR | Zbl

[31] Y. M. Eliashberg & N. M. Mishachev - "Wrinkling of smooth mappings. II. Wrinkling of embeddings and K. Igusa's theorem", Topology 39 (2000), p. 711-732. | DOI | MR | Zbl

[32] F. T. Farrell & W. C. Hsiang - "On the rational homotopy groups of the diffeomorphism groups of discs, spheres and aspherical manifolds", in Algebraic and Geometric Topology (Proc. Sympos. Pure Math., Stanford Univ., Stanford, Calif., 1976), Part 1, Proc. Sympos. Pure Math., XXXII, Amer. Math. Soc., 1978, p. 325-337. | MR | Zbl

[33] W. Franz - "über die Torsion einer überdeckung", J. reine angew. Math. 173 (1935), p. 245-254. | EuDML | JFM | MR | Zbl

[34] D. Fried - "Lefschetz formulas for flows", in The Lefschetz centennial conference, Part III (Mexico City, 1984), Contemp. Math., vol. 58, Amer. Math. Soc., 1987, p. 19-69. | DOI | MR | Zbl

[35] S. Goette - "Equivariant $η$ -invariants and $η$ -forms", J. reine angew. Math. 526 (2000), p. 181-236. | MR | Zbl

[36] S. Goette, "Morse theory and higher torsion invariants I", preprint arXiv:math/0111222, 2001.

[37] S. Goette, "Morse theory and higher torsion invariants II", preprint arXiv:math/0305287, 2003.

[38] S. Goette, "Morse theory and higher torsion invariants III", in preparation.

[39] A. Hatcher & J. B. Wagoner - "Pseudo-isotopies of compact manifolds", Astérisque (1973). | Numdam | MR | Zbl

[40] J. L. Heitsch & C. Lazarov - "Riemann-Roch-Grothendieck and torsion for foliations", J. Geom. Anal. 12 (2002), p. 437-468. | DOI | MR | Zbl

[41] K. Igusa - "Parametrized Morse theory and its applications", in Proceedings of the International Congress of Mathematicians, Vol. I, II (Kyoto, 1990), Math. Soc. Japan, 1991, p. 643-651. | MR | Zbl

[42] K. Igusa, Higher Franz-Reidemeister torsion, Stud. Adv. Math., vol. 31, 2002. | DOI | MR | Zbl

[43] K. Igusa, "Higher complex torsion and the framing principle", Mem. Amer. Math. Soc. 177 (2005). | MR | Zbl

[44] K. Igusa, "Axioms for higher torsion invariants of smooth bundles", J. Topol. 1 (2008), p. 159-186. | DOI | MR | Zbl

[45] K. Igusa & J. Klein - "The Borel regulator map on pictures. II. An example from Morse theory", K-Theory 7 (1993), p. 225-267. | DOI | MR | Zbl

[46] K. Igusa & J. Klein, "Higher Franz-Reidemeister torsion, I. The algebra of filtered chain complexes", preprint, 1995.

[47] F. W. Kamber & P. Tondeur - "Characteristic invariants of foliated bundles", Manuscripta Math. 11 (1974), p. 51-89. | DOI | EuDML | MR | Zbl

[48] J. Klein - "The cell complex construction and higher $R$ -torsion for bundles with framed Morse function", Ph.D. Thesis, Brandeis University, 1989. | MR

[49] K. Köhler - "Equivariant Reidemeister torsion on symmetric spaces", Math. Ann. 307 (1997), p. 57-69. | DOI | MR | Zbl

[50] J. Lott - "Equivariant analytic torsion for compact Lie group actions", J. Funct. Anal. 125 (1994), p. 438-451. | DOI | MR | Zbl

[51] J. Lott, "Diffeomorphisms and noncommutative analytic torsion", Mem. Amer. Math. Soc. 141 (1999). | MR | Zbl

[52] J. Lott, "Secondary analytic indices", in Regulators in analysis, geometry and number theory, Progr. Math., vol. 171, Birkhäuser, 2000, p. 231-293. | DOI | MR | Zbl

[53] X. Ma - "Formes de torsion analytique et familles de submersions. I", Bull. Soc. Math. France 127 (1999), p. 541-621. | DOI | EuDML | Numdam | MR | Zbl

[54] X. Ma, "Formes de torsion analytique et familles de submersions. II", Asian J. Math. 4 (2000), p. 633-667. | DOI | MR | Zbl

[55] X. Ma, "Functoriality of real analytic torsion forms", Israel J. Math. 131 (2002), p. 1-50. | DOI | MR | Zbl

[56] X. Ma & W. Zhang - "Eta-invariants, torsion forms and flat vector bundles", Math. Ann. 340 (2008), p. 569-624. | DOI | MR | Zbl

[57] E. Y. Miller - "The homology of the mapping class group", J. Differential Geom. 24 (1986), p. 1-14. | DOI | MR | Zbl

[58] S. Morita - "Characteristic classes of surface bundles", Invent Math. 90 (1987), p. 551-577. | DOI | EuDML | MR | Zbl

[59] W. Müller - "Analytic torsion and $R$ -torsion of Riemannian manifolds", Adv. in Math. 28 (1978), p. 233-305. | DOI | MR | Zbl

[60] D. Mumford - "Towards an enumerative geometry of the moduli space of curves", in Arithmetic and geometry, Vol. II, Progr. Math., vol. 36, Birkhäuser, 1983, p. 271-328. | DOI | MR | Zbl

[61] D. B. Ray & I. M. Singer - " $R$ -torsion and the Laplacian on Riemannian manifolds", Advances in Math. 7 (1971), p. 145-210. | DOI | MR | Zbl

[62] K. Reidemeister - "Homotopieringe und Linsenräume", Abh. math. Semin. Hamb. Univ. 11 (1935), p. 102-109. | DOI | JFM | MR | Zbl

[63] C. Soulé - Lectures on Arakelov geometry, Cambridge Studies in Advanced Math., vol. 33, Cambridge Univ. Press, 1992. | MR | Zbl

[64] J. B. Wagoner - "Diffeomorphisms, $K_{2}$ , and analytic torsion", in Algebraic and geometric topology (Proc. Sympos. Pure Math., Stanford Univ., Stanford, Calif, 1976), Part 1, Proc. Sympos. Pure Math., XXXII, Amer. Math. Soc., 1978, p. 23-33. | MR | Zbl

[65] F. Waldhausen - "Algebraic $K$ -theory of spaces", in Algebraic and geometric topology (New Brunswick, N.J., 1983), Lecture Notes in Math., vol. 1126, Springer, 1985, p. 318-419. | DOI | MR | Zbl

[66] M. Weiss & B. Williams - "Assembly", in Novikov conjectures, index theorems and rigidity, Vol. 2 (Oberwolfach, 1993), London Math. Soc. Lecture Note Ser., vol. 227, Cambridge Univ. Press, 1995, p. 332-352. | DOI | MR | Zbl

[67] W. Zhang - "Sub-signature operators, $η$ -invariants and a Riemann-Roch theorem for flat vector bundles", Chinese Ann. Math. Ser. B 25 (2004), p. 7-36. | DOI | MR | Zbl