On the map of Bökstedt–Madsen from the cobordism category to A–theory

Raptis, George¹ ; Steimle, Wolfgang²

¹Institut für Mathematik, Universität Osnabrück, Albrechtstr. 28a, D-49076 Osnabrück, Germany
²Institut for Matematiske Fag, University of Copenhagen, Universitetsparken 5, DK-2100 Copenhagen Ø, Denmark

Algebraic and Geometric Topology, Tome 14 (2014) no. 1, pp. 299-347

Résumé

Bökstedt and Madsen defined an infinite loop map from the embedded d–dimensional cobordism category of Galatius, Madsen, Tillmann and Weiss to the algebraic K–theory of BO(d) in the sense of Waldhausen. The purpose of this paper is to establish two results in relation to this map. The first result is that it extends the universal parametrized A–theory Euler characteristic of smooth bundles with compact d–dimensional fibers, as defined by Dwyer, Weiss and Williams. The second result is that it actually factors through the canonical unit map Q(BO(d)+) → A(BO(d)).

Détail

DOI : 10.2140/agt.2014.14.299

Classification : 19D10, 55R12, 57R90
Keywords: cobordism category, bivariant $A$–theory, parametrized Euler characteristic

Bibliographie

[1] B Badzioch, W Dorabiała, B Williams, Smooth parametrized torsion: A manifold approach, Adv. Math. 221 (2009) 660

[2] J C Becker, D H Gottlieb, Transfer maps for fibrations and duality, Compositio Math. 33 (1976) 107

[3] J M Boardman, R M Vogt, Homotopy invariant algebraic structures on topological spaces, Lecture Notes in Mathematics 347, Springer (1973)

[4] M Bökstedt, I Madsen, The cobordism category and Waldhausen's $K$–theory

[5] A K Bousfield, D M Kan, Homotopy limits, completions and localizations, Lecture Notes in Mathematics 304, Springer (1972)

[6] D C Cisinski, Invariance de la $K$–théorie par équivalences dérivées, J. $K$–Theory 6 (2010) 505

[7] W Dorabiała, The double coset theorem formula for algebraic $K$–theory of spaces, $K$–Theory 25 (2002) 251

[8] W Dwyer, M Weiss, B Williams, A parametrized index theorem for the algebraic $K$–theory Euler class, Acta Math. 190 (2003) 1

[9] W Fulton, R Macpherson, Categorical framework for the study of singular spaces, Memoirs AMS 243, Amer. Math. Soc. (1981)

[10] S Galatius, U Tillmann, I Madsen, M Weiss, The homotopy type of the cobordism category, Acta Math. 202 (2009) 195

[11] J Genauer, Cobordism categories of manifolds with corners, Trans. Amer. Math. Soc. 364 (2012) 519

[12] R W Kieboom, A pullback theorem for cofibrations, Manuscripta Math. 58 (1987) 381

[13] J P May, The geometry of iterated loop spaces, Lectures Notes in Mathematics 271, Springer (1972)

[14] G Segal, Configuration-spaces and iterated loop-spaces, Invent. Math. 21 (1973) 213

[15] G Segal, Categories and cohomology theories, Top. 13 (1974) 293

[16] U Tillmann, A splitting for the stable mapping class group, Math. Proc. Cambridge Philos. Soc. 127 (1999) 55

[17] F Waldhausen, Algebraic $K$–theory of topological spaces, II, from: "Algebraic topology" (editors J L Dupont, I H Madsen), Lecture Notes in Math. 763, Springer (1979) 356

[18] F Waldhausen, Algebraic $K$–theory of spaces, a manifold approach, from: "Current trends in algebraic topology, Part 1" (editors R M Kane, S O Kochman, P S Selick, V P Snaith), CMS Conf. Proc. 2, Amer. Math. Soc. (1982) 141

[19] F Waldhausen, Algebraic $K$–theory of spaces, from: "Algebraic and geometric topology" (editors A Ranicki, N Levitt, F Quinn), Lecture Notes in Math. 1126, Springer (1985) 318

[20] M Weiss, B Williams, Assembly, from: "Novikov conjectures, index theorems and rigidity, Vol. 2" (editors S C Ferry, A Ranicki, J Rosenberg), London Math. Soc. Lecture Note Ser. 227, Cambridge Univ. Press (1995) 332

[21] B Williams, Bivariant Riemann Roch theorems, from: "Geometry and topology" (editors K Grove, I H Madsen, E K Pedersen), Contemp. Math. 258, Amer. Math. Soc. (2000) 377