Exotic smooth structures on topological fiber bundles II

Goette, Sebastian; Igusa, Kiyoshi

doi:10.1090/S0002-9947-2013-05858-8

Exotic smooth structures on topological fiber bundles II
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by Sebastian Goette and Kiyoshi Igusa PDF

Trans. Amer. Math. Soc. 366 (2014), 791-832 Request permission

Abstract:

We use a variation of a classical construction of A. Hatcher to construct virtually all stable exotic smooth structures on compact smooth manifold bundles whose fibers have sufficiently large odd dimension (at least twice the base dimension plus 3). Using a variation of the Dwyer-Weiss-Williams smoothing theory which we explain in a separate joint paper with Bruce Williams, we associate a homology class in the total space of the bundle to each exotic smooth structure and show that the image of this class in the homology of the base is the Poincaré dual of the relative higher Igusa-Klein (IK) torsion invariant. This answers the question, in the relative case, of which cohomology classes can occur as relative higher torsion classes.

References

J. F. Adams, On the groups $J(X)$. IV, Topology 5 (1966), 21–71. MR 198470, DOI 10.1016/0040-9383(66)90004-8
Bernard Badzioch, Wojciech Dorabiała, John R. Klein, and Bruce Williams, Equivalence of higher torsion invariants, Adv. Math. 226 (2011), no. 3, 2192–2232. MR 2739777, DOI 10.1016/j.aim.2010.09.017
Bernard Badzioch, Wojciech Dorabiała, and Bruce Williams, Smooth parametrized torsion: a manifold approach, Adv. Math. 221 (2009), no. 2, 660–680. MR 2508934, DOI 10.1016/j.aim.2009.01.005
Jean-Michel Bismut and Sebastian Goette, Families torsion and Morse functions, Astérisque 275 (2001), x+293 (English, with English and French summaries). MR 1867006
Jean-Michel Bismut and John Lott, Flat vector bundles, direct images and higher real analytic torsion, J. Amer. Math. Soc. 8 (1995), no. 2, 291–363 (English, with English and French summaries). MR 1303026, DOI 10.1090/S0894-0347-1995-1303026-5
Raoul Bott, The stable homotopy of the classical groups, Ann. of Math. (2) 70 (1959), 313–337. MR 110104, DOI 10.2307/1970106
W. Dwyer, M. Weiss, and B. Williams, A parametrized index theorem for the algebraic $K$-theory Euler class, Acta Math. 190 (2003), no. 1, 1–104. MR 1982793, DOI 10.1007/BF02393236
Sebastian Goette, Morse theory and higher torsion invariants I, math.DG/0111222.
—, Morse theory and higher torsion invariants II, math.DG/0305287.
Sebastian Goette, Torsion invariants for families, Astérisque 328 (2009), 161–206 (2010) (English, with English and French summaries). MR 2674876
Sebastian Goette, Kiyoshi Igusa and Bruce Williams, Exotic smooth structures on topological fiber bundles I, ArXiv:1203.2203.
André Haefliger, Differential embeddings of $S^{n}$ in $S^{n+q}$ for $q>2$, Ann. of Math. (2) 83 (1966), 402–436. MR 202151, DOI 10.2307/1970475
Dale Husemoller, Fibre bundles, 3rd ed., Graduate Texts in Mathematics, vol. 20, Springer-Verlag, New York, 1994. MR 1249482, DOI 10.1007/978-1-4757-2261-1
Kiyoshi Igusa, On the homotopy type of the space of generalized Morse functions, Topology 23 (1984), no. 2, 245–256. MR 744854, DOI 10.1016/0040-9383(84)90043-0
Kiyoshi Igusa, The space of framed functions, Trans. Amer. Math. Soc. 301 (1987), no. 2, 431–477. MR 882699, DOI 10.1090/S0002-9947-1987-0882699-7
Kiyoshi Igusa, The stability theorem for smooth pseudoisotopies, $K$-Theory 2 (1988), no. 1-2, vi+355. MR 972368, DOI 10.1007/BF00533643
—, Higher Franz-Reidemeister Torsion, AMS/IP Studies in Advance Mathematics, vol. 31, International Press, 2002.
Kiyoshi Igusa, Axioms for higher torsion invariants of smooth bundles, J. Topol. 1 (2008), no. 1, 159–186. MR 2365656, DOI 10.1112/jtopol/jtm011
Kiyoshi Igusa, Higher complex torsion and the framing principle, Mem. Amer. Math. Soc. 177 (2005), no. 835, xiv+94. MR 2155700, DOI 10.1090/memo/0835
Kiyoshi Igusa and John Klein, The Borel regulator map on pictures. II. An example from Morse theory, $K$-Theory 7 (1993), no. 3, 225–267. MR 1244002, DOI 10.1007/BF00961065
John Klein, The cell complex construction and higher R-torsion for bundles with framed Morse function, Ph.D. thesis, Brandeis University, 1989.
J. B. Wagoner, Diffeomorphisms, $K_{2}$, and analytic torsion, Algebraic and geometric topology (Proc. Sympos. Pure Math., Stanford Univ., Stanford, Calif., 1976) Proc. Sympos. Pure Math., XXXII, Amer. Math. Soc., Providence, R.I., 1978, pp. 23–33. MR 520491
Bruce Williams, Stable smoothings of fiber bundles, handwritten notes, April 2006.