Involution on pseudoisotopy spaces and the space of nonnegatively curved metrics

Bustamante, Mauricio; Farrell, Francis; Jiang, Yi

doi:10.1090/tran/8135

Involution on pseudoisotopy spaces and the space of nonnegatively curved metrics
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by Mauricio Bustamante, Francis Thomas Farrell and Yi Jiang PDF

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Abstract:

We prove that certain involutions defined by Vogell and Burghelea-Fiedorowicz on the rational algebraic $K$-theory of spaces coincide. This gives a way to compute the positive and negative eigenspaces of the involution on rational homotopy groups of pseudoisotopy spaces from the involution on rational $S^{1}$-equivariant homology groups of the free loop space of a simply-connected manifold. As an application, we give explicit dimensions of the open manifolds $V$ that appear in Belegradek-Farrell-Kapovitch’s work for which the spaces of complete nonnegatively curved metrics on $V$ have nontrivial rational homotopy groups.

References

Igor Belegradek, F. Thomas Farrell, and Vitali Kapovitch, Space of nonnegatively curved metrics and pseudoisotopies, J. Differential Geom. 105 (2017), no. 3, 345–374. MR 3619306
D. Burghelea, The free loop space. I. Algebraic topology, Algebraic topology (Evanston, IL, 1988) Contemp. Math., vol. 96, Amer. Math. Soc., Providence, RI, 1989, pp. 59–85. MR 1022674, DOI 10.1090/conm/096/1022674
Dan Burghelea, Cyclic homology and the algebraic $K$-theory of spaces. I, Applications of algebraic $K$-theory to algebraic geometry and number theory, Part I, II (Boulder, Colo., 1983) Contemp. Math., vol. 55, Amer. Math. Soc., Providence, RI, 1986, pp. 89–115. MR 862632, DOI 10.1090/conm/055.1/862632
D. Burghelea and Z. Fiedorowicz, Hermitian algebraic $K$-theory of simplicial rings and topological spaces, J. Math. Pures Appl. (9) 64 (1985), no. 2, 175–235. MR 820117
D. Burghelea and Z. Fiedorowicz, Cyclic homology and algebraic $K$-theory of spaces. II, Topology 25 (1986), no. 3, 303–317. MR 842427, DOI 10.1016/0040-9383(86)90046-7
Gerald Dunn, Dihedral and quaternionic homology and mapping spaces, $K$-Theory 3 (1989), no. 2, 141–161. MR 1029956, DOI 10.1007/BF00533376
W. G. Dwyer, Twisted homological stability for general linear groups, Ann. of Math. (2) 111 (1980), no. 2, 239–251. MR 569072, DOI 10.2307/1971200
F. T. Farrell and W. C. Hsiang, On the rational homotopy groups of the diffeomorphism groups of discs, spheres and aspherical manifolds, 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. 325–337. MR 520509
Paul G. Goerss and John F. Jardine, Simplicial homotopy theory, Modern Birkhäuser Classics, Birkhäuser Verlag, Basel, 2009. Reprint of the 1999 edition [MR1711612]. MR 2840650, DOI 10.1007/978-3-0346-0189-4
Thomas G. Goodwillie, Cyclic homology, derivations, and the free loopspace, Topology 24 (1985), no. 2, 187–215. MR 793184, DOI 10.1016/0040-9383(85)90055-2
Thomas G. Goodwillie, Relative algebraic $K$-theory and cyclic homology, Ann. of Math. (2) 124 (1986), no. 2, 347–402. MR 855300, DOI 10.2307/1971283
A. E. Hatcher, Concordance spaces, higher simple-homotopy theory, and applications, 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. 3–21. MR 520490
Allen Hatcher, Algebraic topology, Cambridge University Press, Cambridge, 2002. MR 1867354
Sze-tsen Hu, Homotopy theory, Pure and Applied Mathematics, Vol. VIII, Academic Press, New York-London, 1959. MR 0106454
Thomas Hüttemann, John R. Klein, Wolrad Vogell, Friedhelm Waldhausen, and Bruce Williams, The “fundamental theorem” for the algebraic $K$-theory of spaces. II. The canonical involution, J. Pure Appl. Algebra 167 (2002), no. 1, 53–82. MR 1868117, DOI 10.1016/S0022-4049(01)00067-6
Kiyoshi Igusa, The stability theorem for smooth pseudoisotopies, $K$-Theory 2 (1988), no. 1-2, vi+355. MR 972368, DOI 10.1007/BF00533643
Kiyoshi Igusa, Higher Franz-Reidemeister torsion, AMS/IP Studies in Advanced Mathematics, vol. 31, American Mathematical Society, Providence, RI; International Press, Somerville, MA, 2002. MR 1945530, DOI 10.1090/amsip/031
R. L. Krasauskas and Yu. P. Solov′ev, Rational Hermitian $K$-theory and dihedral homology, Izv. Akad. Nauk SSSR Ser. Mat. 52 (1988), no. 5, 935–969, 1118 (Russian); English transl., Math. USSR-Izv. 33 (1989), no. 2, 261–293. MR 972090, DOI 10.1070/IM1989v033n02ABEH000826
Gerald M. Lodder, Dihedral homology and the free loop space, Proc. London Math. Soc. (3) 60 (1990), no. 1, 201–224. MR 1023809, DOI 10.1112/plms/s3-60.1.201
Jerry M. Lodder, Dihedral homology and Hermitian $K$-theory, $K$-Theory 10 (1996), no. 2, 175–196. MR 1384461, DOI 10.1007/BF00536611
Daniel Quillen, Higher algebraic $K$-theory. I, Algebraic $K$-theory, I: Higher $K$-theories (Proc. Conf., Battelle Memorial Inst., Seattle, Wash., 1972) Lecture Notes in Math., Vol. 341, Springer, Berlin, 1973, pp. 85–147. MR 0338129
Micheline Vigué-Poirrier and Dan Burghelea, A model for cyclic homology and algebraic $K$-theory of $1$-connected topological spaces, J. Differential Geom. 22 (1985), no. 2, 243–253. MR 834279
Wolrad Vogell, The canonical involution on the algebraic $K$-theory of spaces, Algebraic topology, Aarhus 1982 (Aarhus, 1982) Lecture Notes in Math., vol. 1051, Springer, Berlin, 1984, pp. 156–172. MR 764578, DOI 10.1007/BFb0075566
Wolrad Vogell, The involution in the algebraic $K$-theory of spaces, Algebraic and geometric topology (New Brunswick, N.J., 1983) Lecture Notes in Math., vol. 1126, Springer, Berlin, 1985, pp. 277–317. MR 802795, DOI 10.1007/BFb0074448
Friedhelm Waldhausen, Algebraic $K$-theory of topological spaces. I, 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. 35–60. MR 520492
Friedhelm Waldhausen, Algebraic $K$-theory of spaces, a manifold approach, Current trends in algebraic topology, Part 1 (London, Ont., 1981) CMS Conf. Proc., vol. 2, Amer. Math. Soc., Providence, R.I., 1982, pp. 141–184. MR 686115
Friedhelm Waldhausen, Algebraic $K$-theory of spaces, Algebraic and geometric topology (New Brunswick, N.J., 1983) Lecture Notes in Math., vol. 1126, Springer, Berlin, 1985, pp. 318–419. MR 802796, DOI 10.1007/BFb0074449
Friedhelm Waldhausen, Algebraic $K$-theory of spaces, concordance, and stable homotopy theory, Algebraic topology and algebraic $K$-theory (Princeton, N.J., 1983) Ann. of Math. Stud., vol. 113, Princeton Univ. Press, Princeton, NJ, 1987, pp. 392–417. MR 921486
Friedhelm Waldhausen, Bjørn Jahren, and John Rognes, Spaces of PL manifolds and categories of simple maps, Annals of Mathematics Studies, vol. 186, Princeton University Press, Princeton, NJ, 2013. MR 3202834, DOI 10.1515/9781400846528