Kervaire invariants and selfcoincidences

Koschorke, Ulrich¹ ; Randall, Duane²

¹ FB 6 - Mathematik V, Universität Siegen, Emmy Noether Campus, Walter-Flex-Str. 3, D-57068 Siegen, Germany
² Loyola University New Orleans, Mathematical Sciences, 6363 St. Charles Ave., Campus Box 35, New Orleans, LA 70118, USA

Geometry & topology, Tome 17 (2013) no. 2, pp. 621-638.

Résumé

Minimum numbers decide, eg, whether a given map $f:S^m\to S^n∕G$ from a sphere into a spherical space form can be deformed to a map $f^{\prime }$ such that $f\left(x\right)\ne f^{\prime }\left(x\right)$ for all $x\in S^m$. In this paper we compare minimum numbers to (geometrically defined) Nielsen numbers (which are more computable). In the stable dimension range these numbers coincide. But already in the first nonstable range (when $m=2n-2$) the Kervaire invariant appears as a decisive additional obstruction which detects interesting geometric coincidence phenomena. Similar results (involving, eg, Hopf invariants taken mod 4) are obtained in the next seven dimension ranges (when $1<m-2n+3\le 8$). The selfcoincidence context yields also a precise geometric criterion for the open question whether the Kervaire invariant vanishes on the 126–stem or not.

Détail

DOI : 10.2140/gt.2013.17.621

Classification : 55M20, 55P40, 55Q15, 55Q25, 57R99, 55Q45
Keywords: Kervaire invariant, coincidence, Nielsen number

Bibliographie

[1] J F Adams, On the non-existence of elements of Hopf invariant one, Ann. of Math. 72 (1960) 20

[2] M G Barratt, J D S Jones, M E Mahowald, The Kervaire invariant problem, from: "Proceedings of the Northwestern Homotopy Theory Conference" (editors H R Miller, S B Priddy), Contemp. Math. 19, Amer. Math. Soc. (1983) 9

[3] M G Barratt, J D S Jones, M E Mahowald, Relations amongst Toda brackets and the Kervaire invariant in dimension $62$, J. London Math. Soc. 30 (1984) 533

[4] M G Barratt, J D S Jones, M E Mahowald, The Kervaire invariant and the Hopf invariant, from: "Algebraic topology" (editors H R Miller, D C Ravenel), Lecture Notes in Math. 1286, Springer (1987) 135

[5] J M Boardman, B Steer, On Hopf invariants, Comment. Math. Helv. 42 (1967) 180

[6] W Browder, The Kervaire invariant of framed manifolds and its generalization, Ann. of Math. 90 (1969) 157

[7] E H Brown Jr., F P Peterson, Whitehead products and cohomology operations, Quart. J. Math. Oxford Ser. 15 (1964) 116

[8] R F Brown, Wecken properties for manifolds, from: "Nielsen theory and dynamical systems" (editor C K McCord), Contemp. Math. 152, Amer. Math. Soc. (1993) 9

[9] F R Cohen, Fibration and product decompositions in nonstable homotopy theory, from: "Handbook of algebraic topology" (editor I M James), North-Holland (1995) 1175

[10] M C Crabb, The homotopy coincidence index, J. Fixed Point Theory Appl. 7 (2010) 1

[11] A Dold, D L Gonçalves, Self-coincidence of fibre maps, Osaka J. Math. 42 (2005) 291

[12] M Golasiński, J Mukai, Gottlieb groups of spheres, Topology 47 (2008) 399

[13] D L Gonçalves, D Randall, Self-coincidence of maps from $S^q$–bundles over $S^n$ to $S^n$, Bol. Soc. Mat. Mexicana 10 (2004) 181

[14] D Gonçalves, D Randall, Self-coincidence of mappings between spheres and the strong Kervaire invariant one problem, C. R. Math. Acad. Sci. Paris 342 (2006) 511

[15] M Hill, M Hopkins, D Ravenel, On the non-existence of elements of Kervaire invariant one (2010)

[16] I M James, Products on spheres, Mathematika 6 (1959) 1

[17] M A Kervaire, A manifold which does not admit any differentiable structure, Comment. Math. Helv. 34 (1960) 257

[18] M A Kervaire, J W Milnor, Groups of homotopy spheres, I, Ann. of Math. 77 (1963) 504

[19] S O Kochman, M E Mahowald, On the computation of stable stems, from: "The Čech centennial" (editors M Cenkl, H Miller), Contemp. Math. 181, Amer. Math. Soc. (1995) 299

[20] U Koschorke, Selfcoincidences in higher codimensions, J. Reine Angew. Math. 576 (2004) 1

[21] U Koschorke, Nonstabilized Nielsen coincidence invariants and Hopf–Ganea homomorphisms, Geom. Topol. 10 (2006) 619

[22] U Koschorke, Minimizing coincidence numbers of maps into projective spaces, from: "The Zieschang Gedenkschrift" (editors M Boileau, M Scharlemann, R Weidmann), Geom. Topol. Monogr. 14, Geom. Topol. Publ., Coventry (2008) 373

[23] U Koschorke, Minimum numbers and Wecken theorems in topological coincidence theory, I, J. Fixed Point Theory Appl. 10 (2011) 3

[24] K Y Lam, D Randall, Block bundle obstruction to Kervaire invariant one, from: "Recent developments in algebraic topology" (editors A Ádem, J González, G Pastor), Contemp. Math. 407, Amer. Math. Soc. (2006) 163

[25] J Milnor, Differential topology forty-six years later, Notices Amer. Math. Soc. 58 (2011) 804

[26] Y Nomura, Some homotopy groups of real Stiefel manifolds in the metastable range, V, Sci. Rep. College Gen. Ed. Osaka Univ. 29 (1980) 159

[27] H Toda, Composition methods in homotopy groups of spheres, Annals of Mathematics Studies 49, Princeton Univ. Press (1962)

[28] G W Whitehead, Elements of homotopy theory, Graduate Texts in Mathematics 61, Springer (1978)