Involutions on S6 with 3–dimensional fixed point set

Olbermann, Martin¹

¹Max-Planck-Institut für Mathematik, Vivatsgasse 7, 53111 Bonn, Germany

Algebraic and Geometric Topology, Tome 10 (2010) no. 4, pp. 1905-1932

Résumé

In this article, we classify all involutions on S6 with 3–dimensional fixed point set. In particular, we discuss the relation between the classification of involutions with fixed point set a knotted 3–sphere and the classification of free involutions on homotopy ℂ P3’s.

Détail

DOI : 10.2140/agt.2010.10.1905

Keywords: involution, sphere, conjugation, fixed point set, surgery, Smith theory

Bibliographie

[1] G E Bredon, Introduction to compact transformation groups, Pure and Applied Math. 46, Academic Press (1972)

[2] J Cerf, Sur les difféomorphismes de la sphère de dimension trois $(\Gamma _{4}=0)$, Lecture Notes in Math. 53, Springer (1968)

[3] C W Curtis, I Reiner, Representation theory of finite groups and associative algebras, Pure and Applied Math. XI, Wiley-Interscience (1962)

[4] J F Davis, S Weinberger, Characteristic fixed-point sets of semifree actions on spheres, Comm. Pure Appl. Math. 52 (1999) 935

[5] M W Davis, T Januszkiewicz, Convex polytopes, Coxeter orbifolds and torus actions, Duke Math. J. 62 (1991) 417

[6] K H Dovermann, M Masuda, R Schultz, Conjugation involutions on homotopy complex projective spaces, Japan. J. Math. $($N.S.$)$ 12 (1986) 1

[7] M Franz, V Puppe, Steenrod squares on conjugation spaces, C. R. Math. Acad. Sci. Paris 342 (2006) 187

[8] A Haefliger, Knotted $(4k{-}1)$–spheres in $6k$-space, Ann. of Math. $(2)$ 75 (1962) 452

[9] I Hambleton, J C Hausmann, Conjugation spaces and $4$–manifolds

[10] J C Hausmann, T Holm, Conjugation spaces and edges of compatible torus actions

[11] J C Hausmann, T Holm, V Puppe, Conjugation spaces, Algebr. Geom. Topol. 5 (2005) 923

[12] J C Hausmann, A Knutson, The cohomology ring of polygon spaces, Ann. Inst. Fourier (Grenoble) 48 (1998) 281

[13] M Kreck, An extension of results of Browder, Novikov and Wall about surgery on compact manifolds, Preprint (1985)

[14] M Kreck, Surgery and duality, Ann. of Math. $(2)$ 149 (1999) 707

[15] B H Li, Z Lü, Smooth free involution of $H\mathbb{C}\mathrm P^3$ and Smith conjecture for imbeddings of $S^3$ in $S^6$, Math. Ann. 339 (2007) 879

[16] P Löffler, Homotopielineare $Z_{p}$–Operationen auf Sphären, Topology 20 (1981) 291

[17] M Masuda, Review of \citePaw, Math Reviews (2002)

[18] D Montgomery, C T Yang, Differentiable actions on homotopy seven spheres, Trans. Amer. Math. Soc. 122 (1966) 480

[19] M Olbermann, Conjugations on $6$–manifolds with free integral cohomology

[20] M Olbermann, Conjugations on $6$–manifolds, PhD thesis, Universität Heidelberg (2007)

[21] K Pawalowski, Manifolds as fixed point sets of smooth compact Lie group actions, from: "Current trends in transformation groups" (editors A Bak, M Morimoto, F Ushitaki), $K$–Monogr. Math. 7, Kluwer Acad. Publ. (2002) 79

[22] T Petrie, Involutions on homotopy complex projective spaces and related topics, from: "Proceedings of the Second Conference on Compact Transformation Groups (Univ. Massachusetts, Amherst, 1971), Part I" (editors H T Ku, L N Mann, J L Sicks, J C Su), Lecture Notes in Math. 298, Springer (1972) 234

[23] R H Sady, Free involutions of complex projective spaces, Michigan Math. J. 24 (1977) 51

[24] R Schultz, Exotic spheres as stationary sets of homotopy sphere involutions, Michigan Math. J. 29 (1982) 121

[25] A Skopenkov, A classification of smooth embeddings of $3$–manifolds in $6$–space, Math. Z. 260 (2008) 647

[26] Y Su, A note on the $\Z_2$–equivariant Montgomery–Yang correspondence

[27] P Teichner, On the signature of four-manifolds with universal covering spin, Math. Ann. 295 (1993) 745

[28] C T C Wall, Classification problems in differential topology. V. On certain $6$–manifolds, Invent. Math. 1 $(1966)$, 355-374; corrigendum, ibid 2 (1966) 306