The diffeotopy group of the twisted 2-sphere bundle over the circle

Kim, M. Ho; Raymond, Frank

doi:10.1090/S0002-9947-1990-0991965-6

The diffeotopy group of the twisted $2$-sphere bundle over the circle
HTML articles powered by AMS MathViewer

by M. Ho Kim and Frank Raymond

Trans. Amer. Math. Soc. 322 (1990), 159-168

DOI: https://doi.org/10.1090/S0002-9947-1990-0991965-6

PDF | Request permission

Abstract:

The diffeotopy group of the nontrivial $2$-sphere bundle over the circle is shown to be isomorphic to ${\mathbb {Z}_2} \oplus {\mathbb {Z}_2}$. The first generator is induced by a reflection across the base circle, while a second generator comes from rotating the $2$-sphere fiber as one travels around the base circle. The technique employed also shows that homotopic diffeomorphisms are diffeotopic.

References

M. A. Armstrong, Lifting homotopies through fixed points, Proc. Roy. Soc. Edinburgh Sect. A 93 (1982/83), no. 1-2, 123–128. MR 688291, DOI 10.1017/S0308210500031723
Glen E. Bredon, Introduction to compact transformation groups, Pure and Applied Mathematics, Vol. 46, Academic Press, New York-London, 1972. MR 0413144
David Gabai and William H. Kazez, The classification of maps of nonorientable surfaces, Math. Ann. 281 (1988), no. 4, 687–702. MR 958265, DOI 10.1007/BF01456845
Herman Gluck, The embedding of two-spheres in the four-sphere, Trans. Amer. Math. Soc. 104 (1962), 308–333. MR 146807, DOI 10.1090/S0002-9947-1962-0146807-0
Allen E. Hatcher, A proof of the Smale conjecture, $\textrm {Diff}(S^{3})\simeq \textrm {O}(4)$, Ann. of Math. (2) 117 (1983), no. 3, 553–607. MR 701256, DOI 10.2307/2007035
Morris W. Hirsch, Differential topology, Graduate Texts in Mathematics, No. 33, Springer-Verlag, New York-Heidelberg, 1976. MR 0448362
Sze-tsen Hu, Concerning the homotopy groups of the components of the mapping space $Y^{S^{p}}$, Nederl. Akad. Wetensch., Proc. 49 (1946), 1025–1031 = Indagationes Math. 8, 623–629 (1946). MR 19920
Myung Ho Kim, Sadayoshi Kojima, and Frank Raymond, Homotopy invariants of nonorientable $4$-manifolds, Trans. Amer. Math. Soc. 333 (1992), no. 1, 71–81. MR 1028758, DOI 10.1090/S0002-9947-1992-1028758-1
James R. Munkres, Topology: a first course, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1975. MR 0464128
J. H. Rubinstein, On $3$-manifolds that have finite fundamental group and contain Klein bottles, Trans. Amer. Math. Soc. 251 (1979), 129–137. MR 531972, DOI 10.1090/S0002-9947-1979-0531972-6
Norman Steenrod, The Topology of Fibre Bundles, Princeton Mathematical Series, vol. 14, Princeton University Press, Princeton, N. J., 1951. MR 0039258
Yoko Tao, On fixed point free involutions of $S^{1}\times S^{2}$, Osaka Math. J. 14 (1962), 145–152. MR 140092