Bordism of two commuting involutions

Pergher, Pedro

doi:10.1090/S0002-9939-98-04356-1

Bordism of two commuting involutions
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by Pedro L. Q. Pergher PDF

Proc. Amer. Math. Soc. 126 (1998), 2141-2149 Request permission

Abstract:

In this paper we obtain conditions for a Whitney sum of three vector bundles over a closed manifold, $\varepsilon _{1} \oplus \varepsilon _{2} \oplus \varepsilon _{3} \rightarrow F$, to be the fixed data of a $(Z_{2})^{2}$-action; these conditions yield the fact that if $(\varepsilon _{1} \oplus R) \oplus \varepsilon _{2} \oplus \varepsilon _{3} \rightarrow F$ is the fixed data of a $(Z_{2})^{2}$-action, where $R \rightarrow F$ is the trivial one dimensional bundle, then the same is true for $\varepsilon _{1} \oplus \varepsilon _{2} \oplus \varepsilon _{3} \rightarrow F$. The results obtained, together with techniques previously developed, are used to obtain, up to bordism, all possible $(Z_{2})^{2}$-actions fixing the disjoint union of an even projective space and a point.

References

A. Borel and F. Hirzebruch, Characteristic classes and homogeneous spaces. I, Amer. J. Math. 80 (1958), 458–538. MR 102800, DOI 10.2307/2372795
Czes Kosniowski and R. E. Stong, Involutions and characteristic numbers, Topology 17 (1978), no. 4, 309–330. MR 516213, DOI 10.1016/0040-9383(78)90001-0
David C. Royster, Involutions fixing the disjoint union of two projective spaces, Indiana Univ. Math. J. 29 (1980), no. 2, 267–276. MR 563211, DOI 10.1512/iumj.1980.29.29018
P. E. Conner and E. E. Floyd, Differentiable periodic maps, Ergebnisse der Mathematik und ihrer Grenzgebiete, (N.F.), Band 33, Academic Press, Inc., Publishers, New York; Springer-Verlag, Berlin-Göttingen-Heidelberg, 1964. MR 0176478
Pedro L. Q. Pergher, An equivariant construction, Proc. Amer. Math. Soc. 119 (1993), no. 1, 319–320. MR 1152991, DOI 10.1090/S0002-9939-1993-1152991-8
Pedro L. Q. Pergher, Manifolds with $(\textbf {Z}_2)^k$-action, Proc. Amer. Math. Soc. 106 (1989), no. 4, 1091–1094. MR 969320, DOI 10.1090/S0002-9939-1989-0969320-1
Pedro L. Q. Pergher, The union of a connected manifold and a point as fixed set of commuting involutions, Topology Appl. 69 (1996), no. 1, 71–81. MR 1378389, DOI 10.1016/0166-8641(95)00075-5
Pedro L. Q. Pergher, $(Z_2)^k$-actions fixing a product of spheres and a point, Canad. Math. Bull. 38 (1995), no. 3, 366–372. MR 1347311, DOI 10.4153/CMB-1995-053-1
R. E. Stong, Bordism and involutions, Ann. of Math. (2) 90 (1969), 47–74. MR 242170, DOI 10.2307/1970681
R. E. Stong, Equivariant bordism and $(Z_{2})^{k}$ actions, Duke Math. J. 37 (1970), 779–785. MR 271966, DOI 10.1215/S0012-7094-70-03793-2
R. E. Stong, Involutions fixing projective spaces, Michigan Math. J. 13 (1966), 445–447. MR 206979, DOI 10.1307/mmj/1028999602