## $(Z_{2})^{k}$-actions whose fixed data has a section

HTML articles powered by AMS MathViewer

- by Pedro L. Q. Pergher PDF
- Trans. Amer. Math. Soc.
**353**(2001), 175-189 Request permission

## Abstract:

Given a collection of $2^{k}-1$ real vector bundles $\varepsilon _{a}$ over a closed manifold $F$, suppose that, for some $a_{0}, \ \varepsilon _{a_{0}}$ is of the form $\varepsilon _{a_{0}}^{\prime }\oplus R$, where $R\to F$ is the trivial one-dimensional bundle. In this paper we prove that if $\bigoplus _{a} \varepsilon _{a} \to F$ is the fixed data of a $(Z_{2})^{k}$-action, then the same is true for the Whitney sum obtained from $\bigoplus _{a} \varepsilon _{a}$ by replacing $\varepsilon _{a_{0}}$ by $\varepsilon _{a_{0}}^{\prime }$. This stability property is well-known for involutions. Together with techniques previously developed, this result is used to describe, up to bordism, all possible $(Z_{2})^{k}$-actions fixing the disjoint union of an even projective space and a point.## References

- 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,
*Bordism of two commuting involutions*, Proc. Amer. Math. Soc.**126**(1998), no. 7, 2141–2149. MR**1451825**, DOI 10.1090/S0002-9939-98-04356-1 - 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** - R. E. Stong,
*Involutions fixing projective spaces*, Michigan Math. J.**13**(1966), 445–447. MR**206979**

## Additional Information

**Pedro L. Q. Pergher**- Affiliation: Departamento de Matemática, Universidade Federal de São Carlos, Caixa Postal 676, CEP 13.565-905, São Carlos, SP, Brazil
- Email: pergher@dm.ufscar.br
- Received by editor(s): November 11, 1998
- Published electronically: June 21, 2000
- Additional Notes: The present work was partially supported by CNPq
- © Copyright 2000 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**353**(2001), 175-189 - MSC (2000): Primary 57R85; Secondary 57R75
- DOI: https://doi.org/10.1090/S0002-9947-00-02645-3
- MathSciNet review: 1783791