Translator Disclaimer
december 2017 $\mathbb{Z}_2^k$-actions fixing a disjoint union of odd dimensional projective spaces
Allan E. R. de Andrade, Pedro L.Q. Pergher, Sérgio T. Urao
Bull. Belg. Math. Soc. Simon Stevin 24(4): 581-590 (december 2017). DOI: 10.36045/bbms/1515035008

Abstract

Consider the real, complex and quaternionic $n$-dimensional projective spaces, $\mathbb{R}P^n$, $\mathbb{C}P^n$ and $\mathbb{H}P^n$; to unify notation, write $K_dP^n$ for the real ($d=1$), complex ($d=2$) and quaternionic ($d=4$) $n$-dimensional projective space. Consider a pair $(M,\Phi)$, where $M$ is a closed smooth manifold and $\Phi$ is a smooth action of the group $\mathbb{Z}_2^k$ on $M$; here, $\mathbb{Z}_2^k$ is considered as the group generated by $k$ commuting smooth involutions $T_1,T_2,...,T_k$. Write $F$ for the fixed-point set of $\Phi$. In this paper we prove the following two results: i) If $F$ is a disjoint union $F=\mathbb{R}P^{n_1} \sqcup \mathbb{R}P^{n_2} \sqcup ... \sqcup \mathbb{R}P^{n_j}$, where $j \ge 2$, each $n_i$ is odd and $n_i \not=n_t$ if $i \not= t$, then $(M,\Phi)$ bounds equivariantly. ii) If $F= K_dP^n \sqcup K_dP^m$, where $d=1,2$ and $4$ and $n$ and $m$ are odd, then $(M,\Phi)$ bounds equivariantly. These results are found in the literature for $k=1$.

Citation

Download Citation

Allan E. R. de Andrade. Pedro L.Q. Pergher. Sérgio T. Urao. "$\mathbb{Z}_2^k$-actions fixing a disjoint union of odd dimensional projective spaces." Bull. Belg. Math. Soc. Simon Stevin 24 (4) 581 - 590, december 2017. https://doi.org/10.36045/bbms/1515035008

Information

Published: december 2017
First available in Project Euclid: 4 January 2018

zbMATH: 06848702
MathSciNet: MR3743263
Digital Object Identifier: 10.36045/bbms/1515035008

Subjects:
Primary: 57R85
Secondary: 57R75

Rights: Copyright © 2017 The Belgian Mathematical Society

JOURNAL ARTICLE
10 PAGES

This article is only available to subscribers.
It is not available for individual sale.
+ SAVE TO MY LIBRARY

SHARE
Vol.24 • No. 4 • december 2017
Back to Top