Bulletin of the Belgian Mathematical Society - Simon Stevin

$\mathbb{Z}_2^k$-actions fixing a disjoint union of odd dimensional projective spaces

Allan E. R. de Andrade, Pedro L.Q. Pergher, and Sérgio T. Urao

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

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 $k=1$.

Article information

Source
Bull. Belg. Math. Soc. Simon Stevin, Volume 24, Number 4 (2017), 581-590.

Dates
First available in Project Euclid: 4 January 2018

Permanent link to this document
https://projecteuclid.org/euclid.bbms/1515035008

Digital Object Identifier
doi:10.36045/bbms/1515035008

Mathematical Reviews number (MathSciNet)
MR3743263

Zentralblatt MATH identifier
06848702

Subjects
Primary: 57R85: Equivariant cobordism
Secondary: 57R75: O- and SO-cobordism

Keywords
$\mathbb{Z}_2^k$-action Stiefel-Whitney class characteristic number characteristic term real projective bundle fixed data equivariant cobordism simultaneous cobordism

Citation

de Andrade, Allan E. R.; Pergher, Pedro L.Q.; Urao, Sérgio T. $\mathbb{Z}_2^k$-actions fixing a disjoint union of odd dimensional projective spaces. Bull. Belg. Math. Soc. Simon Stevin 24 (2017), no. 4, 581--590. doi:10.36045/bbms/1515035008. https://projecteuclid.org/euclid.bbms/1515035008


Export citation