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


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

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

First available in Project Euclid: 4 January 2018

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

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


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