## Bulletin of the Belgian Mathematical Society - Simon Stevin

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

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

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

#### 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