## Algebraic & Geometric Topology

### Constructing free actions of $p$–groups on products of spheres

Michele Klaus

#### Abstract

We prove that, for $p$ an odd prime, every finite $p$–group of rank $3$ acts freely on a finite complex $X$ homotopy equivalent to a product of three spheres.

#### Article information

Source
Algebr. Geom. Topol., Volume 11, Number 5 (2011), 3065-3084.

Dates
Revised: 11 July 2011
Accepted: 25 August 2011
First available in Project Euclid: 19 December 2017

https://projecteuclid.org/euclid.agt/1513715315

Digital Object Identifier
doi:10.2140/agt.2011.11.3065

Mathematical Reviews number (MathSciNet)
MR2869452

Zentralblatt MATH identifier
1237.57037

Subjects
Primary: 57S17: Finite transformation groups

#### Citation

Klaus, Michele. Constructing free actions of $p$–groups on products of spheres. Algebr. Geom. Topol. 11 (2011), no. 5, 3065--3084. doi:10.2140/agt.2011.11.3065. https://projecteuclid.org/euclid.agt/1513715315

#### References

• A Adem, Torsion in equivariant cohomology, Comment. Math. Helv. 64 (1989) 401–411
• A Adem, Lectures on the cohomology of finite groups, from: “Interactions between homotopy theory and algebra”, (L L Avramov, J D Christensen, W G Dwyer, M A Mandell, B E Shipley, editors), Contemp. Math. 436, Amer. Math. Soc. (2007) 317–334
• A Adem, J H Smith, Periodic complexes and group actions, Ann. of Math. $(2)$ 154 (2001) 407–435
• D J Benson, Representations and cohomology, I: Basic representation theory of finite groups and associative algebras, Cambridge Studies in Advanced Math. 30, Cambridge Univ. Press (1991)
• D J Benson, Representations and cohomology, II: Cohomology of groups and modules, Cambridge Studies in Advanced Math. 31, Cambridge Univ. Press (1991)
• D J Benson, J F Carlson, Complexity and multiple complexes, Math. Z. 195 (1987) 221–238
• K S Brown, Cohomology of groups, Graduate Texts in Math. 87, Springer, New York (1982)
• F X Connolly, S Prassidis, Groups which act freely on ${\bf R}\sp m\times S\sp {n-1}$, Topology 28 (1989) 133–148
• T tom Dieck, Transformation groups and representation theory, Lecture Notes in Math. 766, Springer, Berlin (1979)
• T tom Dieck, Transformation groups, de Gruyter Studies in Math. 8, de Gruyter, Berlin (1987)
• R M Dotzel, G C Hamrick, $p$–group actions on homology spheres, Invent. Math. 62 (1981) 437–442
• D L Ferrario, Self homotopy equivalences of equivariant spheres, from: “Groups of homotopy self-equivalences and related topics (Gargnano, 1999)”, (K-i Maruyama, J W Rutter, editors), Contemp. Math. 274, Amer. Math. Soc. (2001) 105–131
• B Hanke, The stable free rank of symmetry of products of spheres, Invent. Math. 178 (2009) 265–298
• A Heller, A note on spaces with operators, Illinois J. Math. 3 (1959) 98–100
• A G Ilhan, Obstructions for constructing $G$–equivariant fibrations, PhD thesis, Bilkent University (2011)
• M A Jackson, Rank three $p$–groups and free actions on the homotopy product of three spheres, in preparation
• M A Jackson, ${\rm Qd}(p)$–free rank two finite groups act freely on a homotopy product of two spheres, J. Pure Appl. Algebra 208 (2007) 821–831
• I Madsen, C B Thomas, C T C Wall, The topological spherical space form problem–-II existence of free actions, Topology 15 (1976) 375–382
• J Milnor, Groups which act on $S\sp n$ without fixed points, Amer. J. Math. 79 (1957) 623–630
• J-P Serre, Cohomologie des groupes discrets, from: “Prospects in mathematics (Proc. Sympos., Princeton Univ., 1970)”, (F Hirzebruch, L Hörmander, J Milnor, J-P Serre, Z M Singer, editors), Ann. of Math. Studies 70, Princeton Univ. Press (1971) 77–169
• P A Smith, Permutable periodic transformations, Proc. Nat. Acad. Sci. U. S. A. 30 (1944) 105–108
• P A Smith, New results and old problems in finite transformation groups, Bull. Amer. Math. Soc. 66 (1960) 401–415
• M Suzuki, Group theory II, Grund. der Math. Wissenschaften 248, Springer, New York (1986) Translated from the Japanese
• R G Swan, Periodic resolutions for finite groups, Ann. of Math. $(2)$ 72 (1960) 267–291
• O \Unl\Hu, E Yalçin, Fusion systems and constructing free actions on products of spheres