On the constructions of free and locally standard \(\mathbb{Z}_{2}\)-torus actions on manifolds. (English) Zbl 1245.57031

If \((\mathbb{Z}_2)^m\) acts freely and smoothly on a closed manifold \(M^n\), then the orbit space \(Q^n\) is also a closed \(n\)-manifold with \(M^n\) a regular covering of \(Q^n\) with deck transformation group \((\mathbb{Z}_2)^m\). This paper offers a construction which begins with a monochromy homomorphism defined on \(H_{n-1}(Q^n,\mathbb{Z}_2)\) as generated by codimension one submanifolds in general position. The \(\mathbb{Z}_2\)-core of \(Q^n\), \(V^n\), is the complement of tubular neighborhoods of the generators with the natural involutions on the boundaries deformed to commute at intersections. The boundary of \(V^n\) is tessellated by \((n-1)\)-dimensional facets. The union of the facets corresponding to a particular generator is called a panel. Each panel is invariant under the induced involution.
The first major result of this paper asserts that any \(Q^n\) contains a \(\mathbb{Z}_2\)-core and the second describes a glue-back construction in which \(V^n\) and a mapping from the set of panels into \((\mathbb{Z}_2)^m\), called a \((\mathbb{Z}_2)^m\)-coloring, produce an associated \(M^n\).


57R22 Topology of vector bundles and fiber bundles
57R91 Equivariant algebraic topology of manifolds
57S17 Finite transformation groups
Full Text: arXiv Euclid


[1] A. Hatcher: Algebraic Topology, Cambridge Univ. Press, Cambridge, 2002. · Zbl 1044.55001
[2] M.W. Davis and T. Januszkiewicz: Convex polytopes, Coxeter orbifolds and torus actions , Duke Math. J. 62 (1991), 417-451. · Zbl 0733.52006 · doi:10.1215/S0012-7094-91-06217-4
[3] \begingroup Z. Lü and M. Masuda: Equivariant classification of 2-torus manifolds , Colloq. Math. 115 (2009), 171-188. \endgroup · Zbl 1165.57023 · doi:10.4064/cm115-2-3
[4] K. Jänich: On the classification of \(O(n)\)-manifolds , Math. Ann. 176 (1968), 53-76. · Zbl 0153.53801 · doi:10.1007/BF02052956
[5] M.W. Davis: Groups generated by reflections and aspherical manifolds not covered by Euclidean space , Ann. of Math. (2) 117 (1983), 293-324. · Zbl 0531.57041 · doi:10.2307/2007079
[6] V.M. Buchstaber and T.E. Panov: Torus Actions and Their Applications in Topology and Combinatorics, University Lecture Series 24 , Amer. Math. Soc., Providence, RI, 2002. · Zbl 1012.52021
[7] D. Rolfsen: Knots and Links, Publish or Perish, Berkeley, CA, 1976. · Zbl 0339.55004
[9] Z. Lü and L. Yu: Topological types of 3-dimensional small covers , Forum Math. 23 (2011), 245-284. · Zbl 1222.52015 · doi:10.1515/FORM.2011.008
[10] H. Nakayama and Y. Nishimura: The orientability of small covers and coloring simple polytopes , Osaka J. Math. 42 (2005), 243-256. · Zbl 1065.05041
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.