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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 \(M^n\).

MSC:

57R22 Topology of vector bundles and fiber bundles
57R91 Equivariant algebraic topology of manifolds
57S17 Finite transformation groups
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References:

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