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S-cubes. (English) Zbl 0591.57014
Let \(I^ n\) be the n-dimensional cube and \(J^ n_ i\) its i-th ”double face”. Let \(s_ i: \partial I^ n\to \partial I^ n\) be the symmetry of \(\partial I^ n\) with respect to the hyperplane \(x_ i=0\). Denote by G the group generated by the set \(\{s_ 1,...,s_ n\}\). To each n-tuple \((u^ 1,...,u^ n)\in G^ n\) a factor space of \(I^ n\) is assigned as follows: Let S be the binary relation on \(I^ n\) defined via xSy \(\Leftrightarrow x=y\) or there is an index \(i\in \{1,2,...,n\}\) such that \(x,y\in J^ n_ i\) and \(x=u^ i(y)\). The space \(I^ n/T\), where T is the smallest equivalence relation on \(I^ n\) containing S, is denoted by \(I^ n/(u^ 1,...,u^ n)\) and called an s-cube.
In the paper some basic properties of s-cubes are proved. A necessary and sufficient condition for an s-cube to be a manifold is found.

MSC:
57S17 Finite transformation groups
57N15 Topology of the Euclidean \(n\)-space, \(n\)-manifolds (\(4 \leq n \leq \infty\)) (MSC2010)
54B15 Quotient spaces, decompositions in general topology
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References:
[1] DOLD A.: Lectures on Algebraic Topology. Springer Verlag, Berlin-Heidelberg-New York 1972. · Zbl 0234.55001
[2] DUDÁŠIKOVÁ H.: Priestor\( I^n/(s_{12\ldots k, \ldots, s_{12\ldots k)}, }k>1\), je suspenzia nad \(RP^k\). Záverečná správa fakultnej výskumnej úlohy UK 364, Bratislava 1976.
[3] KULICH I., TVAROŽEK J.: Priestor \(I^n/(s_{j_1}, \ldots , s_{j_n})\) je súčinom sfér. Záverečná správa FVÚ UK 364, Bratislava 1976.
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