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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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