The real moment-angle complex (or, more generally, real polyhedral product) and its real toric space have recently attracted much attention in toric topology. The aim of this paper is to give two interesting remarks regarding real polyhedral products and real toric spaces. That is, we first show that Moore"s conjecture holds to be true for certain real polyhedral products. In general, real polyhedral products show some drastic difference between the rational and torsion homotopy groups. Our result shows that at least in terms of the homotopy exponent at a prime this is not the case for real polyhedral products associated to a simplicial complex whose minimal missing faces are all $k$-simplices with $k\ge 2$. Moreover, we also show a structural theorem for a finite group $G$ acting simplicially on the real toric space. In other words, we show that $G$ always contains an element of order $2$, and so the order of $G$ should be even.
The real moment-angle complex (or, more generally, real polyhedral product) and its real toric space have recently attracted much attention in toric topology. The aim of this paper is to give two interesting remarks regarding real polyhedral products and real toric spaces. That is, we first show that Moore"s conjecture holds to be true for certain real polyhedral products. In general, real polyhedral products show some drastic difference between the rational and torsion homotopy groups. Our result shows that at least in terms of the homotopy exponent at a prime this is not the case for real polyhedral products associated to a simplicial complex whose minimal missing faces are all $k$-simplices with $k\ge 2$. Moreover, we also show a structural theorem for a finite group $G$ acting simplicially on the real toric space. In other words, we show that $G$ always contains an element of order $2$, and so the order of $G$ should be even.
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