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Hemant Kumar Singh

Publications and source records attributed to Hemant Kumar Singh.

23 records · Page 2Linked to original sources
arXiv2017

Free Action of Finite Groups on Spaces of Cohomology Type (0, b)

Let G be a finite group acting freely on a finitistic space X having cohomology type (0, b) (for example, S^n x S^{2n} is a space of type (0, 1) and the one-point union S^n V S^{2n} V S^{3n} is a space of type (0, 0)). It is known that a finite group G which contains Zp + Zp + Zp, p a prime, can not act freely on S^n x S^{2n}. In this paper, we show that if a finite group G acts freely on a space of type (0, 1), where n is odd, then G can not contain Zp + Zp, p an odd prime. For spaces of cohomology type (0, 0), we show that every p-subgroup of G is either cyclic or a generalized quaternion group. Moreover, for n even, it is shown that Z2 is the only group which can act freely on X.

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arXiv2013

Free actions of connected Lie groups of rank one on certain spaces

Let $G$ be a connected Lie group of rank one. In this paper the existence of free actions of group $G$ on spheres, real projective spaces and lens spaces has been studied. Most of the results have been obtained for finitistic spaces with cohomology ring isomorphic to the cohomology ring of these spaces. The cohomology algebra of the orbit space has been determined in the case of free $G$-action. The non-existence of $G$-equivariant maps from spheres to cohomology spheres has also been established.

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arXiv2008

Cohomology algebra of the orbit space of some free actions on spaces of cohomology type (a, b)

Let X be a finitistic space with non-trivial cohomology groups H^in(X;Z)=Z with generators v_i, where i = 0, 1, 2, 3. We say that X has cohomology type (a, b) if v_1^2 = av_2 and v_1v_2 = bv_3 . In this note, we determine the mod 2 cohomology ring of the orbit space X/G of a free action of G = Z_2 on X, where both a and b are even. In this case, we observed that there is no equivariant map S^m --> X for m > 3n, where S^m has the antipodal action. Moreover, it is shown that G can not act freely on space X which is of cohomology type (a, b) where a is odd and b is even. We also obtain the mod 2 cohomology ring of the orbit space X/G of free action of G = S^1 on the space X of type (0, b).

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arXiv2008

Cohomology algebra of the orbit space of free circle group actions on lens spaces

Suppose that G=S^1 acts freely on a finitistic space X whose mod p cohomology ring isomorphic to that of a lens space L^{2m-1}(p;q_1,...,q_m). In this paper, we determine the mod p cohomology ring of the orbit space X/G. If the characteristic class α\belongs H^2(X/G;Z_p) of the S^1-bundle S--> X--> X/G is nonzero, then mod p ndex of the action is deined to be the largest integer n such that α^n is nonzero. We also show that the mod p index of a free action of S^1 on a lens space L^(2m-1)(p;q_1,...,q_m) is p-1, provided that αis nonzero.

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