Free Actions of Some Compact Groups on Milnor Manifolds

Abstract

In this paper, we investigate free actions of some compact groups on cohomology real and complex Milnor manifolds. More precisely, we compute the mod 2 cohomology algebra of the orbit space of an arbitrary free ℤ2 and $\mathbb{S}^1$-action on a compact Hausdorff space with mod 2 cohomology algebra of a real or a complex Milnor manifold. As applications, we deduce some Borsuk–Ulam type results for equivariant maps between spheres and these spaces. For the complex case, we obtain a lower bound on the Schwarz genus, which further establishes the existence of coincidence points for maps to the Euclidean plane.

LS-category and topological complexity of Milnor manifolds and corresponding generalized projective product space

“…We list here some results from [7] related to free actions of Z 2 and S 1 on Milnor manifolds that we are going to use to compute lower bounds on the (equivariant) LS-category and topological complexity of Milnor manifolds. Then F H r,s admits a free involution if and only if both r and s are odd integers, where F is either R or C .…”

“…Milnor [18, Lemma 1] shown that the unoriented cobordism algebra of smooth manifolds is generated by the cobordism classes of real projective spaces and real Milnor manifolds. P. Dey and M. Singh [7] characterized milnor manifolds which admits free Z 2 and S 1 -actions with restriction on r and computed the equivarinat cohomology rings in this case. The purpose of this paper is to compute the LS-category and topological complexity of Milnor manifolds.…”

Free Actions of Compact Lie Groups on Manifolds

“…□ In order to investigate the existence of free involutions on a Milnor manifold X ¼ Hm , n ðÞ , Dey and Singh [34] showed that if G ¼  2 acts freely on X, with 1 < n < m and m À 2 mod4 ðÞ , then necessarily m and n must be odd. Furthermore, they construct some examples of such free actions and, in this case, it follows that…”

Cohomology Classification of Spaces with Free S3 Actions

“…There are interesting problems related to transformation groups, for example, to classify the fixed point set X G , the existence of free/semifree actions and the study of the orbit space X/G for free actions of G on X. A number of results has been proved in the literature in this direction [1,3,5,6,10,11]. An another thread of research is to classify X for a given orbit space X/G when G acts freely on X. Su [12] proved that if G = S d , d = 0, 1, acts freely on a space X and the orbit space X/G is cohomology FP n , then space X is the cohomology sphere S (d+1)n+d , when d = 0, F = R with Z 2 coefficients, and when d = 1, F = C with integer coefficients.…”