Free Finite Group Actions on Rational Homology 3-Spheres

Abstract

We use methods from the cohomology of groups to describe the finite groups which can act freely and homologically trivially on closed 3-manifolds which are rational homology spheres.

Minimal Euler characteristics for even-dimensional manifolds with finite fundamental group

“…Examples of Q𝑆 4 -manifolds can be constructed by starting with a rational homology 3-sphere X, forming the product 𝑋 × 𝑆 1 , and then doing surgery on an embedded 𝑆 1 × 𝐷 3 ⊂ 𝑋 × 𝑆 1 representing a generator of 𝜋 1 (𝑆 1 ) = Z. This construction is equivalent to the 'thickened double' construction 𝑍 = 𝑀 (𝐾) for a finite 2-complex of Proposition 3.1 (compare [13,Section 4]).…”

“…This construction is equivalent to the 'thickened double' construction 𝑍 = 𝑀 (𝐾) for a finite 2-complex of Proposition 3.1 (compare [13,Section 4]). Since the quotient of a free finite group action on a rational homology 3-sphere is again a rational homology 3-sphere, one could use the examples 𝑋 = 𝑌 /𝐺 studied by [1], where Y is a Q𝑆 3 and G is a finite group acting freely on Y. However, to obtain a Q𝑆 4 with finite fundamental group by this construction, Y must, itself, have finite fundamental group.…”

Minimal Euler Characteristics for Even-Dimensional Manifolds with Finite Fundamental Group

“…This construction is equivalent to the "thickened double" construction Z = M(K) for a finite 2-complex of Proposition 3.1 (compare [13, §4]). Since the quotient of a free finite group action on a rational homology 3-sphere is again a rational homology 3-sphere, one could use the examples X = Y /G studied by [1], where Y is a QS 3 and G is a finite group acting freely on Y . However, to obtain a QS 4 with finite fundamental group by this construction, Y must itself have finite fundamental group.…”