The topological spherical space form problem—II existence of free actions

Constructing free actions ofp–groups on products of spheres

“…Later Milnor [19] gave a second necessary condition: If a finite group G acts freely on a sphere, then any element in G of order 2 must be central. Finally Thomas, Wall and Madsen [18] were able to prove the these two necessary conditions were in fact sufficient: a finite group G acts freely on a sphere if and only if it has periodic cohomology and all involutions are central.…”

Topological spherical space form problem. III. Dimensional bounds and smoothing

“…As in [3] we now show that the surgery obstruction σ (7) to replacing Y(π) by a smooth manifold vanishes. Note that condition (ii) implies that σ( Y(p)) = 0 for every 2-subgroup of π.…”

“…It now follows that the finiteness obstruction for the π-resolution vanishes, and that there is a finite geometric realisation, Y(π). Following [3] we construct a smooth normal invariant, and show that the surgery obstruction to replacing Y(π) by a homotopy equivalent smooth manifold vanishes. By variation inside the orbit of this manifold under the action of L 2e (π) (Zπ) on § 0 (7(ττ)) we show that the universal cover F(l) may be taken to be the standard sphere.…”

Topological Transformation Groups

“…This problem has been solved when the number of spheres is one; there is the result of Madsen-Thomas-Wall [86] which states that a finite group G acts freely on some sphere if and only if G has no non-cyclic abelian subgroups and no dihedral subgroups. When the number of spheres are greater than one, we discussed algebraic work in section 2, but little geometric work has been done on this problem (but see [105], [67], and [46]).…”

“…For background on classifying spaces and bundle theory see Milgram-Madsen [92]. For information specific to spherical space forms, see DavisMilgram [45], Wall [136] and Madsen-Thomas-Wall [86]. Modern aspects of surgery theory can be found in the books of Ranicki [113] and Weinberger [142], however they were not written with classical surgery theory as their main focus.…”