Semicharacteristic classes

Isospectrality and 3-manifold groups

“…Proofs and generalizations (along the lines of Theorem 2.5) of Milnor's result were given by R. Lee [Lee73] and J. Davis [Dav83] using the concept of the semicharacteristic χ 1 2 . In this vein, it seems quite suggestive that for a 3-manifold, we have…”

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

“…For IIL, IVL and VIL (p = -1 mod4), e(π) = 2d(π), but there is a topological obstruction to the existence of free actions in dimension 2d(π) -1, [2]. In these cases Theorem 5 is the best possible geometric result.…”

“…(This can certainly be done, as in the first paragraph of the proof, since any finite polarised complex is homotopy equivalent to a linear space form, see for example [9,Theorem 6].) Over the subgroups ( j, x 2 ' ) and (xy, x 2 ' ) v 2 reduces to the normal invariant of the unique linear space form with fundamental group Dg. The action defining this extends to Γ* and we are done.…”

On the classification of finite groups acting on homology 3-spheres

“…(iii) Suppose that C G E is isomorphic to a group P 48r of type 2.5. As C G E acts freely, it follows from [Mn,Lemma 2] that r = 3 k , and by [L,Corollary 4.17] the group P 48·3 k does not act freely on a homology 3-sphere, for k ≥ 1.…”

“…By [L,Corollary 4.15] the groups Q(8n, k, l) with n even do not act freely on a homology 3-sphere. This leaves for G exactly the possibilities (ii), (iii) and (iv) of the theorem.…”