The rational symmetric signature of manifolds with finite fundamental group

Knot concordance, Whitney towers and L2-signatures

“…by Proposition 2.11 and Remark 2.8.3, the chain complex of the induced cover of W with coefficients in K is a 4-dimensional symmetric Poincaré complex over K, called the symmetric chain complex of W , and hence represents an element B in L 0 (K), the cobordism classes of such complexes [Ra2,. Since all K-modules are free, this complex is known to be cobordant to one given by the intersection form λ on H 2 (W ; K) (which is nonsingular by the above remarks and is discussed in detail in Section 7) [Da,Lemma 4.4 (ii)]. Moreover, in this case L 0 (K) is known to be isomorphic to the usual Witt group of nonsingular hermitian forms on finitely-generated K modules.…”

Integral cohomology and detection of 𝑤-basic 2-groups

“…The results of this paper are applied by Davis in [3] to classify equivariant intersection forms arising from closed manifolds which are the total space of a finite G-cover. These forms are analyzed by characteristic class formulae involving higher-index homomorphisms whose domain is the homology of G. Quadratic representation theory [7] gives a reduction to the homology of w-basic 2-groups, where an explicit analysis is required.…”