Evaluation of odd-dimensional surgery obstructions with finite fundamental group (Q1082658)

[A preliminary version was published in Math. Gottingensis, Schriftenr. Sonderforschungsbereichs Geom. Anal. 58, 47 pp. (1986).] The key tool for classification of manifolds of dimension greater than four and for determining the existence of a manifold structure within a homotopy type is surgery. The obstructions to surgery lie in Grothendieck groups of quadratic forms - the algebraic L-groups. One needs a method for evaluating a particular element in an L-group effectively for geometric situations. Our eventual aim is to have a formula for the surgery obstruction that could be read off a ''sufficiently good normal map'', and so that sufficiently good normal maps are rather common. We present a two-step obstruction theory. The primary obstruction is the surgery semicharacteristic, which measures the obstruction to doing surgery to a mod 2 homology equivalence. The secondary obstructions are ''numerical invariants'' depending only on the surgery kernels \(K_ *(M)\). This theory has several applications to group actions on spheres. An appendix is included discussing the relationship between Galois cohomology and quadratic forms.