Surgery on Compact Manifolds

Identifying assembly maps in K - and L -theory

“…One can now construct homotopy invariant functors from Spaces to Spectra by composing with the functor X → Rπ(X), where π(X) denotes the fundamental groupoid of a space equipped with the standard involution and R is a ring with unit. For L-theory we will see in § 10 that this functor agrees on homotopy groups in dimensions ≥ 5 with Wall's geometric definition [Wal70] of the surgery obstruction groups L n (Zπ(X)), and with Quinn's construction of the geometric surgery spectra L geom (X) (a full exposition of this construction has been given by Nicas [Nic82]), and also with the algebraic surgery spectra L alg (Zπ(X)) of Ranicki [Ran92a]. We will also see in § 4 that the Loday assembly map for K-theory can be recovered by this process.…”

“…Spectra were introduced into L-theory by F. Quinn [Qui70] in his thesis, based on the foundational Chapter 9 of C. T. C. Wall's book [Wal70]. A tradition of indexing spectra in L-theory in the opposite direction to the usual way was established.…”

“…As explained in § 2, this is constructed from the connective geometric surgery spectrum L geom defined by Quinn [Qui70] whose homotopy groups π n (L geom (X)) ∼ = L n (Zπ 1 (X, x 0 )) are the geometric surgery obstruction groups of C. T. C. Wall [Wal70]. In particular, π n (L geom (•)) = L n (Z) are the surgery obstruction groups for the trivial group.…”

Manifolds homotopy equivalent to P n # P n

“…López de Medrano [LdM71] and C.T.C. Wall [Wal68,Wal99] classified, up to PL homeomorphism, all closed PL manifolds homotopy equivalent to P n when n > 4. This was extended to the topological category by Kirby-Siebenmann [KS77, p. 331].…”

“…It was defined geometrically by Wall (see [Wal99,Theorem 12.9]) as the Z 2 -equivariant defect for handle exchanges in the middle dimension of a certain regular two-fold cover. The map aqk was dubbed the antiquadratic kernel by Ranicki [Ran81, §7.6 pp.…”