## Some applications of algebraic surgery theory: 4-manifolds, triangular matrix rings and braids

##### Abstract

This thesis consists of three applications of Ranicki's algebraic theory of surgery to the topology
of manifolds. The common theme is a decomposition of a global algebraic object into simple local
pieces which models the decomposition of a global topological object into simple local pieces.
Part I: Algebraic reconstruction of 4-manifolds. We extend the product and glueing
constructions for symmetric Poincaré complexes, pairs and triads to a thickening construction
for a symmetric Poincaré representation of a quiver. Gay and Kirby showed that, subject to certain
conditions, the fold curves and fibres of a Morse 2-function F : M4 → Ʃ 2 determine a quiver
of manifold and glueing data which allows one to reconstruct M and F up to diffeomorphism.
The Gay-Kirby method of reconstructing M glues the pre-images of disc neighbourhoods of
cusps and crossings with thickenings of regular fibres and thickenings of cobordisms between
regular fibres. We use our thickening construction for a symmetric Poincaré representation of
a quiver to give an algebraic analogue of the Gay-Kirby result to reconstruct the symmetric
Poincaré complex (C(M); ϕ M) of M from a Morse 2-function.
Part II: The L-theory of triangular matrix rings. We construct a chain duality on the
category of left modules over a triangular matrix ring A = (A1;A2;B) where A1;A2 are rings
with involution and B is an (A1;A2)-bimodule. We describe the resulting L-theory of A and
relate it to the L-theory of A1;A2 and to the change of rings morphism B ⊗A2 − : A2-Mod →
A1-Mod. By examining algebraic surgery over A we define a relative algebraic surgery operation
on an (n+1)-dimensional symmetric Poincaré pair with data an (n+2)-dimensional triad.
This gives an algebraic model for a half-surgery on a manifold with boundary. We then give an
algebraic analogue of Borodzik, Némethi and Ranciki's half-handle decomposition of a relative
manifold cobordism and show that every relative Poincaré cobordism is homotopy equivalent
to a union of traces of elementary relative surgeries.
Part III: Seifert matrices of braids with applications to isotopy and signatures.
Let β be a braid with closure ^β a link. Collins developed an algorithm to find the Seifert
matrix of the canonical Seifert surface Ʃ of ^ β constructed by Seifert's algorithm. Motivated
by Collins' algorithm and a construction of Ghys, we define a 1-dimensional simplicial complex
K(β) and a bilinear form λβ : C1(K(β);Z)×C1(K(β);Z) → Z[ 1/2 ] such that there is an inclusion
K(β) ~ → Ʃ which is a homotopy equivalence inducing an isomorphism H1(Ʃ;Z) ≅ H1(K(β);Z)
such that [λβ] : H1(K(β);Z) × H1(K(β);Z) → Z ⊂ Z[ 1/2 ] is the Seifert form of Ʃ. We show
that this chain level model is additive under the concatenation of braids and then verify that
this model is chain equivalent to Banchoff's combinatorial model for the linking number of two
space polygons and Ranicki's surgery theoretic model for a chain level Seifert pairing. We then
define the chain level Seifert pair (λβ; d β) of a braid β and equivalence relations, called A and
Â-equivalence. Two n-strand braids are isotopic if and only if their chain level Seifert pairs are
A-equivalent and this yields a universal representation of the braid group. Two n-strand braids
have isotopic link closures in the solid torus D2 ×S1 if and only if their chain level Seifert pairs
are Â A-equivalent and this yields a representation of the braid group modulo conjugacy. We use
the first representation to express the ω signature of a braid β in terms of the chain level Seifert
pair (λ β; d β).