Suen, Chun Kit Anthony. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2007. / Includes bibliographical references (leaves 89-92). / Abstracts in English and Chinese. / Abstract --- p.iii / Abstract --- p.iv / Acknowledgements --- p.v / Chapter 1 --- Introduction --- p.1 / Chapter 2 --- Morse Homology --- p.5 / Chapter 2.1 --- Introduction --- p.5 / Chapter 2.2 --- Classical Morse Theory and Morse Functions --- p.5 / Chapter 2.3 --- Handlebody Decomposition for 3-manifold --- p.7 / Chapter 2.4 --- Stable manifold and Unstable manifold --- p.10 / Chapter 2.5 --- Trajectory flows and the Morse-Smale-Witten Complex --- p.11 / Chapter 3 --- Lagrangian Floer Homology --- p.22 / Chapter 3.1 --- Introduction --- p.22 / Chapter 3.2 --- Preliminaries on Symplectic Geometry --- p.23 / Chapter 3.2.1 --- Basic Definitions --- p.23 / Chapter 3.2.2 --- The Symplectic Group --- p.26 / Chapter 3.2.3 --- Maslov index for non-degenerate paths in Sp(2n) --- p.28 / Chapter 3.2.4 --- Maslov index - the analytic aspect --- p.35 / Chapter 3.3 --- Definition of Floer Homology --- p.37 / Chapter 3.4 --- Some Remarks --- p.41 / Chapter 4 --- Heegaard Floer Homology --- p.43 / Chapter 4.1 --- Introduction --- p.43 / Chapter 4.2 --- Basic Set-Up --- p.43 / Chapter 4.3 --- Topological Preliminaries --- p.44 / Chapter 4.3.1 --- Symmetric Product --- p.44 / Chapter 4.3.2 --- The Tori Tα and Tβ --- p.47 / Chapter 4.3.3 --- Intersection Points and Disks --- p.48 / Chapter 4.3.4 --- Domains --- p.52 / Chapter 4.3.5 --- Spinc Structures --- p.54 / Chapter 4.3.6 --- Holomorphic Disks and Maslov Index --- p.63 / Chapter 4.4 --- Definition of Heegaard Floer Homology --- p.65 / Chapter 4.4.1 --- The chain complex CF --- p.66 / Chapter 4.4.2 --- The chain complex CF∞ --- p.67 / Chapter 4.4.3 --- The chain complexes CF+ and CF- --- p.68 / Chapter 4.4.4 --- Some Remarks --- p.70 / Chapter 5 --- Examples and Applications --- p.72 / Chapter 5.1 --- Introduction --- p.72 / Chapter 5.2 --- The homology three-spheres --- p.72 / Chapter 5.2.1 --- The sphere S3 --- p.72 / Chapter 5.2.2 --- The Poincare sphere and the Brieskorn spheres --- p.74 / Chapter 5.2.3 --- Long exact surgery sequence and the absolutely graded Hee- gaard Floer homology --- p.78 / Chapter 5.3 --- More Application --- p.84 / Chapter 5.3.1 --- Knot Floer homology --- p.84 / Chapter 5.3.2 --- Invariants on 4-manifolds --- p.86 / Chapter 5.4 --- Further developments --- p.87 / Bibliography --- p.89

Thesis (Ph.D.)--Kent State University, 2008. / Title from PDF t.p. (viewed Oct. 8, 2009). Advisor: C. Owen Lovejoy. Keywords: lumbar transverse process; vertebral evolution Includes bibliographical references (p. 214-221).

In Part 1, we consider two descriptions of L-homology of a (polyhedron of a) simplicial complex X. The classical approach of Ranicki via (Z,X)-modules (cf. [Ran92]) iswell established and is used in Ranicki’s definition of the total surgery obstruction and his formulation of the algebraic surgery exact sequence (cf. [Ran79], [Ran92],[KMM]). This connection between algebraic surgery and geometric surgery has numerous applications in the theory of (highdimensional) manifolds. The approach described in [RW10] uses a category of homotopy complexes of cosheaves to construct for a manifold M a (rational) orientation class [M]L• in symmetric L-homology which is topologically invariant per construction. This is used to reprove the topological invariance of rational Pontryagin classes. The L-theory of the category of homotopy complexes of sheaves over an ENR X can be naturally identified with L-homology of X. If X is a simplicial complex, both definitions give L-homology, there is no direct comparison however. We close this gap by constructing a functor from the category of (Z,X)-modules to the category of homotopy cosheaves of chain complexes of Ranicki-Weiss inducing an equivalence on L-theory. The work undertaken in Part 1 may be considered as an addendum to [RW10] and suggests some translation of ideas of [Ran92] into the language of [RW10]. Without significant alterations, this work may be generalised to the case of X being a △-set. The L-theory of △-sets is considered in [RW12]. Let A be a unital ring and I a category with objects given by natural numbers and two kinds of morphisms mn → n satisfying certain relations (see Ch.3.4). There is an I-diagram, given by n 7→ ˜K (A[x]/xn) where the tilde indicates the homotopy fiber of the projection induced map on algebraic K-theory (of free modules) K(A[x]/xn) → K(A). In Part 2 we consider the following result by Betley and Schlichtkrull [BS05]. After completion there is an equivalence of spectra TC(A)∧ ≃ holim I ˜K(A[x]/xn)∧ where TC(A) is the topological cyclic homology of A. This is a very important invariant of K-theory (cf. [BHM93], [DGM12]) and comes with the cyclotomic trace map tr : K(A) → TC(A). In [BS05], the authors prove that under the above identification the trace map corresponds to a “multiplication” with an element u∞ ∈ holim I ˜K (Z[x]/xn). In this work we are interested in a generalisation of this result. We construct an element u∞ ∈ holim I ˜K(Cn). where Cn can be viewed as the category of freemodules over the nilpotent extension S[x]/xn of the sphere spectrum S. Let G be a discrete group and S[G] its spherical group ring. Using our lift of u∞ we construct a map trBS : K(S[G]) → holim I ˜K (CG n ) where CG n should be interpreted as the category of free modules over the extension S[G][x]/xn. After linearisation this map coincides with the trace map constructed by Betley and Schlichtkrull. We conjecture but do not prove, that after completion the domain coincides with the topological cyclic homology of S[G]. Some indication is given at the end of the final chapter. To construct the element u∞ we rely on a generalisation of a result of Grayson on the K-theory of endomorphisms (cf. [Gra77]). Denote by EndC the category of endomorphisms of finite CW-spectra and by RC the Waldhausen category of free CW-spectra with an action of N, which are finite in the equivariant sense. Cofibrations are given by cellular inclusions and weak equivalences are given bymaps inducing an equivalence of (reduced) cellular chain complexes of Z[x]-modules, after inverting the set {1 + xZ[x]}. In Chapter 5 we prove (5.8) that there is a homotopy equivalence of spectra ˜K (EndC) ≃ ˜K (RC). where tildes indicate that homotopy fibres of the respective projections are considered. Furthermore, we pursue the goal of constructing an involutive tracemap for themodel of [BS05]. We employ the framework ofWaldhausen categories with duality (cf. [WW98]) to introduce for any G involutions on holim I ˜K (CG n ). We give enough indication for our trace map being involutive, in particular in the last three sections of Chapter 5, we sketch how the generalisation of the theoremof Grayson (5.8) can be improved to an involutive version. In the final chapter, we develop this further. Assuming that the element u∞ ∈ holim I ˜K (Cn) is a homotopy fixed point of the introduced involution, we construct a map from quadratic L-theory of S[G] to the Tate homology spectrum of Z/2 acting on the fibre of trBS (see 6.9) : L•(S[G]) → (hofib(trBS))thZ/2 and discuss the connection of this to a conjecture of Rognes andWeiss. The two parts of the thesis are preluded with their own introduction andmay be read independently. The fewcross references are completely neglectible.

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Thesis (Ph.D.)--University of Delaware, 2008. / Principal faculty advisor: Junghuei Chen, Dept. of Chemistry & Biochemistry. Includes bibliographical references.

Thesis (Ph. D.)--University of Washington, 1997. / Vita. Includes bibliographical references (leaves [117]-121).

Thesis (Ph. D.)--University of California, San Diego, 1998. / Vita. Includes bibliographical references (leaves 167-177).

Thesis (M.S.)--Duquesne University, 2004. / Title from document title page. Abstract included in electronic submission form. Includes bibliographical references (p. 60-65) and abstract.

Thesis (Ph. D.)--Rutgers University, 2009. / "Graduate Program in Mathematics." Includes bibliographical references (p. 91-93).

Thesis (Ph. D.)--Rutgers University, 2008. / "Graduate Program in Computer Science." Includes bibliographical references (p. 103-108).