The inverse Chevalley formula in the equivariant K-theory of semi-infinite flag manifolds of type An−1 is given as a sum over a set of quantum walks in the quantum Bruhat graph, QBG(An−1). We establish bounds on the number of quantum steps and simple stationary steps in these quantum walks. By a result of Kato, we map this formula to the equivariant quantum K-theory of partial flag manifolds G/P to give an alternate proof of [KLNS24, Theorem 8]. / Doctor of Philosophy / The quantum Bruhat graph, is a directed graph with vertex set W . Beginning with an arbitrary element of W , at each position, we may either move to a new element of W along a directed edge (a non-stationary step), or stay at the current element (a stationary step). A quantum walk is the sequence that records the element W at each position. We establish bounds on the number of occurrences of particular kinds of stationary and non-stationary steps called simple stationary steps and quantum steps respectively. These bounds are relevant to calculations of Chevalley formulas in K-Theory.

Bott periodicity plays a fundamental role in the definition and understanding of K-theory, the generalized cohomology theory defined by vector bundles. This paper examines the proof, given by Atiyah and Bott[3], of the periodicity theorem for the complex case. We also describe the long exact sequence for K-cohomology in the category of connected finite CW-complexes. / Master of Science

We construct a geometric model for differential K-theory, and prove it is isomorphic to the model proposed in [25]. We construct differential K-orientations for families and elucidate the pushforward map given in [25] in detail. We prove a geometric index theorem for odd dimensional manifolds. Finally, using this index theorem and the holonomy theorem of Bismut and Freed from [10], we prove what may be considered a special case of a geometric refinement of the Aityah-Singer index theorem. / text

Thesis (Ph. D.)--University of Chicago, Dept. of Mathematics, June 1999. / Includes bibliographical references. Also available on the Internet.

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.

Let <i>G</i> be any compact connected Lie group and let <i>T</i> ≤ <i>G</i> be a maximal torus.  Then for any unstable Adams operation <i>f</i> of degree <i>k</i>, the following diagram commutes up to homotopy «!» And conversely, any map <i>f</i> that makes the above diagram commute must be an unstable Adams operation. Using this characterization, we will construct a self-map of a <i>p</i>-local compact group (<i>S,F,L</i>) in order to define unstable Adams operations on a more general setting. THEOREM.  For any <i>p</i>-local compact group (<i>S,F,L</i>) there is a self-equivalence such that the map on the objects when restricted to the identity component of <i>S</i> is a <i>q^m-th </i>power map.

No description available.

by Leung Chi Wai. / Thesis (M.Ph.)--Chinese University of Hong Kong, 1987. / Bibliography: leaf [51]

Thesis (Ph.D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 1979. / MICROFICHE COPY AVAILABLE IN ARCHIVES AND SCIENCE. / Vita. / Bibliography: leaves 131-133. / Ph.D.

Lam Chi Ming. / Thesis submitted in: August 2000. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2001. / Includes bibliographical references (leaves 47-48). / Abstracts in English and Chinese. / Acknowledgments --- p.i / Abstract --- p.ii / Introduction --- p.3 / Chapter 1 --- Introduction to Real Groups and Real representa- tions --- p.6 / Chapter 1.1 --- Real Groups and Real representations --- p.6 / Chapter 1.2 --- "RR(G, ε)" --- p.10 / Chapter 1.3 --- Examples of Real representations --- p.15 / Chapter 1.3.1 --- Cyclic groups --- p.17 / Chapter 1.3.2 --- Dihedral groups --- p.18 / Chapter 1.3.3 --- Other examples --- p.19 / Chapter 2 --- Brauer induction Theorem on Real representations --- p.22 / Chapter 2.1 --- Real induction --- p.22 / Chapter 2.2 --- p-hyperelementary subgroups --- p.27 / Chapter 2.3 --- Brauer induction Theorem on Real Representations --- p.29 / Chapter 2.4 --- Monomial Real Representations --- p.42 / Bibliography --- p.47