The paper deals with the calculation of the fractional part of the η-invariant for elliptic self-adjoint operators in topological terms. The method used to obtain the corresponding formula is based on the index theorem for elliptic operators in subspaces obtained in [1], [2]. It also utilizes K-theory with coefficients Zsub(n). In particular, it is shown that the group K(T*M,Zsub(n)) is realized by elliptic operators (symbols) acting in appropriate subspaces.

We give the homotopy classification and compute the index of boundary value problems for elliptic equations. The classical case of operators that satisfy the Atiyah-Bott condition is studied first. We also consider the general case of boundary value problems for operators that do not necessarily satisfy the Atiyah-Bott condition.

We construct elliptic theory in the subspaces, determined by pseudodifferential projections. The finiteness theorem as well as index formula are obtained for elliptic operators acting in the subspaces. Topological (K-theoretic) aspects of the theory are studied in detail.

The topological significance of the spectral Atiyah-Patodi-Singer η-invariant is investigated. We show that twice the fractional part of the invariant is computed by the linking pairing in K-theory with the orientation bundle of the manifold. The Pontrjagin duality implies the nondegeneracy of the linking form. An example of a nontrivial fractional part for an even-order operator is presented.

We give a formula for the η-invariant of odd order operators on even-dimensional manifolds, and for even order operators on odd-dimensional manifolds. Geometric second order operators are found with nontrivial η-invariants. This solves a problem posed by P. Gilkey.

No description available.

We study the hyperbolic dynamical systems known as Smale spaces. More specifically we investigate the C*-algebras constructed from these systems. The K group of one of these algebras has a natural ring structure arising from an asymptotically abelian property. The K groups of the other algebras are then modules over this ring. In the case of a shift of finite type we compute these structures explicitly and show that the stable and unstable algebras exhibit a certain type of duality as modules. We also investigate the Bowen measure and its stable and unstable components with respect to resolving factor maps, and prove several results about the traces that arise as integration against these measures. Specifically we show that the trace is a ring/module homomorphism into R and prove a result relating these integration traces to an asymptotic of the usual trace of an operator on a Hilbert space.

Given a C*-algebra $A$ endowed with an action $\alpha$ of $\R$ and an $\alpha$-invariant trace $\tau$, there is a canonical dual trace $\widehat \tau$ on the crossed product $A \rtimes_\alpha \R$. This dual trace induces (as would any suitable trace) a real-valued homomorphism $\widehat \tau_* : K_0(A \rtimes_\alpha \R) \to \R$ on the even $K$-theory group. Recall there is a natural isomorphism $\phi_\alpha^i : K_i(A) \to K_{i+1}(A \rtimes_\alpha \R)$, the Connes-Thom isomorphism. The attraction of describing $\widehat \tau_* \circ \phi_\alpha^1$ directly in terms of the generators of $K_1(A)$ is clear. Indeed, the paper where the isomorphisms $\{\phi_\alpha^0,\phi_\alpha^1\}$ first appear sees Connes show that $\widehat \tau_* \phi_\alpha^1[u] = \frac{1}{2 \pi i} \tau(\delta(u) u^*)$, where $\delta = \frac{d}{dt} \big|_{t=0} \alpha_t(\cdot)$ and $u$ is any appropriate unitary. A careful proof of the aforementioned result occupies a central place in this thesis. To place the result in its proper context, the right-hand side is first considered in its own right, i.e., in isolation from mention of the crossed-product. A study of 1-parameter dynamical systems and exterior equivalence is undertaken, with several useful technical results being proven. A connection is drawn between a lemma of Connes on exterior equivalence and projections, and a quantum-mechanical theorem of Bargmann-Wigner. An introduction to the Connes-Thom isomorphism is supplied and, in the course of this introduction, a refined version of suspension isomorphism $K_1(A) \to K_0(\susp A)$ is formulated and proven. Finally, we embark on a survey of unbounded traces on C*-algebras; when traces are allowed to be unbounded, there is inevitably a certain amount of hard, technical work needed to resolve various domain issues and justify various manipulations. / Graduate / 0280

The thrust of this thesis is to describe base change BC_E/F at the level of chamber homology and K-theory for some p-adic groups, such as SL(2,F), GL(1,F) and GL(2,F). Here F is a non-archimedean local field and E is a Galois extension of F. We have had to master the representation theory of SL(2) and GL(2) including the Langlands parameters. The main result is an explicit computation of the effect of base change on the chamber homology groups, each of which is constructed from cycles. This will have an important connection with the Baum-Connes correspondence for such p-adic groups. This thesis involved the arithmetic of fields such as E and F, geometry of trees, the homology groups and the Weil group W_F.

This thesis contains a study of topological phases of matter, with a strong emphasis on symmetry as a unifying theme. We take the point of view that the "topology" in many examples of what is loosely termed "topological matter", has its origin in the symmetry data of the system in question. From the fundamental work of Wigner, we know that topology resides not only in the group of symmetries, but also in the cohomological data of projective unitary-antiunitary representations. Furthermore, recent ideas from condensed matter physics highlight the fundamental role of charge-conjugation symmetry. With these as physical motivation, we propose to study the topological features of gapped phases of free fermions through a Z₂-graded C*-algebra encoding the symmetry data of their dynamics. In particular, each combination of time reversal and charge conjugation symmetries can be associated with a Clifford algebra. K-theory is intimately related to topology, representation theory, Clifford algebras, and Z₂-gradings, so it presents itself as a powerful tool for studying gapped topological phases. Our basic strategy is to use various K</em-theoretic invariants of the symmetry algebra to classify symmetry-compatible gapped phases. The super-representation group of the algebra classifies such gapped phases, while its K-theoretic difference-group classifies the obstructions in passing between two such phases. Our approach is a noncommutative version of the twisted K-theory approach of Freed--Moore, and generalises the K-theoretic classification first suggested by Kitaev. It has the advantage of conceptual simplicity in its uniform treatment of all symmetries. Physically, it encompasses phenomena which require noncommutative algebras in their description; mathematically, it clarifies and provides rigour to the meaning of "homotopic phases", and easily explains the salient features of Kitaev's Periodic Table.