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Gromov and Lawson developed a codimension 2 index obstruction to positive scalar curvature for a closed spin manifold M, later refined by Hanke, Pape and Schick. Kubota has shown that also this obstruction can be obtained from the Rosenberg index of the ambient manifold M which takes values in the K-theory of the maximal C*-algebra of the fundamental group of M, using relative index constructions. In this note, we give a slightly simplified account of Kubota's work and remark that it also applies to the signature operator, thus recovering the homotopy invariance of higher signatures of codimension 2 submanifolds of Higson, Schick, Xie., Comment: 12 pages. v2 final version to appear in G&T. Small corrections and minor changes of presentation

Using a discrete version of Floquet theory in a space with indefinite metric we reconstruct periodic pseudo-Jacobi matrices from given spectral data. We use two methods to characterize the signature operator H so that the periodic pseudo-Jacobi inverse eigenvalue problem is solvable. As a by-product of the obtained results, we precisely polish some non-rigorous assumptions in the existing literature.

We construct η- and ρ-invariants for Dirac operators, on the universal covering of a closed manifold, that are invariant under the projective action associated to a 2-cocycle of the fundamental group. We prove an Atiyah–Patodi–Singer index theorem in this setting, as well as its higher generalisation. Applications concern the classification of positive scalar curvature metrics on closed spin manifolds. We also investigate the properties of these twisted invariants for the signature operator and the relation to the higher invariants.

We show that and specify how space-time symmetries give rise to corresponding symmetries of the fermionic signature operator and generalized fermionic projector states., 21 pages, LaTeX, 1 figure, minor improvements (published version)

The fermionic signature operator is constructed in Rindler space-time. It is shown to be an unbounded self-adjoint operator on the Hilbert space of solutions of the massive Dirac equation. In two-dimensional Rindler space-time, we prove that the resulting fermionic projector state coincides with the Fulling-Rindler vacuum. Moreover, the fermionic signature operator gives a covariant construction of general thermal states, in particular of the Unruh state. The fermionic signature operator is shown to be well-defined in asymptotically Rindler space-times. In four-dimensional Rindler space-time, our construction gives rise to new quantum states., Comment: 27 pages, LaTeX, more details on self-adjoint extension (published version)

In this paper we prove the existence of a natural mapping from the surgery exact sequence for topological manifolds to the analytic surgery exact sequence of N. Higson and J. Roe. This generalizes the fundamental result of Higson and Roe, but in the treatment given by Piazza and Schick, from smooth manifolds to topological manifolds. Crucial to our treatment is the Lipschitz signature operator of Teleman. We also give a generalization to the equivariant setting of the product defined by Siegel in his Ph.D. thesis. Geometric applications are given to stability results for rho classes. We also obtain a proof of the APS delocalised index theorem on odd dimensional manifolds, both for the spin Dirac operator and the signature operator, thus extending to odd dimensions the results of Piazza and Schick. Consequently, we are able to discuss the mapping of the surgery sequence in all dimensions., 26 pages, accepted in "Journal of Topology and Analysis"

We study characteristic classes arising as the indices of families of elliptic operators acting on the fibers of an oriented M-bundle f : E -->B, M a smooth oriented closed manifold. Given a family of such operators D = {Db} one obtains a family index Ind(D) in K^*(B). If D is "sufficiently natural" (in a sense made precise in [23]) these indices may be viewed as arising from certain universal symbol classes �in K^*(MTSO(n)), where MTSO(n) is the Thom spectrum of the additive inverse of the universal bundle of oriented n-planes over BSO(n). Explicitly, Ind(D) is pulled back from K^*(MTSO(d)) by the Madsen-Tillman-Weiss map associated to f : E --> B. We show Ind(DVo ) = 0 where DVo is the family of odd signature operators on the fibers of f : E --> B with coefficients in a flat Hermitian vector bundle V --> E. DVo is not universal in the sense of [23] however its index can be described in terms of universal symbols. The vanishing relations implied in cohomology show the higher signatures (Novikov [52]) associated to flat Hermitian bundles provide obstructions to fibering as an odd-dimensional manifold bundle. We end by discussing some examples of flat Hermitian vector bundles to verify that these higher signatures provide a more general obstruction than the usual signature of a 4k-dimensional manifold.

We give a non-perturbative construction of a distinguished state for the quantized Dirac field in Minkowski space in the presence of a time-dependent external field of the form of a plane electromagnetic wave. By explicit computation of the fermionic signature operator, it is shown that the Dirac operator has the strong mass oscillation property. We prove that the resulting fermionic projector state is a Hadamard state., 27 pages, LaTeX, 3 figures, minor improvements (published version)

Higher index of signature operator is a far reaching generalization of signature of a closed oriented manifold. When two closed oriented manifolds are homotopy equivalent, one can define a secondary invariant of the relative signature operator called higher rho invariant. The higher rho invariant detects the topological nonrigidity of a manifold. In this paper, we prove product formulas for higher index and higher rho invariant of signature operator on fibered manifolds. Our result implies the classical product formula for numerical signature of fiber manifolds obtained by Chern, Hirzebruch, and Serre in "On the index of a fibered manifold". We also give a new proof of the product formula for higher rho invariant of signature operator on product manifolds, which is parallel to the product formula for higher rho invariant of Dirac operator on product manifolds obtained by Xie and Yu in "Positive scalar curvature, higher rho invariants and localization algebras" and Zeidler in "Positive scalar curvature and product formulas for secondary index invariants"., Comment: 30 pages

The fermionic projector state is a distinguished quasi-free state for the algebra of Dirac fields in a globally hyperbolic spacetime. We construct and analyze it in the four-dimensional de Sitter spacetime, both in the closed and in the flat slicing. In the latter case we show that the mass oscillation properties do not hold due to boundary effects. This is taken into account in a so-called mass decomposition. The involved fermionic signature operator defines a fermionic projector state. In the case of a closed slicing, we construct the fermionic signature operator and show that the ensuing state is maximally symmetric and of Hadamard form, thus coinciding with the counterpart for spinors of the Bunch-Davies state., Comment: 28 pages - accepted in Journal of Mathematical Analysis and Applications

This paper is about the cohomology of certain finite-index subgroups of mapping class groups and its relation to the cohomology of arithmetic groups. For $G=\mathbb{Z}/m\mathbb{Z}$ and for a regular $G$ -cover $S\rightarrow\bar{S}$ (possibly branched), a finite-index subgroup $\Gamma\lt \operatorname{Mod}(\bar{S})$ acts on $H_{1}(S;\mathbb{Z})$ commuting with the deck group action, thus inducing a homomorphism $\Gamma\rightarrow\operatorname{Sp}_{2{g}}^{G}(\mathbb{Z})$ to an arithmetic group. The induced map $H^{*}(\operatorname{Sp}_{2{g}}^{G}(\mathbb{Z});\mathbb{Q})\rightarrow H^{*}(\Gamma;\mathbb{Q})$ can be understood using index theory. To this end, we describe a families version of the $G$ -index theorem for the signature operator and apply this to (i) compute $H^{2}(\operatorname{Sp}_{2{g}}^{G}(\mathbb{Z});\mathbb{Q})\rightarrow H^{2}(\Gamma;\mathbb{Q})$ , (ii) rederive Hirzebruch’s formula for signature of a branched cover, (iii) compute Toledo invariants of surface group representations to $\operatorname{SU}(p,q)$ arising from Atiyah–Kodaira constructions, and (iv) describe how classes in $H^{*}(\operatorname{Sp}_{2{g}}^{G}(\mathbb{Z});\mathbb{Q})$ give equivariant cobordism invariants for surface bundles with a fiberwise $G$ action, following Church–Farb–Thibault.

We extend the study of the de Rham operator with ideal boundary conditions from the case of isolated conic singularities, as analyzed by Cheeger, to the case of arbitrary stratified pseudomanifolds. We introduce a class of ideal boundary operators and the notion of mezzoperversity, which intermediates between the standard lower and upper middle perversities in intersection theory, as interpreted in this de Rham setting, and show that the de Rham operator with these boundary conditions is Fredholm and has compact resolvent. We also prove an isomorphism between the resulting Hodge and L2 de Rham cohomology groups, and that these are independent of the choice of iterated edge metric. On spaces which admit ideal boundary conditions of this type which are also self-dual, which we call `Cheeger spaces', we show that these Hodge/de Rham cohomology groups satisfy Poincare Duality., v2: Slight changes to improve exposition, v3: Improved discussion of core domain, to appear in Crelle's journal

The fermionic signature operator is analyzed on globally hyperbolic Lorentzian surfaces. The connection between the spectrum of the fermionic signature operator and geometric properties of the surface is studied. The findings are illustrated by simple examples and counterexamples., 49 pages, LaTeX, 7 figures, minor improvements (published version)

In Huang and Lee (in The Refined Analytic Torsion and a Well-posed Boundary Condition for the Odd Signature Operator, 2010) introduced a pair of new de Rham complexes on a compact oriented Riemannian manifold with boundary by using a pair of global boundary conditions to discuss the refined analytic torsion on a compact manifold with boundary. In this paper we discuss the Lefschetz fixed point formula on these complexes with respect to a smooth map which is a local isometry on the boundary and has only simple fixed points. For this purpose we are going to use the heat kernel method for the Lefschetz fixed point formula.

We define an L 2 -signature for proper actions on spaces of leaves of transversely oriented foliations with bounded geometry. This is achieved by using the Connes fibration to reduce the problem to the case of Riemannian bifoliations where we show that any transversely elliptic first order operator in an appropriate Beals–Greiner calculus, satisfying the usual axioms, gives rise to a semi-finite spectral triple over the crossed product algebra of the foliation by the action, and hence a periodic cyclic cohomology class through the Connes–Chern character. The Connes–Moscovici hypoelliptic signature operator yields an example of such a triple and gives the differential definition of our “ L 2 -signature”. For Galois coverings of bounded geometry foliations, we also define an Atiyah–Connes semi-finite spectral triple which generalizes to Riemannian bifoliations the Atiyah approach to the L 2 -index theorem. The compatibility of the two spectral triples with respect to Morita equivalence is proven, and by using an Atiyah-type theorem proven in [7], we deduce some integrality results for Riemannian foliations with torsion-free monodromy groupoids.

The structure of the solution space of the Dirac equation in the exterior Schwarzschild geometry is analyzed. Representing the space-time inner product for families of solutions with variable mass parameter in terms of the respective scalar products, a so-called mass decomposition is derived. This mass decomposition consists of a single mass integral involving the fermionic signature operator as well as a double integral which takes into account the flux of Dirac currents across the event horizon. The spectrum of the fermionic signature operator is computed. The corresponding generalized fermionic projector states are analyzed., Comment: 26 pages, LaTeX, 1 figure, minor improvements, references added (published version)

The main result of this paper is the $G$-homotopy invariance of the $G$-index of signature operator of proper co-compact $G$-manifolds. If proper co-compact $G$ manifolds $X$ and $Y$ are $G$-homotopy equivalent, then we prove that the images of their signature operators by the $G$-index map are the same in the $K$-theory of the $C^{*}$-algebra of the group $G$. Neither discreteness of the locally compact group $G$ nor freeness of the action of $G$ on $X$ are required, so this is a generalization of the classical case of closed manifolds. Using this result we can deduce the equivariant version of Novikov conjecture for proper co-compact $G$-manifolds from the Strong Novikov conjecture for $G$., 30 pages, Keywords. Novikov conjecture, Higher signatures, Almost flat bundles

The refined analytic torsion on compact Riemannian manifolds with boundary has been discussed by B. Vertman and the authors, but these two constructions are completely different. Vertman used a double of de Rham complex consisting of the minimal and maximal closed extensions of a flat connection and the authors used well-posed boundary conditions ${\mathcal P}_{-, {\mathcal L}_{0}}$, ${\mathcal P}_{+, {\mathcal L}_{1}}$ for the odd signature operator. In this paper we compare these two constructions by using the BFK-gluing formula for zeta-determinants, the adiabatic method for stretching cylinder part near boundary and the deformation method used in [6], when the odd signature operator comes from a Hermitian flat connection and all de Rham cohomologies vanish., 33 pages. arXiv admin note: text overlap with arXiv:1103.3571

We establish existence of the eta-invariant as well as of the Atiyah-Patodi-Singer and the Cheeger-Gromov rho-invariants for a class of Dirac operators on an incomplete edge space. Our analysis applies in particular to the signature, the Gauss-Bonnet and the spin Dirac operator. We derive an analogue of the Atiyah-Patodi-Singer index theorem for incomplete edge spaces and their non-compact infinite Galois coverings with edge singular boundary. Our arguments employ microlocal analysis of the heat kernel asymptotics on incomplete edge spaces and the classical argument of Atiyah-Patodi-Singer. As an application, we discuss stability results for the two rho-invariants we have defined., 65 pages, 2 figures

Using H. Donnelly result from the article "Eta Invariants for G-Spaces" we calculate the eta invariants of the signature operator for almost all 7-dimensional flat manifolds with cyclic holonomy group. In all cases this eta invariants are an integer numbers. The article was motivated by D. D. Long and A. Reid article "On the geometric boundaries of hyperbolic 4-manifolds, Geom. Topology 4, 2000, 171-178, 18 pages, a new version with referees comments

We relate the spectral flow to the index for paths of selfadjoint Breuer-Fredholm operators affiliated to a semifinite von Neumann algebra, generalizing results of Robbin-Salamon and Pushnitski. Then we prove the vanishing of the von Neumann spectral flow for the tangential signature operator of a foliated manifold when the metric is varied. We conclude that the tangential signature of a foliated manifold with boundary does not depend on the metric. In the Appendix we reconsider integral formulas for the spectral flow of paths of bounded operators., Comment: 23 pages

Nigel Higson and John Roe Abstract: We connect the assembly map in C∗-algebra K-theory to rigidity properties for relative eta invariants that have been investigated by Mathai, Keswani, Weinberger and others. We give a new and conceptual proof of Keswani’s theorem that whenever the C∗-algebra assembly map is an isomorphism, the relative eta invariants associated to the signature operator are homotopy invariants, whereas the relative eta invariants associated to the Dirac operator on a manifold with positive scalar curvature vanish.

The index of the classical Hirzebruch signature operator on a manifold M is equal to the signature of the manifold. The examples of Lusztig ((10), 1972) and Gromov ((4), 1985) present the Hirzebruch signature operator for the cohomology (of a manifold) with coefficients in a flat symmetric or symplectic vector bundle. In (6), we gave a signature operator for the cohomology of transitive Lie algebroids. In this paper, firstly, we present a general approach to the signature operator, and the above four examples become special cases of a single general theorem. Secondly, due to the spectral sequence point of view on the signature of the cohomology algebra of certain filtered DG-algebras, it turns out that the Lusztig and Gromov examples are important in the study of the signature of a Lie algebroid. Namely, under some natural and simple regularity assumptions on the DG-algebra with a decreasing filtration for which the second term lives in a finite rectangle, the signature of the second term of the spectral sequence is equal to the signature of the DG algebra. Considering the Hirzebruch-Serre spectral sequence for a transitive Lie algebroid A over a compact oriented manifold for which the top group of the real cohomology of A is nontrivial, we see that the second term is just identical to the Lusztig or Gromov example (depending on the dimension). Thus, we have a second signature operator for Lie algebroids.

We prove a Kastler-Kalau-Walze type theorem for the Dirac operator and the signature operator for $3,4$-dimensional manifolds with boundary. As a corollary, we give two kinds of operator theoretic explanations of the gravitational action in the case of 4-dimensional manifolds with flat boundary.

We establish a splitting formula for the spectral flow of the odd signature operator on a closed 3-manifold M coupled to a path of SU(2) connections, provided M = S cup X, where S is the solid torus. It describes the spectral flow on M in terms of the spectral flow on S, the spectral flow on X (with certain Atiyah-Patodi-Singer boundary conditions), and two correction terms which depend only on the endpoints. Our result improves on other splitting theorems by removing assumptions on the non-resonance level of the odd signature operator or the dimension of the kernel of the tangential operator, and allows progress towards a conjecture by Lisa Jeffrey in her work on Witten's 3-manifold invariants in the context of the asymptotic expansion conjecture., Published by Geometry and Topology at http://www.maths.warwick.ac.uk/gt/GTVol9/paper52.abs.html

The author presents an alternate proof of the Bismut-Zhang localization formula of η invariants, when the target manifold is a sphere, by using ideas of mod k index theory instead of the difficult analytic localization techniques of Bismut-Lebeau. As a consequence, it is shown that the R/Z part of the analytically defined η invariant of Atiyah-Patodi-Singer for a Dirac operator on an odd dimensional closed spin manifold can be expressed purely geometrically through a stable Chern-Simons current on a higher dimensional sphere. As a preliminary application, the author discusses the relation with the Atiyah-Patodi-Singer R/Z index theorem for unitary flat vector bundles, and proves an R refinement in the case where the Dirac operator is replaced by the Signature operator.

We give geometric constructions leading to analytically controlled Poincarcom- plexes in the sense of the previous paper. In the case of a complete Riemannian manifold we identify the signature of the associated complex with the coarse index of the signature operator. Mathematics Subject Classifications (2000): 19J25, 19K99.

For positive elliptic differential operators Δ, the asymptotic expansion of the heat trace tr(e-tΔ) and its related zeta function ζ(s, Δ) = tr(Δ-s) have numerous applications in geometry and physics. This article discusses the general nature of the boundary conditions that must be considered when there is a singular stratum, and presents three examples in which a choice of boundary conditions at the singularity must be made. The first example concerns the signature operator on a manifold with a singular stratum of conic type. The second concerns the "Zaremba problem" for a nonsingular manifold with smooth boundary, posing Dirichlet conditions on part of the boundary and Neumann conditions on the complement; the intersection of these two regions can be viewed as a singular stratum of conic type, and a boundary condition must be imposed along this stratum. The third example is a one-dimensional manifold where the operator at one end has a singularity like that in conic problems, and the choice of boundary conditions affects not just the residues at the poles of the zeta function, but also the very location of the poles

Let (N, g) be a closed Riemannianmanifold of dimension 2m − 1 and let Γ → N → N be a Galois covering of N. We assumethat Γ is of polynomial growth with respect to a word metric and that ΔN is L2-invertible in degree m. By employing spectral sections with asymmetry property with respect to the ⋆-Hodge operator, we define the higher eta invariant associatedwith the signature operator on N, thus extending previous work of Lott. If π1(M)→ \({\tilde M}\) →M is the universal cover of a compact orientable even-dimensionalmanifold with boundary (∂M = N)then, under the above invertibility assumption on Δ∂\({\tilde M}\), andalways employing symmetric spectral sections, we define acanonical Atiyah–Patodi–Singer index class, in K0(C*r(Γ)), for the signature operator of\({\tilde M}\). Using the higherAPS index theory developed in [6], we express the Chern character ofthis index class in terms of a local integral and of the higher etainvariant defined above, thus establishing a higher APS index theoremfor the signature operator on Galois coverings. We expect the notion ofa symmetric spectral section for the signature operator to have widerimplications in higher index theory for signatures operators.

A mod 2 index theorem for the twisted Signature operator on 4q + 1 dimensional manifolds is established. This result generalizes a result of Farber and Turaev, which was proved for the case of orthogonal flat bundles, to arbitrary real vector bundles. It also provides an analytic interpretation of the sign of the Poincare-Reidemeister scalar product defined by Farber and Turaev.

INVARIANCE PAR HOMOTOPIE DES HAUTES SIGNATURES TWISTEES SUR DES VARIETES A BORD. - Soit M une variete compacte orientee a bord. On suppose que π 1 (M) est le produit d'un groupe fini non trivial F et d'un groupe Γ qui est soit a croissance polynomiale, soit hyperbolique au sens de Gromov. On se donne une representation non triviale ρ:F → U() et on considere le fibre plat unitaire associe E ρ . On designe par M le revetement universel de M et on considere le revetement Γ-galoisien π:M/F → M, le fibre plat releve E ρ = π * (E ρ ) et l'operateur de signature « twiste » associe. Sous l'hypothese supplementaire que l'operateur de signature « twiste » induit sur le bord de M/F est L 2 -inversible, Lott a introduit dans [L2] les hautes signatures « twistees » de M. Notre resultat principal - une reponse positive a une conjecture de type Novikov pour les varietes a bord - est que ce sont des invariants d'homotopie de la paire (M, ∂M). La preuve depend de maniere essentielle du b-B∞-calcul pseudodifferentiel developpe dans [LP1], du theoreme d'indice superieur APS de [LP1] (etendu ici au cas des groupes hyperboliques au sens de Gromov) et du resultat classique de Kaminker-Miller enoncant l'egalite des classes d'indices associees a deux complexes hermitiens de Fredholm homotopiquement equivalents.

Let ( N,g ) be a closed Riemannian manifold of dimension 2 m -1 and Γ → N ˜ → N be a Galois covering of N . We assume that Γ is of polynomial growth and that Δ N ˜ is L 2 -invertible in degree m . By employing spectral sections which are symmetric with respect to the *-Hodge operator, we define the higher eta invariant associated to the signature operator on N ˜ , thus extending previous work of Lott. If π 1 ( M ) → M ˜ → M is the universal cover of a compact orientable even-dimensional manifold with boundary ( ∂M = N ) then, under the above invertibility assumption on Δ ∂ M ˜ , we define a canonical Atiyah-Patodi-Singer signature-index class, in K 0 ( C r * (Γ)). Employing the higher APS index theory developed in [4] we express the Chern character of this index class in terms of a local integral and of the higher eta invariant defined above. We apply these results to the problem of the existence and homotopy invariance of higher signatures on manifolds with boundary.

We compute the periodic cyclic cohomology Chern character of an admissible pseudomanifold Xf with one singular stratum. As a corollary, we obtain the index theorem and spectral flow for signature operators. 0. INTRODUCTION This paper is the sequel to [Chan]. In that paper, by considering the "straight" Chern character, we gave a de-Rham type realization of the Goresky-MacPhersonSiegel L-classes of admissible pseudomanifolds with one singular stratum Xt M U (ct (L) x N) such that 2L has zero oriented cobordism, and we also obtained the index theorem for the twisted signature operators on Xt. In this paper, we will continue to study Xt by using non-commutative geometry. We will compute (Theorem 3.1 and Theorem 3.3) the corresponding periodic cyclic cohomology Chern character by using the infinite temperature limit formula in [CoM]. As in [Chan], we will choose a scaling in the conical direction such that the signature operator is essentially self-adjoint. Also, by using a singular elliptic estimate, we handle the calculation on the singular part by that on model space c0,0, (L) x N. We finish the computation by Getzler's calculus. As a consequence, we recover the index theorem for twisted signature operators on Xt [Chan, Theorem 4.1] in the even case. In the odd case, we obtain (Corollary 3.4) the spectral flow for signature operators on admissible spaces with conical singularity. 1. PRELIMINARIES To fix the notation, let us recall some definitions in [Chan]. Let M be a smooth, oriented, compact and connected m-dimensional manifold with boundary O9M = L x N where L and N are smooth, oriented, closed and connected manifolds of dimensions ? and n respectively. Let c(L) -= (0, 1) x L and ct (L) = [0, 1) x L/{O} x L be the cone and completed cone with link L respectively. Then Xt = MU (ct (L) x N) is called a pseudomanifold with one singular stratum [Chan]. We define a metric g on Xt such that (i) g is a measurable metric on Xt; (ii) glM is a smooth metric on M and is a product near OM; Received by the editors August 30, 1996. 1991 Mathematics Subject Classification. Primary 19D55; Secondary 58G12. ?1998 American Mathematical Society 669 This content downloaded from 157.55.39.127 on Wed, 29 Jun 2016 04:18:19 UTC All use subject to http://about.jstor.org/terms

We define an index of the fermionic signature operator on even-dimensional globally hyperbolic spin manifolds of finite lifetime. The invariance of the index under homotopies is studied. The definition is generalized to causal fermion systems with a chiral grading. We give examples of space-times and Dirac operators thereon for which our index is non-trivial., Comment: 21 pages, LaTeX, 3 figures, minor corrections (published version)

The odd signature operator is a Dirac operator which acts on the space of differential forms of all degrees and whose square is the usual Laplacian. We extend the result of [15] to prove the gluing formula of the zeta-determinants of Laplacians acting on differential forms of all degrees with respect to the boundary conditions ${\mathcal P}_{-, {\mathcal L}_{0}}$, ${\mathcal P}_{+, {\mathcal L}_{1}}$. We next consider a double of de Rham complexes consisting of differential forms of all degrees with the absolute and relative boundary conditions. Using a similar method, we prove the gluing formula of the zeta-determinants of Laplacians acting on differential forms of all degrees with respect to the absolute and relative boundary conditions., Comment: 19 pages

We show how to compute the spectral flow of the odd signature operator $\pm *d_{a_t}-d_{a_t}*$ along an analytic path of flat connections $a_t$ on a bundle over a closed odd-dimensional manifold in terms of Massey products in the DGLA of bundle-valued differential forms. To obtain this information, we set up a sequence of cochain complexes $\{\calg^*_n,\delta_n\}$, for $n=0,1,2,\ldots$ and Hermitian forms $$Q_n:\calg_n\times\calg_n\ra \bbbC$$ whose signatures determine the spectral flow through $t=0$. The complexes and Hermitian forms are constructed using Massey products., Comment: 35 pages, report1

The main result of this paper is a new and direct proof of the natural transformation from the surgery exact sequence in topology to the analytic K-theory sequence of Higson and Roe. Our approach makes crucial use of analytic properties and new index theorems for the signature operator on Galois coverings with boundary. These are of independent interest and form the second main theme of the paper. The main technical novelty is the use of large scale index theory for Dirac type operators that are perturbed by lower order operators., Comment: 29 pages, AMS-LaTeX; v2: small corrections and (hopefully) improved exposition, as suggested by the referee. Final version, to appear in Annals of K-Theory

In this paper, by combining modularity of the Witten genus and the modular forms constructed by Liu and Wang, we establish mod 3 congruence properties of certain twisted signatures of 24 dimensional string manifolds., final version, to appear in Transactions of AMS

In this paper we prove a variety of results about the signature operator on Witt spaces. First, we give a parametrix construction for the signature operator on any compact, oriented, stratified pseudomanifold X which satisfies the Witt condition. This construction, which is inductive over the `depth' of the singularity, is then used to show that the signature operator is essentially self-adjoint and has discrete spectrum of finite multiplicity, so that its index -- the analytic signature of X -- is well-defined. This provides an alternate approach to some well-known results due to Cheeger. We then prove some new results. By coupling this parametrix construction to a C*_r\Gamma-Mishchenko bundle associated to any Galois covering of X with covering group \Gamma, we prove analogues of the same analytic results, from which it follows that one may define an analytic signature index class as an element of the K-theory of C*_r\Gamma. We go on to establish in this setting and for this class the full range of conclusions which sometimes goes by the name of the signature package. In particular, we prove a new and purely topological theorem, asserting the stratified homotopy invariance of the higher signatures of X, defined through the homology L-class of X, whenever the rational assembly map K_* (B\Gamma)\otimes\bbQ \to K_*(C*_r \Gamma)\otimes \bbQ is injective., Comment: Amalgam and replacement of arXiv:0906.1568 and arXiv:0911.0888 with minor corrections