Novikov Conjecture Research Articles

On Kernels of Invariant Schrödinger Operators with Point Interactions. Grinevich–Novikov Conjecture

On Kernels of Invariant Schrödinger Operators with Point Interactions. Grinevich–Novikov Conjecture

Asymptotically flat Fredholm bundles and assembly

Almost flat finitely generated projective Hilbert [Formula: see text]-module bundles were successfully used by Hanke and Schick to prove special cases of the Strong Novikov Conjecture. Dadarlat later showed that it is possible to calculate the index of a [Formula: see text]-homology class [Formula: see text] twisted with an almost flat bundle in terms of the image of [Formula: see text] under Lafforgue’s assembly map and the almost representation associated with the bundle. Mishchenko used flat infinite-dimensional bundles equipped with a Fredholm operator in order to prove special cases of the Novikov higher signature conjecture. We show how to generalize Dadarlat’s theorem to the case of an infinite-dimensional bundle equipped with a continuous family of Fredholm operators on the fibers. Along the way, we show that special cases of the Strong Novikov Conjecture can be proven if there exist sufficiently many almost flat bundles with Fredholm operator. To this end, we introduce the concept of an asymptotically flat Fredholm bundle and its associated asymptotic Fredholm representation, and prove an index theorem which relates the index of the asymptotic Fredholm bundle with the so-called asymptotic index of the associated asymptotic Fredholm representation.

K-theory of relative group $$C^*$$-algebras and the relative Novikov conjecture

K-theory of relative group $$C^*$$-algebras and the relative Novikov conjecture

The strongly quasi-local coarse Novikov conjecture and Banach spaces with Property (H)

The strongly quasi-local coarse Novikov conjecture and Banach spaces with Property (H)

Group boundaries for semidirect products with $\mathbb{Z}$

Bestvina's notion of a \mathcal{Z} -structure provides a general framework for group boundaries that includes Gromov boundaries of hyperbolic groups and visual boundaries of CAT(0) groups as special cases. A refinement, known as an E\mathcal{Z} -structure, has proven useful in attacks on the Novikov conjecture and related problems. Characterizations of groups admitting a \mathcal{Z} - or an E\mathcal{Z} -structure are longstanding open problems. In this paper, we examine groups of the form G\rtimes_{\phi}\mathbb{Z} . For example, we show that if G is torsion-free and admits a \mathcal{Z} -structure, then so does every semidirect product of this type. We prove a similar theorem for E\mathcal{Z} -structures, under an additional hypothesis. As applications, we show that all closed 3-manifold groups admit \mathcal{Z} -structures, as do all strongly polycyclic groups and all groups of polynomial growth. In those latter cases, our \mathcal{Z} -boundaries are always spheres. This allows one to make strong conclusions about the group cohomology and end invariants of those groups. In another direction, we expand upon the notion of an E\mathcal{Z} -structure and discuss new applications to the Novikov conjecture.

A Bott periodicity theorem for ℓp-spaces and the coarse Novikov conjecture at infinity

We formulate and prove a Bott periodicity theorem for an ℓp-space (1≤p<∞). For a proper metric space X with bounded geometry, especially for a coarsely connected space, we introduce a version of K-homology at infinity and the Roe algebra at infinity and show that to prove the coarse Novikov conjecture, it suffices to prove the coarse assembly map at infinity is an injection. As a result, we show that the coarse Novikov conjecture holds for any metric space with bounded geometry which admits a fibred coarse embedding into an ℓp-space. These include all box spaces of a residually finite hyperbolic group, and a large class of warped cones of a compact metric space with an action by a hyperbolic group.

The twisted coarse Baum-Connes conjecture with coefficients in coarsely proper algebras

We introduce a notion of (approximately) coarsely proper algebras for coarse embeddings of metric spaces to formulate and prove a version of twisted coarse Baum-Connes conjecture with coefficients in coarsely proper algebras. This may be regarded as a coarse geometric version of the generalized Green-Julg Theorem in the Baum-Connes conjecture for countable discrete groups. It also provides a conceptual framework for the Dirac-dual-Dirac method to the coarse Novikov conjecture for coarse embeddings into several different spaces, including Hilbert spaces, simply connected complete Riemannian manifolds with non-positive sectional curvature, Banach spaces with property (H) and Hilbert-Hadamard spaces. Moreover, for a group extension 1→N→G→Q→1, we show that if N is coarsely embeddable into Hilbert space and Q is coarsely embeddable into an admissible Hilbert-Hadamard space, then the coarse Novikov conjecture holds for G.

Groupoids decomposition, propagation and operator $K$-theory

In this paper, we streamline the technique of groupoids coarse decomposition for purpose of K -theory computations of groupoids crossed products. This technique was first introduced by Guoliang Yu in his proof of Novikov conjecture for groups with finite asymptotic dimension. The main tool we use for these computations is controlled operator K -theory.

Coarse geometry of Hecke pairs and the Baum–Connes conjecture

We study Hecke pairs using the coarse geometry of their coset space and their Schlichting completion. We prove new stability results for the Baum-Connes and the Novikov conjectures in the case where the pair is co-Haagerup. This allows to generalize previous results, while providing new examples of groups satisfying the Baum-Connes conjecture with coefficients. For instance, we show that for some S-arithmetic subgroups of Sp(5,1) and Sp(3,1) the conjecture with coefficients holds.

Global Fukaya category I

Abstract Let $Ham (M,\omega ) $ denote the Frechet Lie group of Hamiltonian symplectomorphisms of a monotone symplectic manifold $(M, \omega ) $. Let $NFuk (M, \omega )$ be the $A _{\infty } $-nerve of the Fukaya category $Fuk (M, \omega )$, and let $(|\mathbb {S}|, NFuk (M, \omega ))$ denote the $NFuk (M, \omega )$ component of the “space of $\infty $-categories” $|\mathbb {S}| $. Using Floer-Fukaya theory for a monotone $(M, \omega ),$ we construct a natural up to homotopy classifying map $$ \begin{align*}&amp; BHam (M, \omega) \to (|\mathbb{S}|, NFuk (M, \omega)). \end{align*}$$This verifies one sense of a conjecture of Teleman on existence of action of $Ham (M, \omega )$ on the Fukaya category of $(M, \omega ) $. This construction is very closely related to the theory of the Seidel homomorphism and the quantum characteristic classes of the author, and this map is intended to be the deepest expression of their underlying geometric theory. In part II, the above map is shown to be nontrivial by an explicit calculation. In particular, we arrive at a new non-trivial “quantum” invariant of any smooth manifold, which motives the statement of a kind of “quantum” Novikov conjecture.

Weak Z-structures and one-relator groups

Motivated by the notion of boundary for hyperbolic and C A T ( 0 ) groups, Bestvina [2] introduced the notion of a (weak) Z -structure and (weak) Z -boundary for a group G of type F (i.e., having a finite K ( G , 1 ) complex), with implications concerning the Novikov conjecture for G . Since then, some classes of groups have been shown to admit a weak Z -structure (see [15] for example), but the question whether or not every group of type F admits such a structure remains open. In this paper, we show that every torsion free one-relator group admits a weak Z -structure, by showing that they are all properly aspherical at infinity; moreover, in the 1-ended case the corresponding weak Z -boundary has the shape of either a circle or a Hawaiian earring depending on whether the group is a virtually surface group or not. Finally, we extend this result to a wider class of groups still satisfying a Freiheitssatz property.

The coarse Novikov conjecture for coarse fibrations over Banach spaces with Property (H)

The coarse Novikov conjecture for coarse fibrations over Banach spaces with Property (H)

The Novikov conjecture and extensions of coarsely embeddable groups

Let 1\to N\to G\to G/N\to 1 be a short exact sequence of countable discrete groups and let B be any G - C^* -algebra. In this paper, we show that the strong Novikov conjecture with coefficients in B holds for such a group G when the normal subgroup N and the quotient group G/N are coarsely embeddable into Hilbert spaces. As a result, the group G satisfies the Novikov conjecture under the same hypothesis on N and G/N .

Boundary amenability of $Out(F_N)$

We prove that $Out(F_N)$ is boundary amenable. This also holds more generally for $Out(G)$, where $G$ is either a toral relatively hyperbolic group or a right-angled Artin group. As a consequence, all these groups satisfy the Novikov conjecture on higher signatures.

The coarse geometric $\ell^p$-Novikov conjecture for subspaces of nonpositively curved manifolds

In this paper, we prove the coarse geometric \ell^p -Novikov conjecture for metric spaces with bounded geometry which admit a coarse embedding into a simply connected complete Riemannian manifold of nonpositive sectional curvature.

$G$-homotopy invariance of the analytic signature of proper co-compact $G$-manifolds and equivariant Novikov conjecture

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 $G$.

The Novikov conjecture, the group of volume preserving diffeomorphisms and Hilbert-Hadamard spaces

We prove that the Novikov conjecture holds for any discrete group admitting an isometric and metrically proper action on an admissible Hilbert-Hadamard space. Admissible Hilbert-Hadamard spaces are a class of (possibly infinite-dimensional) non-positively curved metric spaces that contain dense sequences of closed convex subsets isometric to Riemannian manifolds. Examples of admissible Hilbert-Hadamard spaces include Hilbert spaces, certain simply connected and non-positively curved Riemannian-Hilbertian manifolds and infinite-dimensional symmetric spaces. Thus our main theorem can be considered as an infinite-dimensional analogue of Kasparov’s theorem on the Novikov conjecture for groups acting properly and isometrically on complete, simply connected and non-positively curved manifolds. As a consequence, we show that the Novikov conjecture holds for geometrically discrete subgroups of the group of volume preserving diffeomorphisms of a closed smooth manifold. This result is inspired by Connes’ theorem that the Novikov conjecture holds for higher signatures associated to the Gelfand-Fuchs classes of groups of diffeormorphisms.

Incidence coefficients in the Novikov Complex for Morse forms: rationality and exponential growth properties

Let f : M → S 1 be a Morse map, v a transverse f -gradient. Theconstruction of the Novikov complex associates to these data a free chain complexC ∗ (f, v) over the ring Z[t]][t −1 ], generated by the critical points of f and computingthe completed homology module of the corresponding infinite cyclic covering of M .Novikov’s Exponential Growth Conjecture says that the boundary operators in thiscomplex are power series of non-zero convergence raduis.In [12] the author announced the proof of the Novikov conjecture for the case ofC 0 -generic gradients together with several generalizations. The proofs of the firstpart of this work were published in [13]. The present article contains the proofs ofthe second part.There is a refined version of the Novikov complex, defined over a suitable com-pletion of the group ring of the fundamental group. We prove that for a C 0 -genericf -gradient the corresponding incidence coefficients belong to the image in the Novikovring of a (non commutative) localization of the fundamental group ring.The Novikov construction generalizes also to the case of Morse 1-forms. In thiscase the corresponding incidence coefiicients belong to a certain completion of thering of integral Laurent polynomials of several variables. We prove that for a givenMorse form ω and a C 0 -generic ω-gradient these incidence coefficients are rationalfunctions.The incidence coefficients in the Novikov complex are obtained by counting thealgebraic number of the trajectories of the gradient, joining the zeros of the Morseform. There is V.I.Arnold’s version of the exponential growth conjecture, whichconcerns the total number of trajectories. We confirm this stronger form of theconjecture for any given Morse form and a C 0 -dense set of its gradients.We give an example of explicit computation of the Novikov complex.

Strong Novikov conjecture for low degree cohomology and exotic group C*-algebras

We strengthen a result of Hanke-Schick about the strong Novikov conjecture for low degree cohomology by showing that their non-vanishing result for the maximal group C*-algebra holds for many other exotic group C*-algebras, in particular the one associated to the smallest strongly Morita compatible and exact crossed product functor used in the new version of the Baum-Connes conjecture. To achieve this we provide a Fell absorption principle for certain exotic crossed product functors.

Positive Scalar Curvature due to the Cokernel of the Classifying Map

This paper contributes to the classification of positive scalar curvature metrics up to bordism and up to concordance. Let $M$ be a closed spin manifold of dimension $\ge 5$ which admits a metric with positive scalar curvature. We give lower bounds on the rank of the group of psc metrics over $M$ up to bordism in terms of the corank of the canonical map $KO_*(M)\to KO_*(B\pi_1(M))$, provided the rational analytic Novikov conjecture is true for $\pi_1(M)$.