Moulay-Tahar Benameur
This is a survey paper which gathers some results related with the study of Positive Scalar Curvature metrics in connection with the Baum-Connes assembly map.
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In an appendix to an earlier paper (cf. arXiv:1703.00984) we showed we showed how to construct tunnels of positive scalar curvature and of arbitrarily small length and volume connecting points in a \emph{three dimensional} manifold of \emph{constant sectional curvature}. Here we generalize the construction to arbitrary dimensions and require only positivity of the scalar curvature. This version contains an added section on existence of arbitrarily narrow tunnels of prescribed length.
Let $N$ be a closed enlargeable manifold in the sense of Gromov-Lawson and $M$ a closed spin manifold of equal dimension, a famous theorem of Gromov-Lawson states that the connected sum $M\# N$ admits no metric of positive scalar curvature. We present a potential generalization of this result to the case where $M$ is nonspin. We use index theory for Dirac operators to prove our result.
We present a history of the Baum-Connes conjecture, the methods involved, the current status, and the mathematics it generated.
Lawson cones and the Allen-Cahn equation
本文研究了渐近于Lawson锥的特殊极小超曲面的非退化性与稳定性，并利用这些超曲面构造了在$\R^{N+1}$中具有$k\ge 2$个连通分支零水平集及无穷Morse指标的Allen-Cahn方程解。
These notes cover the contents of three survey lectures held at the ICTP Trieste Summer school on High dimensional manifold theory 2001. They introduce techniques coming from the theory of operator algebras. We will focus on the basic definitions and properties, and on their relevance to the geometry and topology of manifolds. An central pillar of work in the theory of C*-algebras is the Baum-Connes conjecture. It implies the Novikov conjecture. In the first talk, the Baum-Connes conjecture will be explained and put into our context. One application of the Baum-Connes conjecture is to the posi…
The Gromov-Lawson-Rosenberg Conjecture for Z/4xZ/4
本研究通过 Adams 谱序列计算分类空间连通实 k-同调，结合两类α不变量核检测定理，证明了Z/4×Z/4群的Gromov-Lawson-Rosenberg猜想，同时确定了相关谱序列微分并研究了对应子Hopf代数的谱序列上积结构。
Let G be a finitely generated discrete group. In this paper we establish vanishing results for rho-invariants associated to (i) the spin-Dirac operator of a spin manifold with positive scalar curvature (ii) the signature operator of the disjoint union of a pair of homotopy equivalent oriented manifolds with fundamental group G. The invariants we consider are more precisely - the Atiyah-Patodi-Singer rho-invariant associated to a pair of finite dimensional unitary representations. - the L2-rho invariant of Cheeger-Gromov - the delocalized eta invariant of Lott for a finite conjugacy class of G.…
We discuss a conjecture of Gromov and Lawson, later modified by Rosenberg, concerning the existence of metrics of positive scalar curvature. It says that a closed spin manifold $M$ of dimension $n\ge 5$ has such a metric if and only if the index of a suitable ``Dirac" operator in $KO_n(C^* (π_1(M)))$, the real $K$-theory of the group $C^*$-algebra of the fundamental group of $M$, vanishes. It is known that the vanishing of the index is necessary for existence of a positive scalar curvature metric on $M$, but this is known to be a sufficient condition only if $π_1(M)$ is the trivial group, $\Bb…
This is a continuation of our previous work with Botvinnik on the nontriviality of the secondary index invariant on spaces of metrics of positive scalar curvature, in which we take the fundamental group of the manifolds into account. We show that the secondary index invariant associated to the vanishing of the Rosenberg index can be highly nontrivial, for positive scalar curvature Spin manifolds with torsionfree fundamental groups which satisfy the Baum--Connes conjecture. For example, we produce a compact Spin 6-manifold such that its space of positive scalar curvature metrics has each ration…
As shown by Gromov-Lawson and Stolz the only obstruction to the existence of positive scalar curvature metrics on closed simply connected manifolds in dimensions at least five appears on spin manifolds and is given by the non-vanishing of the $α$-genus of Hitchin. When unobstructed we shall realize a positive scalar curvature metric by an immersion into Euclidean space whose dimension is uniformly close to the classical Whitney upper bound for smooth immersions. Our main tool is an extrinsic counterpart of the well-known Gromov-Lawson surgery procedure for constructing positive scalar curvatur…
We give a complete solution, for discrete countable groups, to the problem of defining and computing a geometric pairing between the left hand side of the Baum-Connes assembly map, given in terms of geometric cycles associated to proper actions on manifolds, and cyclic periodic cohomology of the group algebra. Indeed, for any such group $Γ$ (without any further assumptions on it) we construct an explicit morphism from the Left-Hand side of the Baum-Connes assembly map to the periodic cyclic homology of the group algebra. This morphism, called here the Chern-Baum-Connes assembly map, allows to …
We extend the deep and important results of Lichnerowicz, Connes, and Gromov-Lawson which relate geometry and characteristic numbers to the existence and non-existence of metrics of positive scalar curvature (PSC). In particular, we show: that a spin foliation with Hausdorff homotopy groupoid of an enlargeable manifold admits no PSC metric; that any metric of PSC on such a foliation is bounded by a multiple of the reciprocal of the foliation K-area of the ambient manifold; and that Connes' vanishing theorem for characteristic numbers of PSC foliations extends to a vanishing theorem for Haeflig…
We use the symbol calculus for foliations developed in our previous paper to derive a cohomological formula for the Connes-Chern character of the semi-finite spectral triple. The same proof works for the Type I spectral triple of Connes-Moscovici. The cohomology classes of the two Connes-Chern characters induce the same map on the image of the maximal Baum-Connes map in K-theory, thereby proving an Atiyah $L^2$ covering index theorem.
It is well known that isotopic metrics of positive scalar curvature are concordant. Whether or not the converse holds is an open question, at least in dimensions greater than four. We show that for a particular type of concordance, constructed using the surgery techniques of Gromov and Lawson, this converse holds in the case of closed simply connected manifolds of dimension at least five.
In this paper we prove that the moduli space of metrics with positive scalar curvature of an orientable compact 3-manifold is path-connected. The proof uses the Ricci flow with surgery, the conformal method, and the connected sum construction of Gromov and Lawson. The work of Perelman on Hamilton's Ricci flow is fundamental. As one of the applications we prove the path-connectedness of the space of trace-free asymptotically flat solutions to the vacuum Einstein constraint equations on $\mathbb{R}^3$.
The equivariant coarse index is well-understood and widely used for actions by discrete groups. We extend the definition of this index to general locally compact groups. We use a suitable notion of admissible modules over $C^*$-algebras of continuous functions to obtain a meaningful index. Inspired by work by Roe, we then develop a localised variant, with values in the $K$-theory of a group $C^*$-algebra. This generalises the Baum-Connes assembly map to non-cocompact actions. We show that an equivariant index for Callias-type operators is a special case of this localised index, obtain results …
We show how a suitably twisted Spin-cobordism spectrum connects to the question of existence of metrics of positive scalar curvature on closed, smooth manifolds by building on fundamental work of Gromov, Lawson, Rosenberg, Stolz and others. We then investigate this parametrised spectrum, compute its $mod~2$-cohomology and generalise the Anderson-Brown-Peterson splitting of the regular Spin-cobordism spectrum to the twisted case. Along the way we also describe the $mod~2$-cohomology of various twisted, connective covers of real K-theory.
We prove the inequality $$ \dim_{mc}\Wi M\le n-2$$ for the macroscopic dimension of the universal covers $\Wi M$ of almost spin $n$-manifolds $M$ with positive scalar curvature whose fundamental group $π_1(M)$ is a virtual duality group that satisfies the coarse Baum-Connes conjecture.
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