Papers by Boris Botvinnik
Journal of Topology and Analysis
Let [Formula: see text] be an [Formula: see text]-dimensional Thom–Mather stratified space of dep... more Let [Formula: see text] be an [Formula: see text]-dimensional Thom–Mather stratified space of depth [Formula: see text]. We denote by [Formula: see text] the singular locus and by [Formula: see text] the associated link. In this paper, we study the problem of when such a space can be endowed with a wedge metric of positive scalar curvature. We relate this problem to recent work on index theory on stratified spaces, giving first an obstruction to the existence of such a metric in terms of a wedge [Formula: see text]-class [Formula: see text]. In order to establish a sufficient condition, we need to assume additional structure: we assume that the link of [Formula: see text] is a homogeneous space of positive scalar curvature, [Formula: see text], where the semisimple compact Lie group [Formula: see text] acts transitively on [Formula: see text] by isometries. Examples of such manifolds include compact semisimple Lie groups and Riemannian symmetric spaces of compact type. Under these a...
Positive Scalar Curvature and Homotopy Theory
WORLD SCIENTIFIC eBooks, Jan 8, 2023
Journal of Topology
We study families of diffeomorphisms detected by trivalent graphs via the Kontsevich classes. We ... more We study families of diffeomorphisms detected by trivalent graphs via the Kontsevich classes. We specify some recent results and constructions of the second named author to show that those non-trivial elements in homotopy groups π * (BDiff ∂ (D d )) ⊗ Q are lifted to homotopy groups of the moduli space of h -cobordisms π * (BDiff (D d × I)) ⊗ Q . As a geometrical application, we show that those elements in π * (BDiff ∂ (D d )) ⊗ Q for d ≥ 4 are also lifted to the rational homotopy groups π * (M psc ∂ (D d ) h 0 ) ⊗ Q of the moduli space of positive scalar curvature metrics. Moreover, we show that the same elements come from the homotopy groups π * (M psc (D d × I; g0) h 0 ) ⊗ Q of moduli space of concordances of positive scalar curvature metrics on D d with fixed round metric h0 on the boundary S d-1 .
arXiv (Cornell University), Aug 17, 2018
A (compact) manifold with fibered P-singularities is a (possibly) singular pseudomanifold MΣ with... more A (compact) manifold with fibered P-singularities is a (possibly) singular pseudomanifold MΣ with two strata: an open nonsingular stratumM (a smooth open manifold) and a closed stratum βM (a closed manifold of positive codimension), such that a tubular neighborhood of βM is a fiber bundle with fibers each looking like the cone on a fixed closed manifold P. We discuss what it means for such an MΣ with fibered P-singularities to admit an appropriate Riemannian metric of positive scalar curvature, and we give necessary and sufficient conditions (the necessary conditions based on suitable versions of index theory, the sufficient conditions based on surgery methods and homotopy theory) for this to happen when the singularity type P is either Z/k or S 1 , and M and the boundary of the tubular neighborhood of the singular stratum are simply connected and carry spin structures. Along the way, we prove some results of perhaps independent interest, concerning metrics on spin c manifolds with positive "twisted scalar curvature," where the twisting comes from the curvature of the spin c line bundle.
arXiv (Cornell University), May 23, 2013
The Schouten tensor A of a Riemannian manifold (M, g) provides important scalar curvature invaria... more The Schouten tensor A of a Riemannian manifold (M, g) provides important scalar curvature invariants σ k , that are the symmetric functions on the eigenvalues of A, where, in particular, σ 1 coincides with the standard scalar curvature Scal(g) . Our goal here is to study compact manifolds with positive Γ 2 -curvature, i.e., when σ 1 (g) > 0 and σ 2 (g) > 0 . In particular, we prove that a 3-connected non-string manifold M admits a positive Γ 2 -curvature metric if and only if it admits a positive scalar curvature metric. Also we show that any finitely presented group π can always be realised as the fundamental group of a closed manifold of positive Γ 2 -curvature and of arbitrary dimension greater than or equal to six.
Symmetry, Integrability and Geometry: Methods and Applications, 2021
In this paper we study manifolds, X Σ , with fibred singularities, more specifically, a relevant ... more In this paper we study manifolds, X Σ , with fibred singularities, more specifically, a relevant space R psc (X Σ ) of Riemannian metrics with positive scalar curvature. Our main goal is to prove that the space R psc (X Σ ) is homotopy invariant under certain surgeries on X Σ .
Symmetry, Integrability and Geometry: Methods and Applications, 2021
In this paper we continue the study of positive scalar curvature (psc) metrics on a depth-1 Thom-... more In this paper we continue the study of positive scalar curvature (psc) metrics on a depth-1 Thom-Mather stratified space M Σ with singular stratum βM (a closed manifold of positive codimension) and associated link equal to L, a smooth compact manifold. We briefly call such spaces manifolds with L-fibered singularities. Under suitable spin assumptions we give necessary index-theoretic conditions for the existence of wedge metrics of positive scalar curvature. Assuming in addition that L is a simply connected homogeneous space of positive scalar curvature, L = G/H, with the semisimple compact Lie group G acting transitively on L by isometries, we investigate when these necessary conditions are also sufficient. Our main result is that our conditions are indeed sufficient for large classes of examples, even when M Σ and βM are not simply connected. We also investigate the space of such psc metrics and show that it often splits into many cobordism classes.
arXiv: Differential Geometry, 2019
Let $M_\Sigma$ be an $n$-dimensional Thom-Mather stratified space of depth $1$. We denote by $\be... more Let $M_\Sigma$ be an $n$-dimensional Thom-Mather stratified space of depth $1$. We denote by $\beta M$ the singular locus and by $L$ the associated link. In this paper we study the problem of when such a space can be endowed with a wedge metric of positive scalar curvature. We relate this problem to recent work on index theory on stratified spaces, giving first an obstruction to the existence of such a metric in terms of a wedge $\alpha$-class $\alpha_w (M_\Sigma)\in KO_n$. In order to establish a sufficient condition we need to assume additional structure: we assume that the link of $M_\Sigma$ is a homogeneous space of positive scalar curvature, $L=G/K$, where the semisimple compact Lie group $G$ acts transitively on $L$ by isometries. Examples of such manifolds include compact semisimple Lie groups and Riemannian symmetric spaces of compact type. Under these assumptions, when $M_\Sigma$ and $\beta M$ are spin, we reinterpret our obstruction in terms of two $\alpha$-classes associa...
arXiv: Differential Geometry, 2019
Let $W$ be a manifold with boundary $M$ given together with a conformal class $\bar C$ which rest... more Let $W$ be a manifold with boundary $M$ given together with a conformal class $\bar C$ which restricts to a conformal class $C$ on $M$. Then the relative Yamabe constant $Y_{\bar C}(W,M;C)$ is well-defined. We study the short-time behavior of the relative Yamabe constant $Y_{[\bar g_t]}(W,M;C)$ under the Ricci flow $\bar g_t$ on $W$ with boundary conditions that mean curvature $H_{\bar g_t}\equiv 0$ and $\bar{g}_t|_M\in C = [\bar{g}_0]$. In particular, we show that if the initial metric $\bar{g}_0$ is a Yamabe metric, then, under some natural assumptions, $\left.\frac{d}{dt}\right|_{t=0}Y_{[\bar g_t]}(W,M;C)\geq 0$ and is equal to zero if and only the metric $\bar{g}_0$ is Einstein.
It is well-known that spin structures and Dirac operators play a crucial role in the study of pos... more It is well-known that spin structures and Dirac operators play a crucial role in the study of positive scalar curvature metrics (pscmetrics) on compact manifolds. Here we consider a class of non-spin manifolds with “almost spin” structure, namely those with spin or Pin-structures. It turns out that in those cases (under natural assumptions on such a manifold M), the index of a relevant Dirac operator completely controls existence of a psc-metric which is Sor C2-invariant near a “special submanifold” B of M . This submanifold B ⊂ M is dual to the complex (respectively, real) line bundle L which determines the spin or pin structure on M . We also show that these manifold pairs (M,B) can be interpreted as “manifolds with fibered singularities” equipped with “well-adapted psc-metrics”. This survey is based on our recent work as well as on our joint work with Paolo Piazza.
The connective K-theory groups ko∗(Bπ) of a group π appear in many contexts; for example, they ar... more The connective K-theory groups ko∗(Bπ) of a group π appear in many contexts; for example, they are the building blocks for equivariant spin bordism at the prime 2. They also play an important role in the Gromov–Lawson–Rosenberg conjecture which was the starting point of our original investigation [5]. The second author first studied the eta invariant, which is an analytic invariant, whilst a graduate student under the direction of L. Nirenberg so this is perhaps a fitting subject for this volume. In this paper, we will use the eta invariant to determine the additive structure of ko4ν−1(BQ8), where Q8 = {±1,±i,±j,±k}
Proceedings of the American Mathematical Society, 2004
The statement often called the Gromov-Lawson-Rosenberg Conjecture asserts that a manifold with fi... more The statement often called the Gromov-Lawson-Rosenberg Conjecture asserts that a manifold with finite fundamental group should admit a metric of positive scalar curvature except when the K O ∗ KO_* -valued index of some Dirac operator with coefficients in a flat bundle is non-zero. We prove spin and oriented non-spin versions of this statement for manifolds (of dimension ≥ 5 \ge 5 ) with elementary abelian fundamental groups π \pi , except for “toral” classes, and thus our results are automatically applicable once the dimension of the manifold exceeds the rank of π \pi . The proofs involve the detailed structure of B P ∗ ( B π ) BP_*(B\pi ) , as computed by Johnson and Wilson.
Proceedings of the American Mathematical Society, 2020
We show that the space R p R c ( W g 2 n ) \mathcal {R}^{\mathrm {pRc}}(W_g^{2n}) of metrics with... more We show that the space R p R c ( W g 2 n ) \mathcal {R}^{\mathrm {pRc}}(W_g^{2n}) of metrics with positive Ricci curvature on the manifold W g 2 n ≔ ♯ g ( S n × S n ) W^{2n}_g \coloneq \sharp ^g (S^n \times S^n) has nontrivial rational homology if n ≢ 3 ( mod 4 ) n \not \equiv 3 \pmod 4 and g g are both sufficiently large. The same argument applies to R p R c ( W g 2 n ♯ N ) \mathcal {R}^{\mathrm {pRc}}(W_g^{2n} \sharp N) provided that N N is spin and W g 2 n ♯ N W_g^{2n} \sharp N admits a Ricci positive metric.
Geometry & Topology, 2019
The observer moduli space of Riemannian metrics is the quotient of the space R(M ) of all Riemann... more The observer moduli space of Riemannian metrics is the quotient of the space R(M ) of all Riemannian metrics on a manifold M by the group of diffeomorphisms Diffx 0 (M ) which fix both a basepoint x0 and the tangent space at x0 . The group Diffx 0 (M ) acts freely on R(M ) providing M is connected. This offers certain advantages over the classic moduli space, which is the quotient by the full diffeomorphism group. Results due to Botvinnik, Hanke, Schick and Walsh, and to Hanke, Schick and Steimle have demonstrated that the higher homotopy groups of the observer moduli space M s>0 x 0 (M ) of positive scalar curvature metrics are, in many cases, non-trivial. The aim in the current paper is to establish similar results for the moduli space M Ric>0 x 0 (M ) of metrics with positive Ricci curvature. In particular we show that for a given k , there are infinite order elements in the homotopy group π 4k M Ric>0 x 0 (S n ) provided the dimension n is odd and sufficiently large. In establishing this we make use of a gluing result of Perelman. We provide full details of the proof of this gluing theorem, which we believe have not appeared before in the literature. We also extend this to a family gluing theorem for Ricci positive manifolds.
Journal of Topology, 2017
We study the moduli space of handlebodies diffeomorphic to (D n+1 × S n ) ♮g , i.e. the classifyi... more We study the moduli space of handlebodies diffeomorphic to (D n+1 × S n ) ♮g , i.e. the classifying space BDiff((D n+1 × S n ) ♮g , D 2n ) of the group of diffeomorphisms that restrict to the identity near an embedded disk D 2n ⊂ ∂(D n+1 × S n ) ♮g . We prove that there is a natural map which induces an isomorphism in integral homology when n ≥ 4 . Above, BO(2n + 1) n denotes the n -connective cover of BO(2n + 1) .
Topological Methods in Nonlinear Analysis, 1995
Journal of Differential Geometry, 1997
The geometrical point of view on the Adams-Novikov spectral sequence
Communications in Analysis and Geometry, 2002
We define a relative Yamabe invariant of a smooth manifold with given conformal class on its boun... more We define a relative Yamabe invariant of a smooth manifold with given conformal class on its boundary. In the case of empty boundary the invariant coincides with the classic Yamabe invariant. We develop approximation technique which leads to gluing theorems of two manifolds along their boundaries for the relative Yamabe invariant. We show that there are many examples of manifolds with both positive and non-positive relative Yamabe invariants.
Uploads
Papers by Boris Botvinnik