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Boris Botvinnik

Publications and source records attributed to Boris Botvinnik.

At least 19 recordsLinked to original sources
arXiv2026

Brunnian links and Kontsevich graph complex I

We construct a natural chain map from the Kontsevich graph complex to the rational singular chain complex of $B\mathrm{Diff}_\partial(D^{2k})$ when the dimension $2k$ is sufficiently large, generalizing Goussarov and Habiro's theories of surgery on 3-valent graphs in 3-manifolds. Our construction can be considered as a topological realization of the Kontsevich graph complex. We also give new constructions of elements in the rational homotopy groups of $B\mathrm{Diff}_\partial(D^{2k})$ which are determined by well-known cycles in the graph complex.

math.GT↗
arXiv2025

Classification of spin$^c$ manifolds with generalized positive scalar curvature

Suppose $M$ is a closed $n$-dimensional spin$^c$ manifold with spin$^c$ structure $\sigma$ and associated spin$^c$ line bundle $L$. If one fixes a Riemannian metric $g$ on $M$ and a connection $\nabla_L$ on $L$, the generalized scalar curvature $R^{\text{gen}}$ of $(M,L)$ is $R_g - 2|\Omega_L|_{\text{op}}$, where $|\Omega_L|_{\text{op}}$ is the pointwise operator norm of the curvature $2$-form $\Omega_L$ of $\nabla_L$, acting on spinors. In a previous paper, we showed that positivity of $R^{\text{gen}}$ is obstructed by the non-vanishing of the index of the spin$^c$ Dirac operator on $(M,g,L,\nabla_L)$, and that in some cases, the vanishing of this index guarantees the existence of a pair $(g,\nabla_L)$ with positive generalized scalar curvature. Building on this and on surgery techniques inspired by those that have been developed in the theory of positive scalar curvature on spin manifolds, we show that if $\dim M = n \ge 5$, if the fundamental group $\pi$ of $M$ is in a large class including surface groups and finite groups with periodic cohomology, and if $M$ is totally non-spin (meaning that the universal cover is not spin), then $(M,L)$ admits positive generalized scalar curvature if and only if the generalized $\alpha$-invariant of $(M,L)$ vanishes in the $K$-homology group $K_n(B\pi)$. We also develop an analogue of Stolz's sequence for computing the group of concordance classes of positive generalized scalar curvature metrics, and connect this to the analytic surgery sequence of Roe and Higson. Finally, we give a number of applications to moduli spaces of positive generalized scalar curvature metrics.

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arXiv2024

On the topology of the moduli space of positive scalar curvature concordances

Let $M$ be a manifold which admits a metric with positive scalar curvature (or a positive intermediate curvature in a suitable sense). We study the moduli space ${\mathscr{M}}^{{\mathsf{pos}}_*}_{\sqcup}(M\times I)_g$ of concordances of such metrics (with appropriate boundary conditions) which restrict to a given metric $g$ on $M \times \{0\} \cup\partial M \times I$. We show that $\pi_{4*}{\mathscr{M}}^{{\mathsf{pos}}_*}_{\sqcup}(M \times I)_g \otimes {\mathbb Q} \neq 0$ in a stable range provided $\dim M$ is even. We obtain analogous results when positive scalar curvature is replaced by $k$-positive Ricci curvature for $k \ge 2$.

math.DG↗
arXiv2024

Generalized positive scalar curvature on spin$^c$ manifolds

Let $(M,L)$ be a (compact) non-spin spin$^c$ manifold. Fix a Riemannian metric $g$ on $M$ and a connection $A$ on $L$, and let $D_L$ be the associated spin$^c$ Dirac operator. Let $R^{tw}_{(g,A)}:=R_g + 2ic(\Omega)$ be the twisted scalar curvature (which takes values in the endomorphisms of the spinor bundle), where $R_g$ is the scalar curvature of $g$ and $2ic(\Omega)$ comes from the curvature $2$-form $\Omega$ of the connection $A$. Then the Lichnerowicz-Schr\"odinger formula for the square of the Dirac operator takes the form $D_L^2 =\nabla^*\nabla+\frac{1}{4}R^{tw}_{(g,A)}$. In a previous work we proved that a closed non-spin simply-connected spin$^c$-manifold $(M,L)$ of dimension $n\geq 5$ admits a pair $(g,A)$ such that $R^{tw}_{(g,A)}>0$ if and only if the index $\alpha^c(M,L):=\text{ind}\, D_L$ vanishes in $K_n$. In this paper we introduce a scalar-valued generalized scalar curvature $R^{gen}_{(g,A)}:=R_g - 2|\Omega|_{op}$, where $|\Omega|_{op}$ is the pointwise operator norm of Clifford multiplication $c(\Omega)$, acting on spinors. We show that the positivity condition on the operator $R^{tw}_{(g,A)}$ is equivalent to the positivity of the scalar function $R^{gen}_{g,A}$. We prove a corresponding trichotomy theorem concerning the curvature $R^{gen}_{(g,A)}$, and study its implications. We also show that the space $\mathcal{R}^{gen+}(M,L)$ of pairs $(g,A)$ with $R^{gen}_{(g,A)}>0$ has non-trivial topology, and address a conjecture about non-triviality of the ``index difference'' map.

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arXiv2023

Symmetries of exotic spheres via complex and quaternionic Mahowald invariants

We use new homotopy-theoretic tools to prove the existence of smooth $U(1)$- and $Sp(1)$-actions on infinite families of exotic spheres. Such families of spheres are propagated by the complex and quaternionic analogues of the Mahowald invariant (also known as the root invariant). In particular, we prove that the complex (respectively, quaternionic) Mahowald invariant takes an element of the $k$-th stable stem $\pi_k^s$ represented by a homotopy sphere $\Sigma^k$ to an element of a higher stable stem $\pi_{k+\ell}^s$ represented by another homotopy sphere $\Sigma^{k+\ell}$ equipped with a smooth $U(1)$- (respectively, $Sp(1)$-) action with fixed points the original homotopy sphere $\Sigma^k\subset \Sigma^{k+\ell}$.

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arXiv2022

Families of diffeomorphisms and concordances detected by trivalent graphs

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 $\pi_*(B\mathrm{Diff}_{\partial}(D^d))\otimes \mathbb{Q}$ are lifted to homotopy groups of the moduli space of $h$-cobordisms $\pi_*(B\mathrm{Diff}_{\sqcup}(D^d\times I))\otimes \mathbb{Q}$. As a geometrical application, we show that those elements in $\pi_*(B\mathrm{Diff}_{\partial}(D^d))\otimes \mathbb{Q}$ for $d\geq 4$ are also lifted to the rational homotopy groups $\pi_*(\mathcal{M}^{\mathrm{psc}}_{\partial}(D^d)_{h_0})\otimes \mathbb{Q}$ of the moduli space of positive scalar curvature metrics. Moreover, we show that the same elements come from the homotopy groups $\pi_*(\mathcal{M}^{\mathrm{psc}}_{\sqcup} (D^d\times I; g_0)_{h_0})\otimes \mathbb{Q}$ of moduli space of concordances of positive scalar curvature metrics on $D^d$ with fixed round metric $h_0$ on the boundary $S^{d-1}$.

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arXiv2021

Positive scalar curvature on $\mathbf{Pin}^\pm$- and $\mathbf{Spin}^c$-manifolds

It is well-known that spin structures and Dirac operators play a crucial role in the study of positive scalar curvature metrics (psc-metrics) on compact manifolds. Here we consider a class of non-spin manifolds with "almost spin" structure, namely those with spin$^c$ or pin$^\pm$-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 $S^1$- or $C_2$-invariant near a "special submanifold" $B$ of $M$. This submanifold $B\subset M$ is dual to the complex (respectively, real) line bundle $L$ which determines the spin$^c$ or pin$^\pm$ 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.

math.DG↗
arXiv2020

Positive Scalar Curvature on Spin Pseudomanifolds: the Fundamental Group and Secondary Invariants

In this paper we continue the study of positive scalar curvature (psc) metrics on a depth-1 Thom-Mather stratified space $M_\Sigma$ with singular stratum $\beta M$ (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 $M_\Sigma$ and $\beta M$ are not simply connected. We also investigate the space of such psc metrics and show that it often splits into many cobordism classes.

math.DG↗
arXiv2019

Positive scalar curvature on simply connected spin pseudomanifolds

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 associated to the resolution of $M_\Sigma$, $M$, and to the singular locus $\beta M$. Finally, when $M_\Sigma$, $\beta M$, $L$, and $G$ are simply connected and $\dim M$ is big enough, and when some other conditions on $L$ (satisfied in a large number of cases) hold, we establish the main result of this article, showing that the vanishing of these two $\alpha$-classes is also sufficient for the existence of a well-adapted wedge metric of positive scalar curvature.

math.DG↗
arXiv2019

Evolution of relative Yamabe constant under Ricci Flow

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.

math.DG↗
arXiv2018

On the topology of the space of Ricci-positive metrics

We show that the space $\mathcal{R}^{\mathrm{pRc}}(W_g^{2n})$ of metrics with positive Ricci curvature on the manifold $W^{2n}_g := \sharp^g (S^n \times S^n)$ has nontrivial rational homology if $n \not \equiv 3 \pmod 4$ and $g$ are both sufficiently large. The same argument applies to $\mathcal{R}^{\mathrm{pRc}}(W_g^{2n} \sharp N)$ provided that $N$ is spin and $W_g^{2n} \sharp N$ admits a Ricci positive metric.

math.AT↗
arXiv2018

Concordance and isotopy of metrics with positive scalar curvature, II

In this article, the author provides full details of the proof of the concordance/isotopy problem. The first published proof, [5], accomplished this task only partially since there was an error, see the erratum [6], which damaged the main argument of [5, Theorem 2.9], and, consequently, the proof of [5, Theorem A].

math.DG↗
arXiv2018

Positive scalar curvature on manifolds with fibered singularities

A (compact) manifold with fibered $P$-singularities is a (possibly) singular pseudomanifold $M_\Sigma$ with two strata: an open nonsingular stratum $\mathring M$ (a smooth open manifold) and a closed stratum $\beta M$ (a closed manifold of positive codimension), such that a tubular neighborhood of $\beta M$ 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 $M_{\Sigma}$ 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 $\mathbb Z/k$ 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.

math.DG↗
arXiv2018

Cheeger-Gromov convergence in a conformal setting

For a sequence $\{(M_i, g_i, x_i)\}$ of pointed Riemannian manifolds with boundary, the sequence $\{(M_i,\tilde g_i,x_i)\}$ is its conformal satellite if the metric $\tilde g_i$ is conformal to $g_i$, that is, $\tilde g_i=u^{\frac{4}{n-2}}_ig_i$. Assuming the manifolds $(M_i,g_i,x_i)$ have uniformly bounded geometry, we show that both sequences have smoothly Cheeger-Gromov convergent subsequences provided the conformal factors $u_i$ are principal eigenfunctions of an appropriate elliptic operator. Part of our result is a Cheeger-Gromov compactness for manifolds with boundary. We use stable versions of classical elliptic estimates and inequalities found in the recently established 'flatzoomer' method.

math.DG↗
arXiv2017

Homotopy groups of the observer moduli space of Ricci positive metrics

The observer moduli space of Riemannian metrics is the quotient of the space $\mathcal{R}(M)$ of all Riemannian metrics on a manifold $M$ by the group of diffeomorphisms $\mathrm{Diff}_{x_0}(M)$ which fix both a basepoint $x_0$ and the tangent space at $x_0$. The group $\mathrm{Diff}_{x_0}(M)$ acts freely on $\mathcal{R}(M)$ 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 $\mathcal{M}_{x_0}^{s>0}(M)$ 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 $\mathcal{M}_{x_0}^{\mathrm{Ric}>0}(M)$ of metrics with positive Ricci curvature. In particular we show that for a given $k$, there are infinite order elements in the homotopy group $\pi_{4k}\mathcal{M}_{x_0}^{\mathrm{Ric}>0}(S^n)$ 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.

math.DG↗
arXiv2017

Minimal hypersurfaces and bordism of positive scalar curvature metrics

Let $(Y,g)$ be a compact Riemannian manifold of positive scalar curvature (psc). It is well-known, due to Schoen-Yau, that any closed stable minimal hypersurface of $Y$ also admits a psc-metric. We establish an analogous result for stable minimal hypersurfaces with free boundary. Furthermore, we combine this result with tools from geometric measure theory and conformal geometry to study psc-bordism. For instance, assume $(Y_0,g_0)$ and $(Y_1,g_1)$ are closed psc-manifolds equipped with stable minimal hypersurfaces $X_0 \subset Y_0$ and $X_1\subset Y_1$. Under natural topological conditions, we show that a psc-bordism $(Z,\bar g) : (Y_0,g_0)\rightsquigarrow (Y_1,g_1)$ gives rise to a psc-bordism between $X_0$ and $X_1$ equipped with the psc-metrics given by the Schoen-Yau construction.

math.DG↗
arXiv2016

Stable Moduli Spaces of High Dimensional Handlebodies

We study the moduli space of handlebodies diffeomorphic to $(D^{n+1}\times S^{n})^{\natural g}$, i.e. the classifying space $BDiff((D^{n+1}\times S^n)^{\natural g}, D^{2n})$ of the group of diffeomorphisms that restrict to the identity near a $2n$-dimensional disk embedded in the boundary, $\partial(D^{n+1}\times S^n)^{\natural g}$. We construct a map $colim_{g\to\infty}BDiff((D^{n+1}\times S^n)^{\natural g}, D^{2n}) \longrightarrow Q_{0}BO(2n+1)\langle n \rangle_{+}$ and prove that it induces an isomorphism on integral homology in the case that $2n+1 \geq 9$. Above, $BO(2n+1)\langle n \rangle$ denotes the $n$-connective cover of $BO(2n+1)$. The (co)homology of the space $Q_{0}BO(2n+1)\langle n \rangle_{+}$ is well understood and so our results enable one to compute the homology groups $H_{k}(BDiff((D^{n+1}\times S^n)^{\natural g}, D^{2n}))$ in a range of degrees when $k << g$. Our main theorem can be viewed as an analogue of the Madsen-Weiss theorem for the moduli spaces of surfaces and the recent theorem of Galatius and Randal-Williams for the moduli spaces of manifolds of dimension $2n \geq 6$.

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