56 research outputs found
Homotopy groups of the moduli space of metrics of positive scalar curvature
No full textWe show by explicit examples that in many degrees in a stable range the homotopy groups of the moduli spaces of Riemannian metrics of positive scalar curvature on closed smooth manifolds can be non-trivial. This is achieved by further developing and then applying a family version of the surgery construction of Gromov–Lawson to certain nonlinear smooth sphere bundles constructed by Hatcher
Positive scalar curvature on simply connected spin pseudomanifolds
No full textLet 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(omega)(M-Sigma) is an element of KOn. 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 MS and beta M are spin, we reinterpret our obstruction in terms of two a-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 dimM 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 paper, showing that the vanishing of these two a-classes is also sufficient for the existence of a well-adapted wedge metric of positive scalar curvature
Positive Scalar Curvature on Spin Pseudomanifolds: the Fundamental Group and Secondary Invariants
No full textIn 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
Metrics of Positive Ricci Curvature on Connected Sums: Projective Spaces, Products, and Plumbings
No full textThe classification of simply connected manifolds admitting metrics of positive scalar curvature of initiated by Gromov-Lawson, at its core, relies on a careful geometric construction that preserves positive scalar curvature under surgery and, in particular, under connected sum. For simply connected manifolds admitting metrics of positive Ricci curvature, it is conjectured that a similar classification should be possible, and, in particular, there is no suspected obstruction to preserving positive Ricci curvature under connected sum. Yet there is no general construction known to take two Ricci-positive Riemannian manifolds and form a Ricci-positive metric on their connected sums. In this work, we utilize and extend Perelman’s construction of Ricci-positive metrics on connected sums of complex projective planes, to give an explicit construction of Ricci-positive metrics on connected sums given that the individual summands admit very specific Ricci- positive metrics, which we call core metrics. Working towards the new goal of constructing core metrics on manifolds known to support metrics of positive Ricci curvature: we show how to generalize Perelman’s construction to all projective spaces, we show that the existence of core metrics is preserved under iterated sphere bundles, and we construct core metrics on certain boundaries of plumbing disk bundles over spheres. These constructions come together to give many new examples of Ricci-positive connected sums, in particular on the connected sum of arbitrary products of spheres and on exotic projective spaces
Scalar Curvature and Transfer Maps in Spin and Spin^c Bordism
No full textIn 1992, Stolz proved that, among simply connected Spin-manifolds of dimension5 or greater, the vanishing of a particular invariant α is necessary and sufficient
for the existence of a metric of positive scalar curvature. More precisely, there is
a map α: ΩSpin → ko (which may be realized as the index of a Dirac operator) ∗
which Hitchin established vanishes on bordism classes containing a manifold with
a metric of positive scalar curvature. Stolz showed kerα is the image of a transfer
map ΩSpinBPSp(3) → ΩSpin. In this paper we prove an analogous result for Spinc- ∗−8 ∗
manifolds and a related invariant αc : ΩSpinc → ku. We show that ker αc is the ∗
sum of the image of Stolz’s transfer ΩSpinBPSp(3) → ΩSpinc and an analogous map ∗−8 ∗
ΩSpinc BSU(3) → ΩSpinc . Finally, we expand on some details in Stolz’s original paper ∗−4 ∗
and provide alternate proofs for some parts
Perturbed Special Lagrangian Submanifolds
No full textThis thesis investigates perturbed special Lagrangian submanifolds with the aim of developing a Floer theory analogous to existing theories in symplectic geometry. Special Lagrangians arise naturally in Calabi--Yau manifolds as submanifolds calibrated by the real part of the holomorphic volume form.
Following a proposal of Donaldson and Segal, we view special Lagrangians as solutions to an infinite-dimensional Lagrange multipliers problem. Perturbations of pairs of stable forms which give rise to SU(3)-structures yield perturbed special Lagrangian equations, whose solutions generalize classical special Lagrangian submanifolds.
Chapter 2 introduces a finite-dimensional Morse-theoretic model, linking the Morse homology of a constrained function to the Lagrange function of the associated Lagrange-multipliers problem. Guided loosely by analogy with the finite-dimensional case, Chapter 3 explores the Donaldosn--Segal Lagrange multipliers problem whose solutions are perturbed special Lagrangians. We prove ellipticity results, basic compactness under tameness conditions, and a natural volume bound. The main theorem is a transversality result which states that the moduli space of solutions is generically a set of isolated points
Linking Forms, Singularities, and Homological Stability for Diffeomorphism Groups of Odd Dimensional Manifolds
No full textLet n > 1. We prove a homological stability theorem for the
diffeomorphism groups of (4n+1)-dimensional manifolds, with respect
to forming the connected sum with (2n-1)-connected,
(4n+1)-dimensional manifolds that are stably parallelizable.
Our techniques involve the study of the action of the diffeomorphism group of a manifold M on the linking form associated to the homology groups of M.
In order to study this action we construct a geometric model for the linking form using the intersections of embedded and immersed Z/k-manifolds.
In addition to our main homological stability theorem, we prove several results regarding disjunction for embeddings and immersions of Z/k-manifolds that could be of independent interest
Gluing manifolds with boundary and bordisms of positive scalar curvature metrics
No full textThis thesis presents two main results on analytic and topological aspects of scalar curvature. The first is a gluing theorem for scalar-flat manifolds with vanishing mean curvature on the boundary. Our methods involve tools from conformal geometry and perturbation techniques for nonlinear elliptic PDE. The second part studies bordisms of positive scalar curvature metrics. We present a modification of the Schoen-Yau minimal hypersurface technique to manifolds with boundary which allows us to prove a hereditary property for bordisms of positive scalar curvature metrics. The main technical result is a convergence theorem for stable minimal hypersurfaces with free boundary in bordisms with long collars which may be of independent interest
Administrative apparatus of Stalin era and Alekhin - Botvinnik failed match (1939-1940)
No full textThis article examines the fate of the well-known chess players of the middle of the 20th century - the “expatriate defector” Alexander Alekhine and the Soviet champion Mikhail Botvinnik - as one of the little-known stories related to the history of the contacts between the representatives of the Russian diaspora and the Soviet state of the Stalin era. The author examines the history of the failed match between these two outstanding chess masters in 1939-1940 and shows why the Alekhine-Botvinnik match, which had been initially approved at the highest party and state level, was not held, and find out what role the Soviet administrative apparatus played in this. The author comes to conclusion that under the conditions of strict authoritarian leadership, with the directives of V.M. Molotov, N.A. Bulganin and A.Ya. Vyshinsky, and possibly Joseph Stalin, the managers had a sufficient set of bureaucratic methods that allowed delaying the process of preparing the match up to a favourable occasion which led to the final breakdown in the negotiations. Such methods include precaution, prolonging pauses in interdepartmental communication, requesting for “instructions”, recalculating estimates, using rumours as arguments, using erroneous addresses and redirecting correspondence. The reason for the officials’ inactivity was the fear of personal responsibility for the defeat of the Soviet champion by the “expatriate defector”, especially in the situation when some leaders of the USSR chess movement were repressed. The author’s analysis provides insight into the problems of the functioning of the executive power in the conditions of the political regime established in the USSR by the beginning of the Second World War
Cobordism Obstructions to Complex Sections
No full textThere is a classical problem to determine whether a manifold admits linearly independent tangent vector fields. The first results, due to Poincare and Hopf, show that an oriented manifold admits an everywhere non-zero vector field if and only if its Euler characteristic is zero. Thomas, Mayer, Atiyah and Dupont did further work showing the existence of obstructions whose vanishing was a necessary condition for a manifold to admit vector fields. To solve this problem up to cobordism, B\"okstedt and Svane defined and studied a notion of vector field cobordism for oriented manifolds and a corresponding cobordism category.
The main goal of this thesis is to study the complex version of this problem, namely finding linearly independent complex tangent sections of almost complex manifolds. We define the complex section cobordism groups and the related cobordism categories. We identify an obstruction to finding a manifold in the same complex cobordism class as a given manifold with complex sections. This obstruction is an element of a relevant bordism group. The vanishing of this obstruction is both necessary and sufficient to show a cobordism class contains a manifold which can be equipped with linearly independent complex sections. Up to torsion, we completely describe this obstruction in terms of the Chern characteristic numbers. Further, calculations with the Adams-Novikov spectral sequence for particular Thom spectra allow us to show the torsion in the obstruction vanishes for low values of . For prime , we show that torsion obstructions of order for finding complex sections vanish for and that all torsion obstructions for finding or linearly independent complex sections vanish. Finally, we show that this obstruction vanishes for certain multiplicative generators in the complex cobordism ring
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