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Rustam Sadykov

Publications and source records attributed to Rustam Sadykov.

27 records · Page 2Linked to original sources
arXiv2010

Fold maps, framed immersions and smooth structures

For each integer q>0 there is a cohomology theory such that the zero cohomology group of a manifold N of dimension n is a certain group of cobordism classes of proper fold maps of manifolds of dimension n+q into N. We prove a splitting theorem for the spectrum representing the cohomology theory of fold maps. For even q, the splitting theorem implies that the cobordism group of fold maps to a manifold N is a sum of q/2 cobordism groups of framed immersions to N and a group related to diffeomorphism groups of manifolds of dimension q+1. Similarly, in the case of odd q, the cobordism group of fold maps splits off (q-1)/2 cobordism groups of framed immersions. The proof of the splitting theorem gives a partial splitting of the homotopy cofiber sequence of Thom spectra in the Madsen-Weiss approach to diffeomorphism groups of manifolds.

math.GT↗
arXiv2010

Topological properties of manifolds admitting a $Y^x$-Riemannian metric

A complete Riemannian manifold $(M, g)$ is a $Y^x_l$-manifold if every unit speed geodesic $γ(t)$ originating at $γ(0)=x\in M$ satisfies $γ(l)=x$ for $0\neq l\in \R$. Bérard-Bergery proved that if $(M^m,g), m>1$ is a $Y^x_l$-manifold, then $M$ is a closed manifold with finite fundamental group, and the cohomology ring $H^*(M, \Q)$ is generated by one element. We say that $(M,g)$ is a $Y^x$-manifold if for every $ε>0$ there exists $l>ε$ such that for every unit speed geodesic $γ(t)$ originating at $x$, the point $γ(l)$ is $ε$-close to $x$. We use Low's notion of refocussing Lorentzian space-times to show that if $(M^m, g), m>1$ is a $Y^x$-manifold, then $M$ is a closed manifold with finite fundamental group. As a corollary we get that a Riemannian covering of a $Y^x$-manifold is a $Y^x$-manifold. Another corollary is that if $(M^m,g), m=2,3$ is a $Y^x$-manifold, then $(M, h)$ is a $Y^x_l$-manifold for some metric $h.$

math.DG↗
arXiv2010

Invariants of singular sets of smooth maps

A singular point of a smooth map F: M -> N of manifolds is a point in M at which the rank of the differential dF is less than the minimum of dimensions of M and N. The classical invariant of the set S of singular points of F of a given type is defined by taking the fundamental class [\bar{S}]\in H_*(M) of the closure of S. We introduce and study new invariants of singular sets for which the classical invariants may not be defined, i.e., for which \bar{S} may not possess the fundamental class. The simplest new invariant is defined by carefully choosing the fundamental class of the intersection of \bar{S} and its slight perturbation in M. Surprisingly, for certain singularity types such an invariant is well-define (and not trivial) despite the fact that \bar{S} does not possess the fundamental class. We determine new invariants for maps with Morin singularities---i.e., singularities of types A_k for k>0 in the ADE-classification of simple singularities by Dynkin diagrams---and, as an application, show that these invariants together with generalized Miller-Morita-Mumford classes form a commutative graded algebra of characteristic classes that completely determine the cobordism groups of maps with at most A_k-singularities for each k>0.

math.GT↗
arXiv2010

Obstructions to the existence of fold maps

We study smooth maps between smooth manifolds with only fold points as their singularities, and clarify the obstructions to the existence of such a map in a given homotopy class for certain dimensions. The obstructions are described in terms of characteristic classes, which arise as Postnikov invariants, and can be interpreted as primary and secondary obstructions to the elimination of certain singularities. We also discuss the relationship between the existence problem of fold maps and that of vector fields of stabilized tangent bundles.

math.AT↗
arXiv2010

Singular cobordism categories

Recently Galatius, Madsen, Tillmann and Weiss identified the homotopy type of the classifying space of the cobordism category of embedded d-dimensional manifolds [7] for each positive integer d. Their result lead to a new proof of the generalized standard Mumford conjecture. We extend the main theorem of [7] to the case of cobordism categories of embedded d-dimensional manifolds with prescribed singularities, and explain the relation of singular cobordism categories to the bordism version of the Gromov h-principle.

math.AT↗
arXiv2009

Bordism groups of solutions to differential relations

In terms of category theory, the Gromov homotopy principle for a set valued functor $F$ asserts that the functor $F$ can be induced from a homotopy functor. Similarly, we say that the bordism principle for an abelian group valued functor $F$ holds if the functor $F$ can be induced from a (co)homology functor. We examine the bordism principle in the case of functors given by (co)bordism groups of maps with prescribed singularities. Our main result implies that if a family $R$ of prescribed singularity types satisfies certain mild conditions, then there exists an infinite loop space $B(R)$ such that for each smooth manifold $N$ the cobordism group of maps into $N$ with only $R$-singularities is isomorphic to the group of homotopy classes of maps $[N, B(R)]$.

math.AT↗
arXiv2009

Stable characteristic classes of smooth manifold bundles

Characteristic classes of oriented vector bundles can be identified with cohomology classes of the disjoint union of classifying spaces BSO_n of special orthogonal groups SO_n with n=0,1,... A characteristic class is stable if it extends to a cohomology class of a homotopy colimit BSO of classifying spaces BSO_n. Similarly, characteristic classes of smooth oriented manifold bundles with fibers given by oriented closed smooth manifolds of a fixed dimension d\ge 0 can be identified with cohomology classes of the disjoint union of classifying spaces BDiff M of orientation preserving diffeomorphism groups of oriented closed manifolds of dimension d. A characteristic class is stable if it extends to a cohomology class of a homotopy colimit of spaces BDiff M. We show that each rational stable characteristic class of oriented manifold bundles of even dimension d is tautological, e.g., if d=2, then each rational stable characteristic class is a polynomial in terms of Miller-Morita-Mumford classes.

math.GT↗
arXiv2003

The Chess conjecture

We prove that the homotopy class of a Morin mapping f: P^p --> Q^q with p-q odd contains a cusp mapping. This affirmatively solves a strengthened version of the Chess conjecture [DS Chess, A note on the classes [S_1^k(f)], Proc. Symp. Pure Math., 40 (1983) 221-224] and [VI Arnol'd, VA Vasil'ev, VV Goryunov, OV Lyashenko, Dynamical systems VI. Singularities, local and global theory, Encyclopedia of Mathematical Sciences - Vol. 6 (Springer, Berlin, 1993)]. Also, in view of the Saeki-Sakuma theorem [O Saeki, K Sakuma, Maps with only Morin singularities and the Hopf invariant one problem, Math. Proc. Camb. Phil. Soc. 124 (1998) 501-511] on the Hopf invariant one problem and Morin mappings, this implies that a manifold P^p with odd Euler characteristic does not admit Morin mappings into R^{2k+1} for p > 2k not equal to 1,3 or 7.

math.GT↗