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Ekaterina Pervova

Publications and source records attributed to Ekaterina Pervova.

At least 19 recordsLinked to original sources
arXiv2025

Differential forms on diffeological spaces and diffeological gluing, I

This paper aims to describe the behavior of diffeological differential forms under the operation of gluing of diffeological spaces along a smooth map. In the diffeological context, two ways of looking at diffeological forms are available, that of the vector space of all diffeological forms on a given space, and that of the pseudo-bundle of values of these forms. We describe the behavior of the former under a gluing of diffeological spaces.

math.DG↗
arXiv2020

Linear and multilinear algebra on diffeological vector spaces

This paper deals with some basic constructions of linear and multilinear algebra on finite-dimensional diffeological vector spaces. We consider the diffeological dual formally checking that the assignment to each space of its dual defines a covariant functor from the category of finite-dimensional diffeological vector spaces to the category of standard (that is, carrying the usual smooth structure) vector spaces. We verify that the diffeological tensor product enjoys the typical properties of the usual tensor product, after which we focus on the so-called smooth direct sum decompositions, a phenomenon exclusive to the diffeological setting. We then consider the so-called pseudo-metrics (diffeological analogues of scalar products), and discuss the consequent decompositions of vector spaces into a smooth direct sum of the maximal isotropic subspace and a characteristic subspace; we show that such a decomposition actually depends only on the choice of the coordinate system. Furthermore, we show that, contrary to what was erroneously claimed (by me) elsewhere, a characteristic subspace is not unique and is not invariant under the diffeomorphisms of the space on itself. The maximal isotropic subspace is on the other hand an invariant of the space itself, and this fact allows to assign to each its well-defined characteristic quotient, obtaining another functor, this time a contravariant one, to the category of standard vector spaces. After discussing the diffeological analogues of isometries, we end with some remarks concerning diffeological algebras and diffeological Clifford algebras. Perhaps a larger than usual part of the paper recalls statements that already appear elsewhere, but when this is the case, we try to accompany them with new proofs and examples.

math.DG↗
arXiv2017

On the notion of scalar product for finite-dimensional diffeological vector spaces

It is known that the only finite-dimensional diffeological vector space that admits a diffeologically smooth scalar product is the standard space of appropriate dimension. In this note we consider a way to circumnavigate this issue, by introducing a notion of pseudo-metric, which, said informally, is the least-degenerate symmetric bilinear form on a given space. We apply this notion to make some observation on subspaces which split off as smooth direct summands (providing examples which illustrate that not all subspaces do), and then to show that the diffeological dual of a finite-dimensional diffeological vector space always has the standard diffeology and in particular, any pseudo-metric on the initial space induces, in the obvious way, a smooth scalar product on the dual.

math.DG↗
arXiv2017

Diffeological De Rham operators

We consider the notion of the De Rham operator on finite-dimensional diffeological spaces such that the diffeological counterpart Λ^1(X) of the cotangent bundle, the so-called pseudo-bundle of values of differential 1-forms, has bounded dimension. The operator is defined as the composition of the Levi-Civita connection on the exterior algebra pseudo-bundle \bigwedge(Λ^1(X)) with the standardly defined Clifford action by Λ^1(X); the latter is therefore assumed to admit a pseudo-metric for which there exists a Levi-Civita connection. Under these assumptions, the definition is fully analogous the standard case, and our main conclusion is that this is the only way to define the De Rham operator on a diffeological space, since we show that there is not a straightforward counterpart of the definition of the De Rham operator as the sum d+d^* of the exterior differential with its adjoint. We show along the way that other connected notions do not have full counterparts, in terms of the function they are supposed to fulfill, either; this regards, for instance, volume forms, the Hodge star, and the distinction between the $k$-th exterior degree of Λ^1(X) and the pseudo-bundle of differential k-forms Λ^k(X).

math.DG↗
arXiv2017

Pseudo-bundles of exterior algebras as diffeological Clifford modules

We consider the diffeological pseudo-bundles of exterior algebras, and the Clifford action of the corresponding Clifford algebras, associated to a given finite-dimensional and locally trivial diffeological vector pseudo-bundle, as well as the behavior of the former three constructions (exterior algebra, Clifford action, Clifford algebra) under the diffeological gluing of pseudo-bundles. Despite these being our main object of interest, we dedicate significant attention to the issues of compatibility of pseudo-metrics, and the gluing-dual commutativity condition, that is, the condition ensuring that the dual of the result of gluing together two pseudo-bundles can equivalently be obtained by gluing together their duals (this is not automatic in the diffeological context). We show that, assuming that the dual of the gluing map, which itself does not have to be a diffeomorphism, on the total space is one, the commutativity condition is satisfied, via a very natural map, which in addition turns out to be an isometry for the natural pseudo-metrics on the pseudo-bundles involved.

math.DG↗
arXiv2017

Diffeological connections on diffeological vector pseudo-bundles

We consider one possible definition of a diffeological connection on a diffeological vector pseudo-bundle. It is different from the one proposed in [7] and is in fact simpler, since it is obtained by a straightforward adaption of the standard definition of a connection as an operator on the space of all smooth sections. One aspect prominent in the diffeological context has to do with the choice of an appropriate substitute for tangent vectors and smooth vector fields, since there are not yet standard counterparts for these notions. In this respect we opt for the simplest possibility; since there is an established notion of the (pseudo-)bundle of differential forms on a diffeological space, we take the corresponding dual pseudo-bundle to play the role of the tangent bundle. Smooth vector fields are then smooth sections of this dual pseudo-bundle; this is one reason why we devote a particular attention to the space of smooth sections of an arbitrary diffeological vector pseudo-bundle (one curiosity is that it might easily turn out to be infinite-dimensional, even when the pseudo-bundle itself has a trivial finite-dimensional vector bundle as the underlying map). We concentrate a lot on how this space interacts with the gluing construction for diffeological vector pseudo-bundles (described in [10]). We then deal with the same question for the proposed notion of a diffeological connection.

math.DG↗
arXiv2017

Diffeological Dirac operators and diffeological gluing

This manuscript attempts to present a way in which the classical construction of the Dirac operator can be carried over to the setting of diffeology. A more specific aim is to describe a procedure for gluing together two usual Dirac operators and to explain in what sense the result is again a Dirac operator. Since versions of cut-and-paste (surgery) operations have already appeared in the context of Atiyah-Singer theory, we specify that our gluing procedure is designed to lead to spaces that are not smooth manifolds in any ordinary sense, and since much attention has been paid in recent years to Dirac operators on spaces with singularities, we also specify that our approach is more of a piecewise-linear nature (although, hopefully, singular spaces in a more analytic sense will enter the picture sooner or later; but this work is not yet about them). Most of it is devoted to the diffeological versions of the components that go into the standard definition of a Dirac operator as the composition of a Clifford connection with Clifford action by sections of the cotangent bundle; a diffeological Dirac operator is then standardly defined.

math.DG↗
arXiv2017

Diffeological Levi-Civita connections

A diffeological connection on a diffeological vector pseudo-bundle is defined just the usual one on a smooth vector bundle; this is possible to do, because there is a standard diffeological counterpart of the cotangent bundle. On the other hand, there is not yet a standard theory of tangent bundles, although there are many suggested and promising versions, such as that of the internal tangent bundle, so the abstract notion of a connection on a diffeological vector pseudo-bundle does not automatically provide a counterpart notion for Levi-Civita connections. In this paper we consider the dual of the just-mentioned counterpart of the cotangent bundle in place of the tangent bundle (without making any claim about its geometrical meaning). To it, the notions of compatibility with a pseudo-metric and symmetricity can be easily extended, and therefore the notion of a Levi-Civita connection makes sense as well. In the case when $Λ^1(X)$, the counterpart of the cotangent bundle, is finite-dimensional, there is an equivalent Levi-Civita connection on it as well.

math.DG↗
arXiv2016

Diffeological gluing of vector pseudo-bundles and pseudo-metrics on them

Although our main interest here is developing an appropriate analog, for diffeological vector pseudo-bundles, of a Riemannian metric, a significant portion is dedicated to continued study of the gluing operation for pseudo-bundles introduced in arXiv:1509.03023. We give more details regarding the behavior of this operation with respect to gluing, also providing some details omitted from arXiv:1509.03023, and pay more attention to the relations with the spaces of smooth maps. We also show that a usual smooth vector bundle over a manifold that admits a finite atlas can be seen as a result of a diffeological gluing, and thus deduce that its usual dual bundle is the same as its diffeological dual. We then consider the notion of a pseudo-metric, the fact that it does not always exist (which seems to be related to non-local-triviality condition), construction of an induced pseudo-metric on a pseudo-bundle obtained by gluing, and finally, the relation between the spaces of all pseudo-metrics on the factors of a gluing, and on its result. We conclude by commenting on the induced pseudo-metric on the pseudo-bundle dual to the given one.

math.DG↗
arXiv2016

Diffeological Clifford algebras and pseudo-bundles of Clifford modules

We consider the diffeological version of the Clifford algebra of a (diffeological) finite-dimensional vector space; we start by commenting on the notion of a diffeological algebra (which is the expected analogue of the usual one) and that of a diffeological module (also an expected counterpart of the usual notion). After considering the natural diffeology of the Clifford algebra, and its expected properties, we turn to our main interest, which is constructing pseudo-bundles of diffeological Clifford algebras and those of diffeological Clifford modules, by means of the procedure called diffeological gluing. The paper has a significant expository portion, regarding mostly diffeological algebras and diffeological vector pseudo-bundles.

math.DG↗
arXiv2016

Groups of tree automorphisms as diffeological groups

We consider certain groups of tree automorphisms as so-called diffeological groups. The notion of diffeology, due to Souriau, allows to endow non-manifold topological spaces, such as regular trees that we look at, with a kind of a differentiable structure that in many ways is close to that of a smooth manifold; a suitable notion of a diffeological group follows. We first study the question of what kind of a diffeological structure is the most natural to put on a regular tree in a way that the underlying topology be the standard one of the tree. We then proceed to consider the group of all automorphisms of the tree as a diffeological space, with respect to the functional diffeology, showing that this diffeology is actually the discrete one, the fact that therefore is true for its subgroups as well.

math.DG↗
arXiv2015

Diffeological vector pseudo-bundles

We consider a diffeological counterpart of the notion of a vector bundle (we call this counterpart a pseudo-bundle, although in the other works it is called differently; among the existing terms there are a "regular vector bundle" of Vincent and "diffeological vector space over X" of Christensen-Wu). The main difference of the diffeological version is that (for reasons stemming from the independent appearance of this concept elsewhere), diffeological vector pseudo-bundles may easily not be locally trivial (and we provide various examples of such, including those where the underlying topological bundle is even trivial). Since this precludes using local trivializations to carry out many typical constructions done with vector bundles (but not the existence of constructions themselves), we consider the notion of diffeological gluing of pseudo-bundles, which, albeit with various limitations that we indicate, provides when applicable a substitute for said local trivializations. We quickly discuss the interactions between the operation of gluing and typical operations on vector bundles (direct sum, tensor product, taking duals) and then consider the notion of a pseudo-metric on a diffeological vector pseudo-bundle.

math.DG↗
arXiv2011

Notes on the complexity of 3-valent graphs in 3-manifolds

A theory of complexity for pairs (M,G) with M an arbitrary closed 3-manifold and G a 3-valent graph in M was introduced by the first two named authors, extending the original notion due to Matveev. The complexity c is known to be always additive under connected sum away from the graphs, but not always under connected sum along (unknotted) arcs of the graphs. In this article we prove the slightly surprising fact that if in M there is a sphere intersecting G transversely at one point, and this point belongs to an edge e of G, then e can be canceled from G without affecting the complexity. Using this fact we completely characterize the circumstances under which complexity is additive under connected sum along graphs. For the set of pairs (M,K) with K a knot in M, we also prove that any function that is fully additive under connected sum along knots is actually a function of the ambient manifold only.

math.GT↗
arXiv2011

Generalized Mom-structures and ideal triangulations of 3-manifolds with non-spherical boundary

The so-called Mom-structures on hyperbolic cusped 3-manifolds without boundary were introduced by Gabai, Meyerhoff, and Milley, and used by them to identify the smallest closed hyperbolic manifold. In this work we extend the notion of a Mom-structure to include the case of 3-manifolds with non-empty boundary that does not have spherical components. We then describe a certain relation between such generalized Mom-structures, called protoMom-structures, internal on a fixed 3-manifold N, and ideal triangulations of N; in addition, in the case of non-closed hyperbolic manifolds without annular cusps, we describe how an internal geometric protoMom-structure can be constructed starting from Epstein-Penner or Kojima decomposition. Finally, we exhibit a set of combinatorial moves that relate any two internal protoMom-structures on a fixed N to each other.

math.GT↗
arXiv2009

Lower bounds on complexity of geometric 3-orbifolds

We establish a lower bound on the complexity orientable locally orientable geometric 3-orbifolds in terms of Delzant's T-invariants of their orbifold-fundamental groups, generalizing previously known bounds for complexity of 3-manifolds.

math.GT↗
arXiv2009

On the existence of branched coverings between surfaces with prescribed branch data, I

For the existence of a branched covering Sigma~ --> Sigma between closed surfaces there are easy necessary conditions in terms of chi(Sigma~), chi(Sigma), orientability, the total degree, and the local degrees at the branching points. A classical problem dating back to Hurwitz asks whether these conditions are also sufficient. Thanks to the work of many authors, the problem remains open only when Sigma is the sphere, in which case exceptions to existence are known to occur. In this paper we describe new infinite series of exceptions, in particular previously unknown exceptions with Sigma~ not the sphere and with more than three branching points. All our series come with systematic explanations, based on several different techniques (including dessins d'enfants and decomposability) that we exploit to attack the problem, besides Hurwitz's classical technique based on permutations. Using decomposability we also establish an easy existence result.

math.GT↗
arXiv2008

Complexity of links in 3-manifolds

We introduce a natural-valued complexity c(X) for pairs X=(M,L), where M is a closed orientable 3-manifold and L is a link contained in M. The definition employs simple spines, but for well-behaved X's we show that c(X) equals the minimal number of tetrahedra in a triangulation of M containing L in its 1-skeleton. Slightly adapting Matveev's recent theory of roots for graphs, we carefully analyze the behaviour of c under connected sum away from and along the link. We show in particular that c is almost always additive, describing in detail the circumstances under which it is not. To do so we introduce a certain (0,2)-root for a pair X, we show that it is well-defined, and we prove that X has the same complexity as its (0,2)-root. We then consider, for links in the 3-sphere, the relations of c with the crossing number and with the hyperbolic volume of the exterior, establishing various upper and lower bounds. We also specialize our analysis to certain infinite families of links, providing rather accurate asymptotic estimates.

math.GT↗