Recent zbMATH articles in MSC 57Rhttps://www.zbmath.org/atom/cc/57R2021-04-16T16:22:00+00:00WerkzeugTopology change of level sets in Morse theory.https://www.zbmath.org/1456.370612021-04-16T16:22:00+00:00"Knauf, Andreas"https://www.zbmath.org/authors/?q=ai:knauf.andreas"Martynchuk, Nikolay"https://www.zbmath.org/authors/?q=ai:martynchuk.nikolayThis paper considers Morse functions \(f \in C^2(M, \mathbb{R})\) on a smooth \(m\)-dimensional manifold without boundary. For such functions and for every critical point \(x \in M\) of \(f\), the Hessian of \(f\) at \(x\) is nondegenerate.
The authors are interested in the topology of the level sets \(f^{-1}(a) = \partial{M^a}\) where \(M^a = \{ x \in M : f(x) \le a\}\) is a so-called sublevel set. The authors address the question of under what conditions the topology of \(\partial{M^a}\) can change when the function \(f\) passes a critical level with one or more critical points. ``Change in topology'' here means roughly that if \(a\) and \(b\) are regular values with \(a < b\), then \(H_k(\partial{M^a};G) \ne H_k(\partial{M^b};G)\) where \(k\) is an integer and the \(H_k\) are homology groups over some abelian group \(G\) when the function \(f\) passes through a level with one or more critical points. A motivation for this question derives from the \(n\)-body problem: does the topology of the integral manifolds always change when passing through a bifurcation level?
The first part of the paper considers level sets of abstract Morse functions that satisfy the Palais-Smale condition and whose level sets have finitely generated homology groups. They show that for such functions the topology of \(f^{-1}(a)\) changes when passing a single critical point if the index of the critical point is different from \(m/2\) where \(m\) is the dimension of the manifold \(M\). Then they move on to consider the case where \(M\) is a vector bundle of rank \(n\) over a manifold \(N\) of dimension \(n\), and where (up to a translation) \(f\) is a Morse function that is the sum of a positive definite quadratic form on the fibers and a potential function that is constant on the fibers. Here the authors have in mind Hamiltonian functions \(H\) on the cotangent bundle of the base manifold \(N\). In this case they show that the topology of \(H^{-1}(h)\) always changes when passing a single critical point if the Euler characteristic of \(N\) is not \(\pm 1\).
Finally the authors apply their results to examples from Hamiltonian and celestial mechanics, with emphasis on the planar three-body problem. There they show that the topology always changes for the planar three-body problem provided that the reduced Hamiltonian is a Morse function with at most two critical points on each level set.
Reviewer: William J. Satzer Jr. (St. Paul)A singular radial connection over \(\mathbb B^5\) minimizing the Yang-Mills energy.https://www.zbmath.org/1456.580152021-04-16T16:22:00+00:00"Petrache, Mircea"https://www.zbmath.org/authors/?q=ai:petrache.mirceaSummary: We prove that the pullback of the \(\mathrm{SU}(n)\)-soliton of Chern number \(c_2=1\) over \(\mathbb S^4\) via the radial projection \(\pi :\mathbb B^5{\setminus }\{0\}\to \mathbb S^4\) minimizes the Yang-Mills energy under a topologically fixed boundary trace constraint. In particular this shows that stationary Yang-Mills connections in high dimension can have singular sets of codimension 5.Homotopy properties of smooth functions on the Möbius band.https://www.zbmath.org/1456.570272021-04-16T16:22:00+00:00"Kuznietsova, Iryna"https://www.zbmath.org/authors/?q=ai:kuznietsova.iryna"Maksymenko, Sergiy"https://www.zbmath.org/authors/?q=ai:maksymenko.sergii-ivanovychLet \(B\) be a Möbius band and \(f:B\rightarrow \mathbb{R}\) be a Morse map taking a constant value on the boundary \(\partial B\), and \(S(f;\partial B)\) be the group of diffeomorphisms \(h\) of \(B\) fixed on \(\partial B\) and preserving \(f\) in the sense that \(f\circ h=f\). In this interesting paper the authors compute, under certain assumptions on \(f\), the group \(\pi_0S(f;\partial B)\) of isotopy classes of such diffeomorphisms (Theorem 1.5).
The paper is organized into eight sections dealing with the following aspects: description of the main result, Kronrod-Reeb graph, proof of Theorem 1.5, diffeomorphisms of non-orientable manifolds, Hamiltonian like flows for \(g \in \mathcal{F}(M,P)\), group \(\Delta (f)\), functions on the annulus, completing the proof of Theorem 1.5.
Other papers by the second author directly connected to this topic are [\textit{S. Maksymenko}, Ann. Global Anal. Geom. 29, No. 3, 241--285 (2006; Zbl 1099.37013); Methods Funct. Anal. Topol. 16, No. 2, 167--182 (2010; Zbl 1224.57017); Zb. Pr. Inst. Mat. NAN Ukr. 7, No. 4, 7--66 (2010; Zbl 1240.57019); Ukr. Math. J. 64, No. 9, 1350--1369 (2013; Zbl 1272.58008); translation from Ukr. Mat. Zh. 64, No. 9, 1186--1203 (2012)]; \textit{S. I. Maksymenko} and \textit{B. G. Feshchenko}, ibid. 66, No. 9, 1346--1353 (2015; Zbl 1354.57036); translation from Ukr. Mat. Zh. 66, No. 9, 1205--1212 (2014); Mat. Stud. 44, No. 1, 67--83 (2015; Zbl 1376.37048); Methods Funct. Anal. Topol. 21, No. 1, 22--40 (2015; Zbl 1340.57009)].
Reviewer: Dorin Andrica (Riyadh)Three applications of delooping to \(h\)-principles.https://www.zbmath.org/1456.580092021-04-16T16:22:00+00:00"Kupers, Alexander"https://www.zbmath.org/authors/?q=ai:kupers.alexanderIn his book [Partial differential relations. Berlin etc.: Springer (1986; Zbl 0651.53001)] \textit{M. Gromov} formulated a very general \(h\)-principle for invariant topological sheaves of smooth functions on manifolds, by which many geometric problems can be reduced to more tractable homotopy-theoretic problems.
In the present paper, the author applies the machinery of Gromov to give general conditions under which
an \(h\)-principle holds on closed manifolds, and check that these conditions are satisfied in three examples: (i) a homotopical version of Vassiliev's \(h\)-principle, (ii) the contractibility of the space of framed functions, (iii) a version of Mather-Thurston theory.
Reviewer: Vagn Lundsgaard Hansen (Lyngby)Energy gap for Yang-Mills connections. I: Four-dimensional closed Riemannian manifolds.https://www.zbmath.org/1456.580142021-04-16T16:22:00+00:00"Feehan, Paul M. N."https://www.zbmath.org/authors/?q=ai:feehan.paul-m-nSummary: We extend an \(L^2\) energy gap result due to \textit{M. Min-Oo} [Compos. Math. 47, 153--163 (1982; Zbl 0519.53042), Theorem 2] and \textit{T. H. Parker} [Commun. Math. Phys. 85, 563--602 (1982; Zbl 0502.53022), Proposition 2.2] for Yang-Mills connections on principal \(G\)-bundles, \(P\), over closed, connected, four-dimensional, oriented, smooth manifolds, \(X\), from the case of positive Riemannian metrics to the more general case of good Riemannian metrics, including metrics that are generic and where the topologies of \(P\) and \(X\) obey certain mild conditions and the compact Lie group, \(G\), is \(\operatorname{SU}(2)\) or \(\operatorname{SO}(3)\).
[For Part II see Zbl 1375.58013.]The group of symplectic birational maps of the plane and the dynamics of a family of 4D maps.https://www.zbmath.org/1456.530662021-04-16T16:22:00+00:00"Cruz, Inês"https://www.zbmath.org/authors/?q=ai:cruz.ines"Mena-Matos, Helena"https://www.zbmath.org/authors/?q=ai:mena-matos.helena"Sousa-Dias, Esmeralda"https://www.zbmath.org/authors/?q=ai:sousa-dias.esmeraldaSummary: We consider a family of birational maps \(\varphi_k\) in dimension 4, arising in the context of cluster algebras from a mutation-periodic quiver of period 2. We approach the dynamics of the family \(\varphi_k\) using Poisson geometry tools, namely the properties of the restrictions of the maps \(\varphi_k\) and their fourth iterate \(\varphi^{(4)}_k\) to the symplectic leaves of an appropriate Poisson manifold \((\mathbb{R}^4_+, P)\). These restricted maps are shown to belong to a group of symplectic birational maps of the plane which is isomorphic to the semidirect product \(SL(2, \mathbb{Z})\ltimes\mathbb{R}^2\). The study of these restricted maps leads to the conclusion that there are three different types of dynamical behaviour for \(\varphi_k\) characterized by the parameter values \(k = 1\), \(k = 2\) and \(k\geq 3\).On trivialities of Euler classes of oriented vector bundles over manifolds.https://www.zbmath.org/1456.570252021-04-16T16:22:00+00:00"Naolekar, Aniruddha C."https://www.zbmath.org/authors/?q=ai:naolekar.aniruddha-c"Subhash, B."https://www.zbmath.org/authors/?q=ai:subhash.bhaskaran"Thakur, Ajay Singh"https://www.zbmath.org/authors/?q=ai:thakur.ajay-singhLet \(\mathcal{E}\) be the set of diffeomorphism classes of closed connected smooth manifolds \(X \) such that for every oriented vector bundle \(\alpha\) over \(X\), the Euler class \(e(\alpha) = 0\), and let \(\mathcal{E}_k\) be the subset of \(\mathcal{E}\) of manifolds with dimension \(k\). In the recent literature there are results about the same subject by considering oriented bundles over \(CW\)-complexes.
In this very interesting paper, the authors give a complete description for \(k \leq 5\) and give partial results for \(k= 6\). Let us denote by \(X\) a closed connected smooth \(k\)-manifold. Then one has the following results:
\begin{itemize}
\item \(X \in \mathcal{E}_3\) if and only if \(X\) is a smooth homology \(3\)-sphere.
\item \( \mathcal{E}_4 = \emptyset\).
\item \(X \in \mathcal{E}_5\) if and only if \(H^2(X;\mathbb{Z}) = 0; \, H^4(X;\mathbb{Z} ) = 0\), and the \(2\)-primary component of \(H_2(X; \mathbb{Z})\) is trivial
\end{itemize}
There exists an oriented vector bundle \(\alpha\) over the projective space \(\mathbb{RP}^5\) or over the lens spaces of dimension \( 5\) such that \( e(\alpha) \neq 0\). Then the authors construct examples of \(X \in \mathcal{E}_5 \) in the case where \(X\) is a simply connected closed smooth \(5\)-manifold, for \(X\) a smooth closed non-orientable manifold and also for \(X\) a non-simply connected closed smooth oriented \(5\)-manifold.
The authors also observe that any \(X \in \mathcal{E}_6 \) must have non-positive Euler characteristic.
Reviewer: Alice Kimie Miwa Libardi (São Paulo)The KW equations and the Nahm pole boundary condition with knots.https://www.zbmath.org/1456.813112021-04-16T16:22:00+00:00"Mazzeo, Rafe"https://www.zbmath.org/authors/?q=ai:mazzeo.rafe-r"Witten, Edward"https://www.zbmath.org/authors/?q=ai:witten.edwardIn this detailed technical paper the authors extend further their previous analysis of the Kapustin-Witten (KW) equations with Nahm pole boundary condition now adapted to general 4-manifolds-with-boundary such that the boundary-3-manifold contains a knot or more generally a link.
The motivation is a conjecture of the second author that the coefficients of the Laurent expansion of the Jones polynomial of a link \(L\subset {\mathbb R}^3\) arise by counting solutions of the KW equations on the half-space \({\mathbb R}^4_+\) obeying a generalized Nahm pole boundary condition on \(\partial {\mathbb R}^4_+={\mathbb R}^3\supset L\) i.e. the Nahm pole boundary condition generalized to be compatible with the extra information of containing a link on the boundary. Roughly this means to prescribe further singularities in the Higgs field part of the KW pair along each link component while the connection part is continuous up to the boundary as before. The conjecture is important because it is well-known that computing the Jones polynomial of a link is an exponentially difficult problem in terms of e.g. the crossing number of any plane diagram of the link.
Reviewer: Gabor Etesi (Budapest)Positively curved Killing foliations via deformations.https://www.zbmath.org/1456.530232021-04-16T16:22:00+00:00"Caramello, Francisco C. jun.."https://www.zbmath.org/authors/?q=ai:caramello.francisco-c-jun"Töben, Dirk"https://www.zbmath.org/authors/?q=ai:toben.dirkThis paper contain several interesting results on the (transverse) geometry and topology of compact manifolds with a Killing foliation with positive transverse sectional curvature. Recall that a Killing foliation is a complete Riemannian foliation with globally constant Molino sheaf; a class that includes Riemannian foliations on simply connected manifolds and foliations induced by isometric actions. A key step behind the main results is to deform the foliation into one with closed leaves while keeping transverse geometric properties, using a method of \textit{A. Haefliger} and \textit{E. Salem} [Ill. J. Math. 34, No. 4, 706--730 (1990; Zbl 0701.53053)]. One may then apply an orbifold version of the maximal symmetry-rank classification in positive curvature to show that the manifold fibers over finite quotients of spheres or weighted complex projective spaces if the closure of the original foliation has maximal dimension. Several other consequences involving the basic Euler characteristic are also established.
Reviewer: Renato G. Bettiol (New York)Conley theory for Gutierrez-Sotomayor fields.https://www.zbmath.org/1456.370242021-04-16T16:22:00+00:00"Montúfar, H."https://www.zbmath.org/authors/?q=ai:montufar.h"de Rezende, K. A."https://www.zbmath.org/authors/?q=ai:de-rezende.ketty-abaroaThe authors study, from a topological perspective, continuous flows associated to \(C^1\) structurally stable vector fields tangent to a two-dimensional compact subset \(M\) of \(\mathbb{R}^k\). They call these flows Gutierrez-Sotomayor (shortly denoted by GS) flows on manifolds \(M\) with simple singularities and they use Conley index theory to study them. The Conley indices of all simple singularities are computed and an Euler characteristic formula is obtained. By considering a stratification of \(M\) which decomposes it into a union of its regular and singular strata, certain Euler-type formulas which relate the topology of \(M\) and the dynamics on the strata are obtained. The existence of a Lyapunov function for GS flows without periodic orbits and singular cycles is established. Using long exact sequence analysis of index pairs the authors determine necessary and sufficient conditions for a GS flow to be defined on an isolating block. They organize this information combinatorially with the aid of Lyapunov graphs and using a Poincaré-Hopf equality to construct isolating blocks for all simple singularities.
The main results generalize results of the second author and \textit{R. D. Franzosa} [Trans. Am. Math. Soc. 340, No. 2, 767--784 (1993; Zbl 0806.58042)], where Morse-Smale flows and more generally continuous flows on smooth surfaces are classified.
Reviewer: Dorin Andrica (Riyadh)Chiral algebra, localization, modularity, surface defects, and all that.https://www.zbmath.org/1456.813682021-04-16T16:22:00+00:00"Dedushenko, Mykola"https://www.zbmath.org/authors/?q=ai:dedushenko.mykola"Fluder, Martin"https://www.zbmath.org/authors/?q=ai:fluder.martinThe authors study Lagrangian \(\mathcal{N} = 2\) superconformal field theories in four dimensions.
By employing supersymmetric localization on a rigid background of the form \(S^3 \times S^1_y\) they explicitly localize a given Lagrangian superconformal field theory and obtain the corresponding two-dimensional vertex operator algebra VOA (chiral algebra) on the torus \(S^1\times S^1_y\subset S^3\times S^1_y\). To derive the VOA the authors define the appropriate rigid supersymmetric \(S^3 \times S^1_y\) background reproducing the superconformal index. They analyze the supersymmetry algebra and classify the possible fugacities and their preserved subalgebras. Although the minimal amount of supersymmetry needed to retain the VOA construction is \(\mathfrak{su}(1|1)_\ell\times \mathfrak{su}(1|1)_r\) it appears that it is possible to turn on fugacities preserving an \(\mathfrak{su}(1|1)_\ell\times \mathfrak{su}(2|1)_r\) subalgebra which can be further broken to the minimal one by defects. Specifically, discrete fugacities \(M,N \in \mathbb{Z}\) can be turned on. The authors argue that these deformations do not affect the VOA construction but change the complex structure of the
torus and affect the boundary conditions (spin structure) upon going around one of the cycles, \(S^1_y\)
The authors address the two-dimensional theory corresponding to the localization of the \(\mathcal{N} = 2\) vector multiplets and hypermultiplets. In the latter case they show that the remnant classical piece in the localization precisely reduces to the two-dimensional symplectic boson theory on the boundary torus \(S^1\times S^1_y\). The authors show that in the presence of flavor holonomies, which appear as mass-like central charges in the supersymmetry algebra, vertex operators charged under the flavor symmetries fail to remain holomorphic while the sector that remains holomorphic is formed by flavor-neutral operators.
The authors study the modular properties of the four-dimensional Schur index. They introduce formal partition functions \(Z^{(\nu_1,\nu_2)}_{(m,n)}\), which are defined as the partition function in the given spin structure \((\nu_1,\nu_2)\), but with the modified contour of the holonomy integral in the localization formula, labeled by two integers \(m\) and \(n\). The authors suggest that the objects \(Z^{(\nu_1,\nu_2)}_{(m,n)}\) furnish an infinite-dimensional projective representation of \(\mathrm{SL}(2,\mathbb{Z})\).
Finally the authors comment on the flat \(\Omega\)-background underlying the chiral algebra.
Reviewer: Farhang Loran (Isfahan)Marao, about Hopf fibration.https://www.zbmath.org/1456.550082021-04-16T16:22:00+00:00"Berishvili, Guram"https://www.zbmath.org/authors/?q=ai:berishvili.guramSummary: A marao is a cover of a vector space by a set of equidimensional subspaces with pairwise trivial intersections. Such structures give rise to fibrations of particular kind. Naturally occurring examples are described. In particular, it is explained how the classical Hopf fibrations can be uniformly obtained from maraos.Virtual classes of parabolic \(\operatorname{SL}_2(\mathbb{C})\)-character varieties.https://www.zbmath.org/1456.140652021-04-16T16:22:00+00:00"González-Prieto, Ángel"https://www.zbmath.org/authors/?q=ai:gonzalez-prieto.angelLet \(\Sigma_g\) be the closed orientable surface of genus \(g\) and \(Q\) a parabolic structure on \(\Sigma_g\). In this paper, the author completes his study of the virtual classes of the \(\operatorname{SL}_2({\mathbb C})\)-character varieties of \((\Sigma_g,Q)\) by considering the case where there are parabolic points of semi-simple type. More precisely, let \({\mathfrak X}_{\operatorname{SL}_2({\mathbb C})}(\Sigma_g,Q)\) denote the representation variety of \((\Sigma_g,Q)\) and \({\mathcal R}_{\operatorname{SL}_2({\mathbb C})}(\Sigma_g,Q):={\mathfrak X}_{\operatorname{SL}_2({\mathbb C})}(\Sigma_g,Q)//\operatorname{SL}_2({\mathbb C})\) the corresponding character variety. Now let \(\operatorname{K\mathbf{Var}}_{\mathbb C}\) be the Grothendieck ring of complex algebraic varieties and \(\operatorname{\widetilde{K}\mathbf{Var}}_{\mathbb C}\) the localisation of this ring with respect to the multiplicative set generated by \(q\), \(q+1\) and \(q-1\), where \(q\) is the class of \({\mathbb C}\) in \(\operatorname{K\mathbf{Var}}_{\mathbb C}\). Then the author computes explicitly the virtual class of \({\mathfrak X}_{\operatorname{SL}_2({\mathbb C})}(\Sigma_g,Q)\) in \(\operatorname{\widetilde{K}\mathbf{Var}}_{\mathbb C}\) when there is at least one parabolic point with semi-simple holonomy and possibly some additional parabolic points with holonomy of Jordan type \(J_+\) (Theorem 5.6). From this, he deduces a formula for the virtual class of \({\mathcal R}_{\operatorname{SL}_2({\mathbb C})}(\Sigma_g,Q)\), valid for all holonomies (Theorem 6.1).
Character varieties have been much studied in recent years by both arithmetic and geometric methods. Both methods have limitations when there are parabolic points. In his thesis, the author developed a method involving TQFTs to avoid these limitations and used this method to compute the classes of \({\mathfrak X}_{\operatorname{SL}_2({\mathbb C})}(\Sigma_g,Q)\) and \({\mathcal R}_{\operatorname{SL}_2({\mathbb C})}(\Sigma_g,Q)\) in \(\operatorname{K\mathbf{MHS}}\) where the punctures are of Jordan type or type \(-\operatorname{Id}\). Here, \(\operatorname{K\mathbf{MHS}}\) is the Grothendieck ring of the category of mixed Hodge structures. (The relevant part of the author's et al. [Bull. Sci. Math. 161, Article ID 102871, 33 p. (2020; Zbl 1441.57031)]). However, there are new complications when parabolic points of semi-simple type are involved. In particular, the ``core submodule'' constructed by the author is no longer invariant under the TQFT. Moreover, if the punctures are non-generic, a new interaction phenomenon arises. These problems are addressed in the current paper.
In section 2, the author sketches the construction of the TQFT mentioned above together with a modification which allows computations in \(\operatorname{\widetilde{K}\mathbf{Var}}_{\mathbb C}\). The key section 3 is concerned with \(\operatorname{SL}_2({\mathbb C})\)-representation varieties and is preliminary to the computation of the geometric TQFT in section 4. The interaction phenomenon is described in section 5, culminating in Theorem 5.6. Section 6 is directed towards proving Theorem 6.1.
The author comments that there is much work still to be done in extending his results to groups other than \(\operatorname{SL}_2({\mathbb C})\) and to more general spaces, for example singular and non-orientable surfaces or knot complements.
Reviewer: P. E. Newstead (Liverpool)Brownian motion on foliated complex surfaces, Lyapunov exponents and applications.https://www.zbmath.org/1456.370012021-04-16T16:22:00+00:00"Deroin, Bertrand"https://www.zbmath.org/authors/?q=ai:deroin.bertrandFrom the text: These lectures are motivated by the dynamical study of differential
equations in the complex domain. Most of the topic will concern holomorphic foliations on complex surfaces, and their connections with the
theory of complex projective structures on curves. In foliation theory,
the interplay between geometry and dynamics is what makes the beauty
of the subject. In these lectures, we will try to develop this relationship
even more.
For the entire collection see [Zbl 1378.53005].Invariants and TQFT's for cut cellular surfaces from finite groups.https://www.zbmath.org/1456.570262021-04-16T16:22:00+00:00"Bragança, Diogo"https://www.zbmath.org/authors/?q=ai:braganca.diogo"Picken, Roger"https://www.zbmath.org/authors/?q=ai:picken.roger-fSummary: We introduce the notion of a cut cellular surface (CCS), being a surface with boundary, which is cut in a specified way to be represented in the plane, and is composed of 0-, 1- and 2-cells. We obtain invariants of CCS's under Pachner-like moves on the cellular structure, by counting colourings of the 1-cells with elements of a finite group, subject to a ``flatness'' condition for each 2-cell. These invariants are also described in a TQFT setting, which is not the same as the usual 2-dimensional TQFT framework. We study the properties of functions which arise in this context, associated to the disk, the cylinder and the pants surface, and derive general properties of these functions from topology, including properties which come from invariance under the Hatcher-Thurston moves on pants decompositions.The geometry of synchronization problems and learning group actions.https://www.zbmath.org/1456.051052021-04-16T16:22:00+00:00"Gao, Tingran"https://www.zbmath.org/authors/?q=ai:gao.tingran"Brodzki, Jacek"https://www.zbmath.org/authors/?q=ai:brodzki.jacek"Mukherjee, Sayan"https://www.zbmath.org/authors/?q=ai:mukherjee.sayanSummary: We develop a geometric framework, based on the classical theory of fibre bundles, to characterize the cohomological nature of a large class of synchronization-type problems in the context of graph inference and combinatorial optimization. We identify each synchronization problem in topological group \(G\) on connected graph \(\Gamma\) with a flat principal \(G\)-bundle over \(\Gamma\), thus establishing a classification result for synchronization problems using the representation variety of the fundamental group of \(\Gamma\) into \(G\). We then develop a twisted Hodge theory on flat vector bundles associated with these flat principal \(G\)-bundles, and provide a geometric realization of the graph connection Laplacian as the lowest-degree Hodge Laplacian in the twisted de Rham-Hodge cochain complex. Motivated by these geometric intuitions, we propose to study the problem of learning group actions -- partitioning a collection of objects based on the local synchronizability of pairwise correspondence relations -- and provide a heuristic synchronization-based algorithm for solving this type of problems. We demonstrate the efficacy of this algorithm on simulated and real datasets.Three dimensional Saito free divisors and singular curves.https://www.zbmath.org/1456.140082021-04-16T16:22:00+00:00"Sekiguchi, Jiro"https://www.zbmath.org/authors/?q=ai:sekiguchi.jiroSummary: The purpose of the present study is to find out examples of Saito free divisors by constructing Lie algebras generated by logarithmic vector fields along them. In the course of the study, the author recognized a deep connection between Saito free divisors and deformations of curve singularities. In this paper, we will explain a method of constructing three dimensional Saito free divisors and show some examples.Homology theory formulas for generalized Riemann-Hurwitz and generalized monoidal transformations.https://www.zbmath.org/1456.570222021-04-16T16:22:00+00:00"Glazebrook, James F."https://www.zbmath.org/authors/?q=ai:glazebrook.james-f"Verjovsky, Alberto"https://www.zbmath.org/authors/?q=ai:verjovsky.albertoAuthors' abstract: In the context of orientable circuits and subcomplexes of these as representing certain singular spaces, we consider characteristic class formulas generalizing those classical results as seen for the Riemann-Hurwitz formula for regulating the topology of branched covering maps and that for monoidal transformations which include the standard blowing-up process. Here the results are presented as cap product pairings, which will be elements of a suitable homology theory, rather than characteristic numbers as would be the case when taking Kronecker products once Poincaré duality is defined. We further consider possible applications and examples including branched covering maps, singular varieties involving virtual tangent bundles, the Chern-Schwartz-MacPherson class, the homology L-class, generalized signature, and the cohomology signature class.
Reviewer: Vagn Lundsgaard Hansen (Lyngby)