Differentiable Montgomery-Samelson Fiberings with Finite Singular Sets

1969 ◽  
Vol 21 ◽  
pp. 1489-1495 ◽  
Author(s):  
Peter L. Antonelli
Keyword(s):  
Fixed Points ◽  
Group Action ◽  
Singular Set ◽  
Differentiable Structure ◽  
Singular Sets ◽  
Complete Bibliography

In 1946 Montgomery and Samelson (11) introduced a generalization of the notion of a differentiable group action with one type of orbit besides fixed points. Such an object is essentially a locally trivial fibering except on a certain singular set over which fibres are pinched to points. In recent years there has been a fair amount of research on these MS-fiberings and similar singular fiberings. This paper is another effort in this direction. For a fairly complete bibliography of the literature, the reader should consult the references, and in particular, (5).Let f: Mn → Sp, with Mn a closed connected n-manifold and Sp the unit p-sphere with standard differentiable structure, be the projection map of a smooth MS-fibering with finite non-empty singular set.

scholarly journals Singular Sets of Some Kleinian Groups (II)

1967 ◽  
Vol 29 ◽  
pp. 145-162 ◽  
Author(s):  
Tohru Akaza
Keyword(s):  
Kleinian Groups ◽  
Singular Set ◽  
Fundamental Domains ◽  
Dimensional Measure ◽  
Singular Sets ◽  
Discontinuous Groups ◽  
Positive Capacity ◽  
Automorphic Functions

In the theory of automorphic functions it is important to investigate the properties of the singular sets of the properly discontinuous groups. But we seem to know nothing about the size or structure of the singular sets of Kleinian groups except the results due to Myrberg and Akaza [1], which state that the singular set has positive capacity and there exist Kleinian groups whose singular sets have positive 1-dimensional measure. In our recent paper [2], we proved the existence of Kleinian groups with fundamental domains bounded by five circles whose singular sets have positive 1-dimensional measure and presented the problem whether there exist or not such groups in the case of four circles. The purpose of this paper is to solve this problem. Here we note that, by Schottky’s condition [4], the 1-dimensional measure of the singular set is always zero in the case of three circles.

scholarly journals Singular sets and the Lavrentiev phenomenon

2015 ◽  
Vol 145 (3) ◽  
pp. 513-533 ◽  
Author(s):  
Richard Gratwick
Keyword(s):  
Variational Problem ◽  
Singular Set ◽  
Lavrentiev Phenomenon ◽  
Smooth Functions ◽  
Strictly Convex ◽  
Superlinear Growth ◽  
Singular Sets

We show that non-occurrence of the Lavrentiev phenomenon does not imply that the singular set is small. Precisely, given a compact Lebesgue null subsetE⊆ ℝ and an arbitrary superlinearity, there exists a smooth strictly convex Lagrangian with this superlinear growth such that all minimizers of the associated variational problem have singular set exactlyEbut still admit approximation in energy by smooth functions.

scholarly journals Cuspidal quintics and surfaces with , and 5-torsion

2016 ◽  
Vol 19 (1) ◽  
pp. 42-53
Author(s):  
Carlos Rito
Keyword(s):  
Computer Algebra ◽  
General Type ◽  
Free Action ◽  
The Other ◽  
Singular Set ◽  
Canonical Map ◽  
Singular Sets ◽  
Surface Of General Type ◽  
Quintic Threefold ◽  
Hyperplane Sections

If $S$ is a quintic surface in $\mathbb{P}^{3}$ with singular set 15 3-divisible ordinary cusps, then there is a Galois triple cover ${\it\phi}:X\rightarrow S$ branched only at the cusps such that $p_{g}(X)=4$, $q(X)=0$, $K_{X}^{2}=15$ and ${\it\phi}$ is the canonical map of $X$. We use computer algebra to search for such quintics having a free action of $\mathbb{Z}_{5}$, so that $X/\mathbb{Z}_{5}$ is a smooth minimal surface of general type with $p_{g}=0$ and $K^{2}=3$. We find two different quintics, one of which is the van der Geer–Zagier quintic; the other is new.We also construct a quintic threefold passing through the 15 singular lines of the Igusa quartic, with 15 cuspidal lines there. By taking tangent hyperplane sections, we compute quintic surfaces with singular sets $17\mathsf{A}_{2}$, $16\mathsf{A}_{2}$, $15\mathsf{A}_{2}+\mathsf{A}_{3}$ and $15\mathsf{A}_{2}+\mathsf{D}_{4}$.

scholarly journals Poincaré Theta Series and Singular Sets of Schottky Groups

1964 ◽  
Vol 24 ◽  
pp. 43-65 ◽  
Author(s):  
Tohru Akaza
Keyword(s):  
Essential Role ◽  
Theta Series ◽  
Schottky Group ◽  
Singular Set ◽  
Schottky Groups ◽  
Linear Transformations ◽  
Convergence Problem ◽  
Discontinuous Group ◽  
Singular Sets ◽  
Metrical Property

In the theory of automorphic functions for a properly discontinuous group G of linear transformations, the Poincaré theta series plays an essential role, since the convergence problem of the series occupies an important part of the theory. This problem was treated by many mathematicians such as Poincaré, Burnside [2], Fricke [4], Myrberg [6], [7] and others. Poincaré proved that the (-2m)-dimensional Poincaré theta series always converges if m is a positive integer greater than 2, and Burnside treated the problem and conjectured that ( -2)-dimensional Poincaré theta series always converges if G is a Schottky group. This conjecture was solved negatively by Myrberg. As is shown later (Theorem A), the convergence of Poincaré theta series gives an information on a metrical property of the singular set of the group.

Fundamental Vector Fields

2020 ◽  
pp. 87-96
Author(s):  
Loring W. Tu
Keyword(s):  
Vector Field ◽  
Fixed Points ◽  
Lie Algebra ◽  
Group Action ◽  
Lie Group ◽  
Vector Fields ◽  
Principal Bundle ◽  
Left Action ◽  
Fundamental Vector ◽  
Vertical Tangent

This chapter addresses fundamental vector fields. The concept of a connection on a principal bundle is essential in the construction of the Cartan model. To define a connection on a principal bundle, one first needs to define the fundamental vector fields. When a Lie group acts smoothly on a manifold, every element of the Lie algebra of the Lie group generates a vector field on the manifold called a fundamental vector field. On a principal bundle, the fundamental vectors are precisely the vertical tangent vectors. In general, there is a relation between zeros of fundamental vector fields and fixed points of the group action. Unless specified otherwise (such as on a principal bundle), a group action is assumed to be a left action.

scholarly journals Structure of singular sets of some classes of subharmonic functions

2021 ◽  
Vol 31 (4) ◽  
pp. 519-535
Author(s):  
B.I. Abdullaev ◽  
S.A. Imomkulov ◽  
R.A. Sharipov
Keyword(s):  
Harmonic Functions ◽  
Singular Set ◽  
Test Functions ◽  
Subharmonic Functions ◽  
Singular Sets ◽  
Basic Functions

In this paper, we survey the recent results on removable singular sets for the classes of $m$-subharmonic ($m-sh$) and strongly $m$-subharmonic ($sh_m$), as well as $\alpha$-subharmonic functions, which are applied to study the singular sets of $sh_{m}$ functions. In particular, for strongly $m$-subharmonic functions from the class $L_{loc}^{p}$, it is proved that a set is a removable singular set if it has zero $C_{q,s}$-capacity. The proof of this statement is based on the fact that the space of basic functions, supported on the set $D\backslash E$, is dense in the space of test functions defined in the set $D$ on the $L_{q}^{s}$-norm. Similar results in the case of classical (sub)harmonic functions were studied in the works by L. Carleson, E. Dolzhenko, M. Blanchet, S. Gardiner, J. Riihentaus, V. Shapiro, A. Sadullaev and Zh. Yarmetov, B. Abdullaev and S. Imomkulov.

Overview

2020 ◽  
pp. 5-10
Author(s):  
Loring W. Tu
Keyword(s):  
Twentieth Century ◽  
Fixed Points ◽  
Group Action ◽  
Equivariant Cohomology ◽  
Topological Spaces ◽  
Computational Tool ◽  
Geometric Problem ◽  
Localization Formula ◽  
Algebraic Problem ◽  
Finite Sum

This chapter provides an overview of equivariant cohomology. Cohomology in any of its various forms is one of the most important inventions of the twentieth century. A functor from topological spaces to rings, cohomology turns a geometric problem into an easier algebraic problem. Equivariant cohomology is a cohomology theory that takes into account the symmetries of a space. Many topological and geometrical quantities can be expressed as integrals on a manifold. Integrals are vitally important in mathematics. However, they are also rather difficult to compute. When a manifold has symmetries, as expressed by a group action, in many cases the localization formula in equivariant cohomology computes the integral as a finite sum over the fixed points of the action, providing a powerful computational tool.

Structure Theory for Montgomery-Samelson Fiberings between Manifolds. I

1969 ◽  
Vol 21 ◽  
pp. 170-179 ◽  
Author(s):  
Peter L. Antonelli
Keyword(s):  
Total Space ◽  
Structure Theory ◽  
Homology Sphere ◽  
Singular Set ◽  
Connected Manifold ◽  
Singular Sets ◽  
Subsequent Publication

In (12), Montgomery and Samelson conjectured that an MS-fibering of polyhedra with total space an n-sphere must have a homology sphere as its singular set. Mahowald (11) has shown that, indeed, an orientable fibering with n ≧ 4 must have a Z2-cohomology sphere as its singular set, while Conner and Dyer (4) have shown this for n arbitrary provided the fiber itself is a Z2-cohomology sphere. We show that if the singular set is tame, then it is a Z-homology sphere if the fiber is also one. This result together with those of Stallings (15), Gluck (7), and Newman and Connell (13) are applied in the case where the singular sets are locally flat and tame. It is shown (Theorem 5.2) that MS-fiberings of spheres on spheres, with closed connected manifold fibers and singular sets, are topologically just suspensions of (Hopf) sphere bundles. In a subsequent publication, the case where the singular sets are finite shall be considered. The reader is invited to consult (3) and (18) in this case.

Maps with Locally Flat Singular Sets

1970 ◽  
Vol 22 (5) ◽  
pp. 916-921
Author(s):  
J. G. Timourian
Keyword(s):  
Singular Set ◽  
Singular Sets

A map f : M → N is topologically equivalent tog: X → Y if there exist homeomorphisms α: M → X and β: N → Y such that βfα–1 = g. At x ∊ M, f is locally topologically equivalent to g if, for every neighbourhood W ⊂ M of x, there exist neighbourhoods U ⊂ W of x and V of f(x) such that f|U: U → V is topologically equivalent to g.1.1. Definition. Given a map f: M → N and x ∊ M, let F be the component of f–1(f(x)) containing x. The singular set Af is defined as follows: x ∊ M – Af if and only if there are neighbourhoods U of F and V of f(x) such that f| U: U → F is topologically equivalent to the product projection map of V × F onto V.

scholarly journals Algebraic intersection spaces

2019 ◽  
Vol 12 (04) ◽  
pp. 1157-1194 ◽  
Author(s):  
Christian Geske
Keyword(s):  
Tubular Neighborhood ◽  
Space Theory ◽  
Singular Set ◽  
Projective Varieties ◽  
Local Duality ◽  
Chain Complexes ◽  
Singular Sets ◽  
Significant Extension ◽  
Real Analytic ◽  
Complex Projective

We define a variant of intersection space theory that applies to many compact complex and real analytic spaces [Formula: see text], including all complex projective varieties; this is a significant extension to a theory which has so far only been shown to apply to a particular subclass of spaces with smooth singular sets. We verify existence of these so-called algebraic intersection spaces and show that they are the (reduced) chain complexes of known topological intersection spaces in the case that both exist. We next analyze “local duality obstructions,” which we can choose to vanish, and verify that algebraic intersection spaces satisfy duality in the absence of these obstructions. We conclude by defining an untwisted algebraic intersection space pairing, whose signature is equal to the Novikov signature of the complement in [Formula: see text] of a tubular neighborhood of the singular set.

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