Topology I - School of Mathematics

... out: the global calculus of variations, global geometry, the topology of Lie groups and homogeneous spaces, the topology of complex manifolds and algebraic varieties, the qualitative (topological) theory of dynamical systems and foliations, the topology of elliptic and hyperbolic partial differentia ...

... out: the global calculus of variations, global geometry, the topology of Lie groups and homogeneous spaces, the topology of complex manifolds and algebraic varieties, the qualitative (topological) theory of dynamical systems and foliations, the topology of elliptic and hyperbolic partial differentia ...

as a PDF - Universität Bonn

... In the last decades, the theory of Riemann surfaces and their moduli spaces has seen a rapid development. In particular, there is now an elaborate homotopy theory of moduli spaces of Riemann surfaces. This homotopy theory of moduli spaces is based on Teichmüller theory. Teichmüller theory tells us ...

... In the last decades, the theory of Riemann surfaces and their moduli spaces has seen a rapid development. In particular, there is now an elaborate homotopy theory of moduli spaces of Riemann surfaces. This homotopy theory of moduli spaces is based on Teichmüller theory. Teichmüller theory tells us ...

MAT1360: Complex Manifolds and Hermitian Differential Geometry

... is not to give a thorough treatment of the algebraic and differential geometry of complex manifolds, but to introduce the reader to material of current interest as quickly as possible. As a glance at the table of contents indicates, Part I treats standard introductory analytic material on complex ma ...

... is not to give a thorough treatment of the algebraic and differential geometry of complex manifolds, but to introduce the reader to material of current interest as quickly as possible. As a glance at the table of contents indicates, Part I treats standard introductory analytic material on complex ma ...

Cohomology of cyro-electron microscopy

... of noisy 2-dimensional (2D) projected images, reconstruct the 3-dimensional (3D) structure of the molecule that gave rise to these images. Viewed from a high level, it takes the form of an inverse problem similar to those in medical imaging, remote sensing, or underwater acoustics, except that for c ...

... of noisy 2-dimensional (2D) projected images, reconstruct the 3-dimensional (3D) structure of the molecule that gave rise to these images. Viewed from a high level, it takes the form of an inverse problem similar to those in medical imaging, remote sensing, or underwater acoustics, except that for c ...

Vector Bundles and K

... Definition A family E of vector spaces over X is said to be locally trivial if every x ∈ X has a neighbourhood U ⊂ X such that E|U is trivial : any isomorphism between E|U and a product family over U will be called a trivialisation of E over U . In this case the family will be called a vector bundle ...

... Definition A family E of vector spaces over X is said to be locally trivial if every x ∈ X has a neighbourhood U ⊂ X such that E|U is trivial : any isomorphism between E|U and a product family over U will be called a trivialisation of E over U . In this case the family will be called a vector bundle ...

introduction to algebraic topology and algebraic geometry

... exists for k ≥ k0 , then H k (M, d) = 0 for k ≥ k0 . In the case of complexes bounded below zero (i.e., M = ⊕k∈N M k ) often a homotopy is defined only for k ≥ 1, and it ...

... exists for k ≥ k0 , then H k (M, d) = 0 for k ≥ k0 . In the case of complexes bounded below zero (i.e., M = ⊕k∈N M k ) often a homotopy is defined only for k ≥ 1, and it ...

Lecture Notes

... Proposition 4.4. There is no vector field s ∈ Γ (TS2n ) which is everywhere nonzero. This result is colloquially known as the “Hairy Ball Theorem”. With TS1 and TS2n squared away for the moment, we turn to the tangent bundles of higher dimensional odd spheres. Classification here becomes much more n ...

... Proposition 4.4. There is no vector field s ∈ Γ (TS2n ) which is everywhere nonzero. This result is colloquially known as the “Hairy Ball Theorem”. With TS1 and TS2n squared away for the moment, we turn to the tangent bundles of higher dimensional odd spheres. Classification here becomes much more n ...

PICARD GROUPS OF MODULI PROBLEMS

... (1) If any of the objects X we are parametrizing admits an automorphism of finite order, there exist non-isomorphic étale-locally trivial families, all of whose fibers are isomorphic to X, over, say, Gm . But these families would correspond to the same (constant) map to the moduli space. (2) There ...

... (1) If any of the objects X we are parametrizing admits an automorphism of finite order, there exist non-isomorphic étale-locally trivial families, all of whose fibers are isomorphic to X, over, say, Gm . But these families would correspond to the same (constant) map to the moduli space. (2) There ...

9. Sheaf Cohomology Definition 9.1. Let X be a topological space

... Noetherian ring and let OX (1) be a very ample line bundle on X. Let F be a coherent sheaf. (1) H i (X, F) are finitely generated A-modules. (2) There is an integer n0 such that H i (X, F(n)) = 0 for all n ≥ n0 and i > 0. Proof. By assumption there is an immersion i : X −→ PrA such that OX (1) = i∗ ...

... Noetherian ring and let OX (1) be a very ample line bundle on X. Let F be a coherent sheaf. (1) H i (X, F) are finitely generated A-modules. (2) There is an integer n0 such that H i (X, F(n)) = 0 for all n ≥ n0 and i > 0. Proof. By assumption there is an immersion i : X −→ PrA such that OX (1) = i∗ ...

Equivariant K-theory

... where N is the kernel of the action ofH on E(). H/N is a Lie group, so we are reduced to the earlier case. Of course if M is any G-module (finite-dimensional complex representation space ofG) and X is any G-space one can form the G-vector bundle X x M on X, which I shall call trivial, and denote by ...

... where N is the kernel of the action ofH on E(). H/N is a Lie group, so we are reduced to the earlier case. Of course if M is any G-module (finite-dimensional complex representation space ofG) and X is any G-space one can form the G-vector bundle X x M on X, which I shall call trivial, and denote by ...

(pdf)

... for me to attend the REU program at the University of Chicago as well as making it such a pleasurable experience, and providing many helpful comments on this paper. I would also like to thank Randy Van Why for his helpful conversations about the material in this paper, and significant contributions ...

... for me to attend the REU program at the University of Chicago as well as making it such a pleasurable experience, and providing many helpful comments on this paper. I would also like to thank Randy Van Why for his helpful conversations about the material in this paper, and significant contributions ...

Note on fiber bundles and vector bundles

... These notes, written for another class, are provided for reference. I begin with fiber bundles. Then I will discuss the particular case of vector bundles and the construction of the tangent bundle. Intuitively, the tangent bundle is the disjoint union of the tangent spaces (see (21)). What we must d ...

... These notes, written for another class, are provided for reference. I begin with fiber bundles. Then I will discuss the particular case of vector bundles and the construction of the tangent bundle. Intuitively, the tangent bundle is the disjoint union of the tangent spaces (see (21)). What we must d ...

Background notes

... These notes, written for another class, are provided for reference. I begin with fiber bundles. Then I will discuss the particular case of vector bundles and the construction of the tangent bundle. Intuitively, the tangent bundle is the disjoint union of the tangent spaces (see (20)). What we must d ...

... These notes, written for another class, are provided for reference. I begin with fiber bundles. Then I will discuss the particular case of vector bundles and the construction of the tangent bundle. Intuitively, the tangent bundle is the disjoint union of the tangent spaces (see (20)). What we must d ...

Lecture 9: Tangential structures We begin with some examples of

... (9.32) Double covers and uniqueness of spin structures. We work in the context of (9.7). Let ρ : H → G be a double cover of the Lie group G. We have in mind G = SO(n) and H = Spin(n). Let P → M be a principal G-bundle and (Q, θ) a reduction along ρ to a principal H-bundle. Suppose R → M is a double ...

... (9.32) Double covers and uniqueness of spin structures. We work in the context of (9.7). Let ρ : H → G be a double cover of the Lie group G. We have in mind G = SO(n) and H = Spin(n). Let P → M be a principal G-bundle and (Q, θ) a reduction along ρ to a principal H-bundle. Suppose R → M is a double ...

Classifying Spaces - School of Mathematics and Statistics

... B −→ E/G is a well defined, continuous, bijective map with inverse π continuous, i.e. a homeomorphism. ...

... B −→ E/G is a well defined, continuous, bijective map with inverse π continuous, i.e. a homeomorphism. ...

Vector bundles over cylinders

... closed intervals, by induction we can conclude that every topological vector bundle over a product of closed intervals — and hence also every topological vector bundle over a closed disk — is a product bundle. 3. Using the preceding, we can conclude that every k – dimensional vector n bundle over th ...

... closed intervals, by induction we can conclude that every topological vector bundle over a product of closed intervals — and hence also every topological vector bundle over a closed disk — is a product bundle. 3. Using the preceding, we can conclude that every k – dimensional vector n bundle over th ...

LECTURE 30: INDUCED MAPS BETWEEN CLASSIFYING SPACES

... {ndimensional complex vector bundles over X}. The ﬁrst correspondence associates to a principle U (n)bundle P over X the vector bundle P ×U (n) Cn → X with a ﬁxed hermitian structure on Cn . Using partitions of unity, every vector bundle admits a hermitian structure, unique up to isomorphism. 3. C ...

... {ndimensional complex vector bundles over X}. The ﬁrst correspondence associates to a principle U (n)bundle P over X the vector bundle P ×U (n) Cn → X with a ﬁxed hermitian structure on Cn . Using partitions of unity, every vector bundle admits a hermitian structure, unique up to isomorphism. 3. C ...