Ph.D. Qualifying examination in topology Charles Frohman and

... A1) Prove or give a counterexample: The product of two regular spaces is regular. A2) De…ne the uniform and box topologies on a product of topological spaces. Let X = RJ be the product of a countable number of copies of the real numbers. Prove that the product, uniform and box topologies yield three ...

... A1) Prove or give a counterexample: The product of two regular spaces is regular. A2) De…ne the uniform and box topologies on a product of topological spaces. Let X = RJ be the product of a countable number of copies of the real numbers. Prove that the product, uniform and box topologies yield three ...

1.6 Smooth functions and partitions of unity

... of the category of rings, respectively, in such a way which respects identities and composition of morphisms. Such a map is called a functor. In this case, it has the peculiar property that it switches the source and target of morphisms. It is therefore a contravariant functor from the category of m ...

... of the category of rings, respectively, in such a way which respects identities and composition of morphisms. Such a map is called a functor. In this case, it has the peculiar property that it switches the source and target of morphisms. It is therefore a contravariant functor from the category of m ...

Rn a vector space over R (or C) with canonical basis {e 1, ...,en

... Remark 3.5. For a “smooth” manifold, M ⊂ Rn , can choose a projection by using the fact that for all p ∈ M there exists a unit normal vector Np and tangent plane Tp (M ) which varies continuously with p. Example: smooth and non-smooth curve. Defn: X is regular if one-point sets are closed in X and i ...

... Remark 3.5. For a “smooth” manifold, M ⊂ Rn , can choose a projection by using the fact that for all p ∈ M there exists a unit normal vector Np and tangent plane Tp (M ) which varies continuously with p. Example: smooth and non-smooth curve. Defn: X is regular if one-point sets are closed in X and i ...

Lecture 3. Submanifolds

... The Möbius strip is not orientable. I will not prove this rigorously yet. Heuristically, the idea is that if we take an oriented pair of vectors at some point (s, 0), and ‘slide’ them around the Möbius strip to (s + 1, 0), then if there were an oriented atlas it would have to be the case that the ...

... The Möbius strip is not orientable. I will not prove this rigorously yet. Heuristically, the idea is that if we take an oriented pair of vectors at some point (s, 0), and ‘slide’ them around the Möbius strip to (s + 1, 0), then if there were an oriented atlas it would have to be the case that the ...

THE REAL DEFINITION OF A SMOOTH MANIFOLD 1. Topological

... Remark. Notice that the Hausdorff property of manifolds then implies that they are locally compact. In other words, every x ∈ X has a neighborhood W with compact closure. 6. An example Real projective space RPn is an n-dimensional smooth manifold which is not be naturally defined as a subset of RN . ...

... Remark. Notice that the Hausdorff property of manifolds then implies that they are locally compact. In other words, every x ∈ X has a neighborhood W with compact closure. 6. An example Real projective space RPn is an n-dimensional smooth manifold which is not be naturally defined as a subset of RN . ...

An introduction to differential topology

... smooth atlas containing those charts. However, there exist topological manifolds for which there does not exist any smooth structure. Also, there exist so-called “exotic 7-spheres”, which are smooth structures for S7 that are not diffeomorphic to each other. There exist uncountably many non-diffeomo ...

... smooth atlas containing those charts. However, there exist topological manifolds for which there does not exist any smooth structure. Also, there exist so-called “exotic 7-spheres”, which are smooth structures for S7 that are not diffeomorphic to each other. There exist uncountably many non-diffeomo ...

Background notes

... bundles of Lie groups. One important special type of fiber bundle is a vector bundle: the fibers are vector spaces. Definition 19. A vector bundle is a fiber bundle as in Definition 3 for which the fibers π −1 (p), p ∈ M are vector spaces, the manifolds F in the local trivialization are vector space ...

... bundles of Lie groups. One important special type of fiber bundle is a vector bundle: the fibers are vector spaces. Definition 19. A vector bundle is a fiber bundle as in Definition 3 for which the fibers π −1 (p), p ∈ M are vector spaces, the manifolds F in the local trivialization are vector space ...

Note on fiber bundles and vector bundles

... bundles of Lie groups. One important special type of fiber bundle is a vector bundle: the fibers are vector spaces. Definition 20. A vector bundle is a fiber bundle as in Definition 3 for which the fibers π −1 (p), p ∈ M are vector spaces, the manifolds F in the local trivialization are vector space ...

... bundles of Lie groups. One important special type of fiber bundle is a vector bundle: the fibers are vector spaces. Definition 20. A vector bundle is a fiber bundle as in Definition 3 for which the fibers π −1 (p), p ∈ M are vector spaces, the manifolds F in the local trivialization are vector space ...

Smooth manifolds - University of Arizona Math

... You are expected to have a good grounding in multivariable calculus and pointset topology already. You may want to review a bit. 2. Topological manifolds We de…ne a topological manifold as follows: De…nition 1. A topological space M is a manifold of dimension n if (1) M is Hausdor¤ , and (2) M is se ...

... You are expected to have a good grounding in multivariable calculus and pointset topology already. You may want to review a bit. 2. Topological manifolds We de…ne a topological manifold as follows: De…nition 1. A topological space M is a manifold of dimension n if (1) M is Hausdor¤ , and (2) M is se ...

Chapter 6 Manifolds, Tangent Spaces, Cotangent Spaces, Vector

... This is a general fact learned from experience: Geometry arises not just from spaces but from spaces and interesting classes of functions between them. In particular, we still would like to “do calculus” on our manifold and have good notions of curves, tangent vectors, differential forms, etc. The s ...

... This is a general fact learned from experience: Geometry arises not just from spaces but from spaces and interesting classes of functions between them. In particular, we still would like to “do calculus” on our manifold and have good notions of curves, tangent vectors, differential forms, etc. The s ...

The bordism version of the h

... Lagrangian immersions [11], [45], [5] and general theorems of Ando [2], Szucs [44] and the author [35]. Our argument is different from those in the mentioned papers in that it does not rely on the h-principle for differential relations over closed manifolds (see § 7) and gives a more general and pre ...

... Lagrangian immersions [11], [45], [5] and general theorems of Ando [2], Szucs [44] and the author [35]. Our argument is different from those in the mentioned papers in that it does not rely on the h-principle for differential relations over closed manifolds (see § 7) and gives a more general and pre ...

Smooth manifolds - IME-USP

... of dimension n if M has the topology induced from N and, for every p ∈ M , there exists a local chart (U, ϕ) of N such that ϕ(U ∩ M ) = ϕ(U ) ∩ Rn , where we view Rn as a subspace of Rn+k in the standard way. We say that (U, ϕ) is a local chart of M adapted to N . Note that an embedded submanifold M ...

... of dimension n if M has the topology induced from N and, for every p ∈ M , there exists a local chart (U, ϕ) of N such that ϕ(U ∩ M ) = ϕ(U ) ∩ Rn , where we view Rn as a subspace of Rn+k in the standard way. We say that (U, ϕ) is a local chart of M adapted to N . Note that an embedded submanifold M ...

2 - Ohio State Department of Mathematics

... Word hyperbolicity We will show below that by using the strict hyperbolization technique of Charney–Davis [4], one can arrange for nontriangulable aspherical manifolds of dimension ≥ 6 to have word hyperbolic fundamental groups. So, in this paragraph h(K) is the strict hyperbolization functor of [4] ...

... Word hyperbolicity We will show below that by using the strict hyperbolization technique of Charney–Davis [4], one can arrange for nontriangulable aspherical manifolds of dimension ≥ 6 to have word hyperbolic fundamental groups. So, in this paragraph h(K) is the strict hyperbolization functor of [4] ...