Differential Topology

A smooth manifold is a topological space that is locally Euclidean, meaning that at each point there is an open neighborhood that is homeomorphic to an open subset of Euclidean space.

A tangent bundle is a fiber bundle whose fibers are tangent spaces. It is a collection of all tangent spaces to a manifold, and each point on the manifold has an associated vector space called the tangent space at that point.

A vector field on a manifold is a smooth assignment of a tangent vector to each point on the manifold. It is a smooth section of the tangent bundle.

A differential form on a manifold is a multilinear map from the tangent bundle to the real numbers. It is a smooth section of the exterior algebra of the tangent bundle.

The Poincaré Duality Theorem is a theorem that relates the homology and cohomology groups of a smooth manifold. It states that the homology groups of a smooth manifold are isomorphic to the cohomology groups of the same manifold with compact support.

The Gauss-Bonnet Theorem is a theorem that relates the curvature of a surface to its Euler characteristic. It states that the integral of the Gaussian curvature over a closed surface is equal to 2π times the Euler characteristic of the surface.

The Hodge Decomposition Theorem is a theorem that decomposes a differential form into a sum of exact, coexact, and harmonic forms. It states that every differential form on a compact Riemannian manifold can be written as a sum of an exact form, a coexact form, and a harmonic form.

The de Rham Cohomology Theorem is a theorem that relates the de Rham cohomology groups of a manifold to its singular cohomology groups. It states that the de Rham cohomology groups of a smooth manifold are isomorphic to the singular cohomology groups of the same manifold.

The Whitney Embedding Theorem is a theorem that states that every smooth manifold can be embedded in Euclidean space. It states that for any smooth manifold M of dimension n, there exists an integer k such that M can be embedded in Euclidean space of dimension k.

The Nash-Moser Theorem is a theorem that states that every smooth manifold can be embedded in Euclidean space with a smooth embedding. It states that for any smooth manifold M of dimension n, there exists an integer k and a smooth embedding of M into Euclidean space of dimension k.

The Novikov Conjecture is a conjecture that states that every smooth manifold admits a Morse function with a finite number of critical points of each index and a non-degenerate Hessian at each critical point. It is one of the most important unsolved problems in differential topology.

The Gromov-Witten Invariant is an invariant of a smooth manifold that counts the number of pseudo-holomorphic curves in the manifold. It is a powerful tool for studying the geometry and topology of smooth manifolds.

The Donaldson Invariant is an invariant of a smooth 4-manifold that is defined using instantons. It is a powerful tool for studying the geometry and topology of smooth 4-manifolds.

The Seiberg-Witten Invariant is an invariant of a smooth 4-manifold that is defined using Seiberg-Witten invariants. It is a powerful tool for studying the geometry and topology of smooth 4-manifolds.

Floer Homology is a homology theory for smooth manifolds that is defined using pseudo-holomorphic curves. It is a powerful tool for studying the geometry and topology of smooth manifolds.