topics on low dimensional topology

List of topic on low dimensional topology that should be here at PM

1 Manifolds in general

• manifold

• homotopy, homotopy groups,

• isotopy, mapping class group, homeotopy

• homology and cohomology

• fiber bundle, characteristic classes

• piecewise linear topology

• topological methods in group theory

• CW complex

2 1-manifolds

• real line vs projective line

• circle from a Euclidean loop to a Lie group

• curve

• knot, link and Jones polynomial. Prime knots

• torus knots

• braid

• loop and string, Witten M-Theory

• fundamental group

• graph and graph theory

3 2-manifolds

• Mobius strip

• surface

• two sphere

• torus

• projective plane

• Klein bottle

• genus (genus of topological surface)

• Euler characteristic (Euler’s polyhedron formula)

• Gaussian curvature

• Nielsen-Thurston classification of surface homeomorphisms

4 3-manifolds

• ${ℝ}^3$ Euclidean three-space

• $S^3$ three sphere, $SO(3)$, $SU(2)$

• surface bundle

• circle bundle, lens space, Seifert fiber space (http://planetmath.org/SeifertFiberSpace)

• homology sphere

• Poincare conjecture

• Dehn’s lemma, loop theorem, sphere theorem

• annulus and torus theorems

• Hopf bundle

5 4-manifolds

• Donalson’s theory

• quaternions

• relativity, space-time

• surface bundles over surfaces

• spin structures

• there are homeomorphic four-manifolds with differentiable structures each, which make them non-diffeomorphic.

• gauge theory: monopoles

• classification of simply connected 4-manifolds

• intersection form

• geometrization

6 Mathematical physics