Algebraic Topology

Algebraic Topology : The Philosophy

  • Homeomorphism Problem
  • Extension Problem
  • Retraction Problem
  • Lifting Problem
  • Functors
  • Brouwer Fixed Point Theorem in Dimension Two
  • Hopf Fibration

Homotopy

  • Homotopic Maps
  • Contractible Spaces
  • Cones and Suspensions
  • Homotopy Type of a Topological Space
  • Deformation Retraction
  • Relative Homotopy
  • Path Homotopy
  • PoincarĂ© Conjecture

Fundamental Group Functor

  • \pi_1(X,x_0)
  • Induced Homomorphisms
  • Functoriality
  • Homotopy Invariance
  • Simple Connectivity
  • Contractibility of the Sphere in Hilbert Space

Fundamental Groups of Spheres and Applications

  • Covering Spaces
  • Path Lifting Theorem
  • Covering Homotopy Theorem
  • \pi_1(S^1)
  • \pi_1(RP^1)
  • Fundamental Theorem of Algebra
  • Borsuk-Ulam Theorem in Dimension Two
  • Seifert-Van Kampen Theorem (Special Case)
  • \pi_1(S^n)
  • Fundamental Group of a Product
  • Covering Spaces (Again)

Singular Homology Theory

  • Chain Complexes and Homology Groups
  • Motivation (Complex Analysis)
  • Singular Homology Groups
  • Other Coefficient Groups
  • Homology of a Point
  • H_0(X) for Pathwise Connected Spaces
  • H_1(X) versus \pi_1(X)
  • Homology of Convex Subspaces of Rn
  • Chain Maps and Chain Homotopy
  • Homotopy Invariance
  • Homology of Contractible Spaces
  • Mayer-Vietoris Sequence
  • Homology of Spheres
  • No-Retraction and Brouwer Fixed Point Theorems
  • Relative Homology
  • Brouwer Degree for Maps S^n\rightarrow S^n
  • Degree of the Antipodal Map
  • Vector Fields on Spheres

Appendix A: Proof of Homotopy Invariance for Singular Homology

Appendix B: Proof of the Mayer-Vietoris Sequence