MU-401 Algebraic Topology

Unit · year 4

MU-401 · Algebraic Topology

Threads space · structure24 lectures4 theorems

Algebraic invariants that distinguish spaces up to deformation.

PREREQUISITES

L01	The Programme: Algebra as a Topological Invariant —
L02	Homotopy of Paths and Maps The fundamental group
L03	The Fundamental Group The fundamental group
L04	Basepoints and Change of Basepoint The fundamental group
L05	Functoriality and Homotopy Invariance The fundamental group
L06	The Fundamental Group of the Circle The fundamental group
L07	Covering Spaces —
L08	The Lifting Criterion and Deck Transformations —
L09	The Universal Cover —
L10	Free Groups and Presentations The Seifert–van Kampen theorem
L11	The Seifert–van Kampen Theorem The Seifert–van Kampen theorem
L12	Computing π₁ of Surfaces The Seifert–van Kampen theorem
L13	Retractions and the No-Retraction Theorem The Brouwer fixed-point theorem
L14	The Brouwer Fixed-Point Theorem The Brouwer fixed-point theorem
L15	Applications of Brouwer The Brouwer fixed-point theorem
L16	Simplicial Complexes —
L17	Simplicial Homology —
L18	Singular Homology —
L19	Homotopy Invariance of Homology —
L20	The Mayer–Vietoris Sequence —
L21	The Euler Characteristic Invariance of the Euler characteristic
L22	Invariance of the Euler Characteristic Invariance of the Euler characteristic
L23	Classification of Surfaces Invariance of the Euler characteristic
L24	Synthesis: Holes, Counted Algebraically The fundamental group, Invariance of the Euler characteristic

Loops up to homotopy form a group invariant of the space.

The fundamental group of a union from those of its pieces.

Every continuous self-map of a disc has a fixed point.

V−E+F is a topological invariant.