Algebraic Topology(D)

ALGEBRAIC TOPOLOGY (D)

Part IB Analysis II is essential and Metric and Topological Spaces is highly desirable.

The fundamental group

Homotopy of continuous functions and homotopy equivalence between topological spaces. The fundamental group of a space, homomorphisms induced by maps of spaces, change of base point, invariance under homotopy equivalence. [3]

Covering spaces

Covering spaces and covering maps. Path-lifting and homotopy-lifting properties, and ther application to the calculation of fundamental groups. The fundamental group of the circle; topological proof of the fundamental theorem of algebra. *Construction of the universal covering of a path-connected, locally simply connected space*. The correspondence between connected coverings of X and conjugacy classes of subgroups of the fundamental group of X. [5]

The Seifert–Van Kampen theorem

Free groups, generators and relations for groups, free products with amalgamation. Statement *and proof* of the Seifert–Van Kampen theorem. Applications to the calculation of fundamental groups. [4]

Simplicial complexes

Finite simplicial complexes and subdivisions; the simplicial approximation theorem. [3]

Homology

Simplicial homology, the homology groups of a simplex and its boundary. Functorial properties for simplicial maps. *Proof of functoriality for continuous maps, and of homotopy invariance*. [4]

Homology calculations

The homology groups of Sn, applications including Brouwer’s ﬁxed-point theorem. The Mayer-Vietoris theorem. *Sketch of the classiﬁcation of closed combinatorical surfaces*; determination of their homology groups. Rational homology groups; the Euler–Poincar´e characteristic and the Lefschetz ﬁxed-point theorem [5]

Appropriate books