Algebraic Topology I 1000-135TA1

Categories, functors, natural transformations, adjoint functors, limits
and colimits.
Examples from set theory, algebra and topology.

Compact-open topology in mapping spaces. Quotient spaces and quotient maps.
Surfaces and spaces arrising from linear algebra (linear groups and
their homogeneous spaces).

Category of (pointed) topological spaces and its homotopy category.
Homotopy equivalences, cofibrations and fibrations. Language of closed model categories.

Suspension and loops on a space. The Puppe exact sequences of sets of homotopy classes.

Homotopy groups. Action of the fundamental group on homotopy groups.
Exact sequence of fibration and of a pair of spaces.

CW-complexes. J.H.C. Whitehead theorem.

Basics of differential topology. Sard theorem. Smooth homotopy
approximation of continous maps.

Homotopy classification of maps from n-dimensional smooth manifold to k-dimensional sphere for n\leq k (Hopf theorem).

Homotopy addition theorem. Spaces with given homotopy groups.
Calculations of homotopy groups.

Eilenberg-MacLane spaces. The Postnikov decomposition of a space.