Algebraic Topology

CW complexes are topological spaces constructed by inductively gluing cells of increasing dimensions. Learners will understand how to build these spaces, compute their homology groups, and use them to simplify homotopy calculations.

Characteristic classes are cohomology classes associated with vector bundles that measure their topological non-triviality. Learners will understand how to compute these invariants to classify bundles and study obstructions to geometric structures.

Cohomology is a mathematical tool that associates algebraic invariants with geometric spaces to measure global topological obstructions. Learners will understand sheaf cohomology, Čech cohomology, and derived functors, enabling them to solve global existence problems in algebraic geometry.

Covering spaces are topological spaces that locally project homeomorphically onto a base space. Learners will understand the relationship between covering spaces, fundamental groups, and deck transformations, enabling them to compute fundamental groups of complex spaces.

Fiber bundles are topological spaces that locally look like product spaces but may have complex global structures. Learners will understand how to analyze these spaces using transition functions, connection theory, and associated vector bundles.

The fundamental group is an algebraic invariant that measures the loops and 1-dimensional holes within a topological space. Learners will understand how to compute this group using the Seifert-van Kampen theorem and classify covering spaces.

Homology is a mathematical procedure that associates a sequence of abelian groups or modules with a topological space. Learners will understand how to compute these groups to detect and classify multi-dimensional holes in spaces.

Homotopy studies the continuous deformation of topological spaces and maps between them. Learners will understand how to determine if two spaces are topologically equivalent and compute homotopy groups to analyze higher-dimensional geometric structures.

K-theory is an algebraic framework used to study vector bundles on topological spaces or projective modules over rings. Learners will understand how to compute K-groups to solve problems in topology, geometry, and operator algebras.

Simplicial complexes are topological spaces constructed by gluing together points, line segments, triangles, and higher-dimensional tetrahedra. Learners will understand how to represent continuous spaces combinatorially and compute their simplicial homology groups.