Algebraic Topology

Turning shape into algebra — cell complexes, the fundamental group by pieces, homology and cohomology, and the classical fixed-point theorems they prove.

Homotopy & Cell Complexes

• Homotopy Equivalence & Deformation Retracts — The same shape up to deformation: retracts, contractible spaces, and homotopy type.
• CW Complexes — Spaces built one cell at a time: skeleta, attaching maps, and the operations that glue them together.
• Van Kampen's Theorem — Cut a space into overlapping pieces and read off a presentation.
• Classification of Covering Spaces — Covers of a space are subgroups of its fundamental group, and nothing less.
• Graphs, Trees & Free Groups — Collapse a spanning tree, cover it, and every subgroup of a free group turns out free.

Homology

• Δ-Complexes & Simplicial Homology — Cycles that do not bound: homology you can compute by hand, one triangle at a time.
• Singular Homology — Homology for every space at once, with no triangulation to choose and nothing left to check.
• Exact Sequences & the Long Exact Sequence — Image equals kernel: the bookkeeping that turns unknown homology groups into arithmetic.
• Excision & Mayer–Vietoris — Cut a space in two and let an exact sequence assemble its homology.
• Cellular Homology — Homology from a handful of cells: the cellular chain complex, degrees of attaching maps, and the computations that finally make homology practical.
• Degree of a Map — One integer per self-map of a sphere — the winding number, computed by homology.

Cohomology & Classical Theorems

• Brouwer & Borsuk–Ulam — Stir the coffee: a group with nothing in it forbids the escape, so a fixed point has to be there.
• Cohomology & Universal Coefficients — Dualize the chains and every arrow turns around: cochains, contravariance, and the theorem that moves torsion up a degree.
• Cup Product & the Cohomology Ring — Multiplying cohomology classes: the cup product on cochains, the graded commutative ring it builds, and the spaces it separates that homology cannot.
• Poincaré Duality — The homology of a closed orientable manifold reads the same forwards and backwards — a symmetry strong enough to rule spaces out.
• Higher Homotopy Groups & Hurewicz — Throw spheres instead of loops: the same construction one dimension up, abelian and almost uncomputable.

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