Homotopy + homology + cohomology theories; CW-complexes; fibrations; spectral sequences; characteristic classes; K-theory; stable + unstable homotopy.
algebraic-topology
Homotopy equivalence + CW complexes
Whitehead 1949 CW-complexes built from cells via attaching-maps. Cellular-approximation theorem. Whitehead theorem: weak…
Fundamental group π₁
Poincaré 1895: loops at basepoint up to homotopy form group under concatenation. Functorial. Van-Kampen theorem (cross-listed L0). Examples…
Higher homotopy groups π_n
Hurewicz 1935 πn(X) abelian for n ≥ 2. Hurewicz theorem π_n(X) → H_n(X). Hard to compute (πn(S^k) for n > k mostly unknown).…
Singular homology + cohomology
Eilenberg-Steenrod 1952 axioms. Singular: simplicial-chains on continuous-maps Δ^n → X. H_n(X) functorial homotopy-invariant. Cohomology +…
Fibrations + fiber bundles (cross-listed)
Cross-listed L0 fiber-bundle. Hurewicz fibration: homotopy-lifting property. Long-exact-sequence in homotopy. Serre fibration weaker.…
Spectral sequences (Leray-Serre)
Leray 1946 / Serre 1951: spectral-sequence E^{p,q}_r → H^{p+q}(filtered) computes homology of filtered space. Bockstein / Adams /…
Characteristic classes (Stiefel-Whitney / Chern / Pontryagin)
Stiefel-Whitney w_i(E) ∈ H^i(B,Z/2); Chern c_i(E) ∈ H^{2i}(B,Z); Pontryagin. Obstruct triviality of vector-bundles. Splitting principle.…
K-theory (topological)
Atiyah-Hirzebruch 1959-61: K^0(X) = Grothendieck-group of vector-bundles on X. Bott-periodicity K^n(pt) period 2 (complex) / 8 (real).…
Eilenberg-MacLane spaces + spectra
K(G,n): π_n(K(G,n))=G, others 0. Represents singular-cohomology. Spectra = sequence (E_n) with structure-maps ΣE_n → E_{n+1}. Generalised…
Cobordism (Thom)
Thom 1954: oriented-cobordism ring Ω*; computed via Thom-Pontryagin construction. Manifolds modulo bordism. Smooth / symplectic / spin /…
Classifying spaces BG
BG = K(G,1) for discrete G; principal G-bundles classified by maps to BG. Milnor-construction E×G/G. Cohomology H*(BG,F_p) = mod-p…
Model categories (Quillen)
Quillen 1967 'Homotopical Algebra': abstract framework for homotopy-theory. Weak-equivalence + fibration + cofibration triples.…
Homotopy equivalence (detail)
Maps f:X->Y, g:Y->X with fg ~ id_Y, gf ~ id_X define homotopy equivalence; X ~ Y; coarser than homeomorphism; preserves homotopy invariants…
Simplicial complex + CW
Simplicial complex = collection of simplices closed under face-taking; CW complex generalizes via attaching cells of increasing dimension;…
Singular vs cellular homology
Singular homology defined via continuous maps from standard simplices; cellular homology computed from cellular chain complex on CW; both…
Hurewicz theorem (1935 detail)
Hurewicz 1935: for (n-1)-connected space X, pi_n(X) iso H_n(X) for n >= 1; bridges homotopy + homology theories.
Eilenberg-MacLane spaces K(G,n)
Eilenberg-MacLane 1945 spaces: pi_n(K(G,n)) iso G, all other homotopy groups trivial; classify ordinary cohomology functorially [X, K(G,n)]…
Spectral sequences (Serre)
Serre 1951 fibration spectral sequence: computes cohomology of fibration F -> E -> B from H*(B) tensor H*(F); foundation of computational…