Alphabeta MathAlgebraic Topology
Pathway
Pages are grouped by how many dependency steps into this group they sit. Everything a page needs from this group appears above it.
Level 0
1 pageThis page fixes the library's foundational simplicial convention at the level of abstract simplicial complexes: faces are determined by their vertex sets, the empty face is…
13 definitions, 9 lemmas, 4 propositions, 3 theorems, 1 corollaryExamples & counterexamples →
Level 1
2 pagesFace chains give geometric barycentric subdivision and its canonical homeomorphism.
7 definitions, 14 lemmas, 4 theoremsExamples & counterexamples →- Singular Chains and Singular Homology27 results
This page replaces a chosen triangulation by the full singular chain complex built from all continuous simplices.
10 definitions, 6 lemmas, 5 propositions, 3 theorems, 3 corollariesExamples & counterexamples →
Level 2
1 pageThis page builds relative singular chains, then establishes excision through barycentric subdivision and cover-small chains before deriving the two-open Mayer–Vietoris sequence.
8 definitions, 6 lemmas, 2 propositions, 11 theorems, 3 corollariesExamples & counterexamples →
Level 3
1 page- Cw Complexes and Cellular Homology26 results
A CW decomposition gives a filtration whose consecutive relative homology groups are free on cells.
8 definitions, 6 lemmas, 6 propositions, 4 theorems, 2 corollariesExamples & counterexamples →
Level 4
1 pageOrdinary homology is characterized on CW pairs by its axioms and a specified coefficient group.
6 definitions, 19 lemmas, 7 propositions, 10 theorems, 2 corollariesExamples & counterexamples →
Level 5
2 pagesBased cubes give the higher homotopy groups, their relative sequence and basepoint transport.
7 definitions, 15 lemmas, 5 propositions, 6 theorems, 1 corollaryExamples & counterexamples →Singular cochains assign coefficient values to singular simplices and evaluate finite chains by finite sums.
5 definitions, 6 lemmas, 3 propositions, 8 theorems, 4 corollariesExamples & counterexamples →
Level 6
2 pagesCup product turns singular cohomology with a commutative unital coefficient ring into a graded ring.
7 definitions, 5 lemmas, 4 propositions, 5 theoremsExamples & counterexamples →A fibration lifts homotopies; a fiber bundle supplies local product charts.
8 definitions, 2 lemmas, 4 propositions, 4 theorems, 1 exampleExamples & counterexamples →
Level 7
2 pagesCellular approximation, Whitehead's theorem, homotopy excision and Hurewicz compare different ways of measuring a space.
4 definitions, 22 lemmas, 1 proposition, 9 theorems, 2 corollariesExamples & counterexamples →Homological orientation begins with the local groups of a Hausdorff, second-countable topological manifold.
9 definitions, 15 lemmas, 2 propositions, 12 theorems, 1 corollaryExamples & counterexamples →
Level 8
4 pagesCohomology operations begin here with the coefficient connecting map.
7 definitions, 14 lemmas, 4 propositions, 5 theoremsExamples & counterexamples →Local coefficients are organized as covariant functors from the fundamental groupoid.
8 definitions, 3 lemmas, 3 propositions, 7 theoremsExamples & counterexamples →Simple homotopy refines homotopy equivalence of finite CW complexes by recording whether a map is built from finitely many elementary cell expansions and collapses.
7 definitions, 16 lemmas, 4 theoremsExamples & counterexamples →- Spectra and Stable Homotopy Groups15 results
A sequential prespectrum records spaces together with coherent suspension maps.
9 definitions, 2 lemmas, 4 propositionsExamples & counterexamples →
Level 9
1 pageCellular obstruction theory turns an extension problem into a cocycle only after its coefficient system, orientations, basepoint transports, and partial map have been fixed.
8 definitions, 2 lemmas, 1 proposition, 11 theorems, 2 corollariesExamples & counterexamples →
Level 10
2 pagesThe construction begins by separating a strict fiber from the homotopy fiber of the mapping-path replacement.
4 definitions, 7 lemmas, 3 propositions, 10 theorems, 1 corollaryExamples & counterexamples →Topological vector bundles can be described locally by linear transition cocycles and globally by maps into stable Grassmannians.
10 definitions, 3 lemmas, 2 propositions, 10 theorems, 1 corollaryExamples & counterexamples →
Level 11
2 pagesComplex topological K-theory begins by group-completing complex vector bundles under Whitney sum.
6 definitions, 6 lemmas, 2 propositions, 5 theorems, 1 corollaryExamples & counterexamples →Leray–Hirsch starts with a hypothesis that cannot be weakened to constant fiber ranks: named global cohomology classes must restrict to a basis on every fiber.
6 definitions, 6 lemmas, 2 propositions, 6 theoremsExamples & counterexamples →
Level 12
2 pagesA reduced generalized cohomology theory is fixed axiomatically: homotopy invariance, exact cofiber sequences with suspension isomorphisms, and the wedge axiom, with no dimension axiom.
5 definitions, 8 lemmas, 3 propositions, 5 theorems, 1 corollaryExamples & counterexamples →This page constructs the real characteristic classes of vector bundles from universal data, and stops at bundle-level obstruction statements.
6 definitions, 3 lemmas, 3 propositions, 9 theoremsExamples & counterexamples →
Level 13
2 pagesThe page builds complex characteristic classes from the projective bundle rather than postulating them.
6 definitions, 10 lemmas, 2 propositions, 14 theorems, 1 corollaryExamples & counterexamples →This page develops the real Clifford-module construction of the KO Thom map and records its consequences for ABS orientations, Spin and Spin-c Thom classes, and the finite-dimensional stable free-fermion extension table.
8 definitions, 18 lemmas, 11 theorems, 1 corollaryExamples & counterexamples →
Level 14
1 pageAssuming AC, this page constructs the shared fixed-coordinate Thom prespectrum of the universal real and oriented bundles and uses it to detect stable unoriented Thom homotopy.
6 definitions, 31 lemmas, 1 proposition, 15 theorems, 2 corollariesExamples & counterexamples →