Algebraic Topology

Hatcher, A. (2002). Algebraic Topology. Cambridge University Press.

Type

Book

Intellectual base

Review or monograph

Year

2002

Link

https://pi.math.cornell.edu/~hatcher/AT/AT.pdf

Cited from

mathematics

Cited by

18 citations across 18 artifacts.

Each citation links to the sentence it supports in the citing article.

Primes

• Cycle
  • Algebra, topology, politics, and ritual. Cycles in chain complexes found homology, with cycles-modulo-boundaries measuring topological holes; precedent-citation rings entrench legal positions; liturgical and festival cycles recur the same closed-loop pattern with different ethical weight.

    This sourceDevelops homology via cycles modulo boundaries in chain complexes, with cycles-not-bounding measuring topological holes.

    Supported in partVerified against the work's full text

    Hatcher defines homology as cycles modulo boundaries in a chain complex and ties it to counting holes; it says nothing on precedent rings or liturgical cycles.

    “Elements of Ker ∂n are called cycles and elements of Im ∂n+1 are called boundaries.”

    Read by an automated reader; not a human review. How support was checked

• Topology
  • The two subjects are sibling cousins rather than identical.

    This sourceThe standard contemporary textbook on algebraic topology; develops simplicial and singular homology, the Euler characteristic, Betti numbers, fundamental group and higher homotopy groups, and CW-complex methods. Freely available from the author's webpage.

    Supported in partVerified against the work's full text

    Hatcher defines the Euler characteristic as an alternating count of cells, but the supplied text never states chi = 2 - 2g or the V - E + F = 2 polyhedron formula as a result.

    “the alternating sum 29 − 51+ 23 equal to 1 . This is the Euler characteristic , which for a cell complex with finitely many cells is defined to be the number of even-dimensional cells minus the number o f odd-dimensional cells.”

    Read by an automated reader; not a human review. How support was checked

Domain-specific

• Alexander Duality
  • Compactness, properness, local contractibility, the ambient dimension, and integer coefficients are part of this base claim rather than background decoration.

    This sourceAuthoritative treatment of Poincaré and Alexander duality, the compact locally contractible sphere-complement theorem, reduced groups, examples, and the Čech-cohomology extension.

• Bockstein homomorphism
  • The coefficient sequence is part of the name: different short exact sequences produce different Bocksteins.
• Categorical Lift
  • Covering maps provide unique path lifts after an initial point is chosen, while many categorical lifting properties assert existence without a unique filler.
• Cellular homology
  • The entry records a descriptive analytical identity; practical use requires the governing domain's evidence, standards, and safety obligations.
• Chain complex
  • … and degree, homomorphisms d_n, equation d_{n-1} d_n equals zero, cycle and boundary subobjects, homology quotient, chain maps and any augmentation boundedness or coefficient convention are explicit. to An applied instance preserves the invariant under changed notation, scale, dataset, jurisdiction, or implementation..
• Change of fiber
  • Path lifting in a fibration carries points or fiber data over one endpoint to the other and produces a map well-defined up to homotopy, with loops inducing monodromy actions.
• Classifying space
  • The entry records a descriptive analytical identity; practical use requires the governing domain's evidence, standards, and safety obligations.
• CW complex
  • Attaching maps determine the space, cells need not form a simplicial complex and regular CW complexes add the stronger requirement that characteristic maps embed closed balls.
• Excisive triad
  • For example, the distinction between constitutive identity and a convenient observable transfers from An open cover X=U union V gives an excisive triad (X;U,V) because each set equals its own interior. to A proof checks relative interiors rather than only A union B=X, since bare set coverage is insufficient..
• Graph Duality
  • The construction generalizes beyond planar graphs to cell complexes via Poincaré duality — a \(k\)-cell in a triangulation corresponds to a \((d-k)\)-cell in the dual cell decomposition of a closed \(d\)-manifold

    This sourcePoincaré duality for closed manifolds, the higher-dimensional setting of the primal-dual cell pairing.

    Supported in partVerified against the source

    Read by an automated reader; not a human review. How support was checked

• Induced homomorphism
  • Basepoints, variance and homotopy or chain conventions determine direction and equality; induced maps compose and preserve identities by functoriality.
• Lefschetz duality
  • The entry records a descriptive analytical identity; practical use requires the governing domain's evidence, standards, and safety obligations.
• Lens space
  • The entry records a descriptive analytical identity; practical use requires the governing domain's evidence, standards, and safety obligations.
• Locally simply connected space
  • A topological space is locally simply connected when each point has a neighborhood basis consisting of open simply connected sets.
• Whitehead Theorem
  • Hatcher states the theorem equivalently: weak homotopy equivalences between CW complexes are homotopy equivalences, while warning that the implication fails for arbitrary spaces.
• Zig-zag lemma
  • Cycles in the quotient complex are lifted into the middle complex, their boundaries descend from the subcomplex and the resulting connecting homomorphisms make the homology sequence exact in any abelian category.

Verification

Does it exist? Not checked yet. This entry carries no identifier to resolve. It was extracted from the citation as written in the article, normalized, and deduplicated against the rest of the registry.

Does it back the claim? Read against the text for 3 of 18 citations: 3 supported in part. Each verdict is shown under its citation below, with what in the work backs the sentence.

Was it audited? Yes. A second, independent pass read the citation against the article text and recorded a verdict.

Support is checked per citation rather than per work — the same source can be cited soundly in one article and wrongly in another. Per-citation recording began recently, so a citation with no recorded check is a gap in the record rather than evidence it went unchecked.

See how references were verified.

Links previously used in the corpus

Before the registry existed this work was also linked 2 other ways.

• https://pi.math.cornell.edu/~hatcher/AT/ATpage.html ×2
• https://pi.math.cornell.edu/~hatcher/AT/ATchapters.html ×1

Registry ID ref:036f9f4fab29 · see in the full table