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

13 citations across 13 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.

• 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.

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.
• 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..
• 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.
• 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.
• 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.

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

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