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  2. Algebraic topology - Wikipedia

    en.wikipedia.org/wiki/Algebraic_topology

    Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism , though usually most classify up to homotopy equivalence .

  3. Homotopy theory - Wikipedia

    en.wikipedia.org/wiki/Homotopy_theory

    "A Concise Course in Algebraic Topology" (PDF). University of Chicago. May, J. Peter; Ponto, Kate. More Concise Algebraic Topology: Localization, completion, and model categories (PDF). University of Chicago Press. p. 215. ISBN 978-022651178-8 – via University of Edinburgh. Whitehead, George William (1978). Elements of homotopy theory ...

  4. Glossary of algebraic topology - Wikipedia

    en.wikipedia.org/wiki/Glossary_of_algebraic_topology

    edit] * The base point of a based space. X + {\displaystyle X_{+}} For an unbased space X, X + is the based space obtained by adjoining a disjoint base point. A absolute neighborhood retract abstract 1. Abstract homotopy theory Adams 1. John Frank Adams. 2. The Adams spectral sequence. 3. The Adams conjecture. 4. The Adams e -invariant. 5. The Adams operations. Alexander duality Alexander ...

  5. Higher category theory - Wikipedia

    en.wikipedia.org/wiki/Higher_category_theory

    While this concept is too strict for some purposes in for example, homotopy theory, where "weak" structures arise in the form of higher categories, [2] strict cubical higher homotopy groupoids have also arisen as giving a new foundation for algebraic topology on the border between homology and homotopy theory; see the article Nonabelian ...

  6. Betti number - Wikipedia

    en.wikipedia.org/wiki/Betti_number

    In algebraic topology, the Betti numbers are used to distinguish topological spaces based on the connectivity of n-dimensional simplicial complexes.For the most reasonable finite-dimensional spaces (such as compact manifolds, finite simplicial complexes or CW complexes), the sequence of Betti numbers is 0 from some point onward (Betti numbers vanish above the dimension of a space), and they ...

  7. Classifying space - Wikipedia

    en.wikipedia.org/wiki/Classifying_space

    An example of a classifying space is that when G is cyclic of order two; then BG is real projective space of infinite dimension, corresponding to the observation that EG can be taken as the contractible space resulting from removing the origin in an infinite-dimensional Hilbert space, with G acting via v going to −v, and allowing for homotopy ...

  8. Real projective space - Wikipedia

    en.wikipedia.org/wiki/Real_projective_space

    Topology and geometry, Graduate Texts in Mathematics, Springer Verlag 1993, 1996; Davis, Donald. "Table of immersions and embeddings of real projective spaces" Hatcher, Allen (2001). Algebraic Topology. Cambridge University Press. ISBN 978-0-521-79160-1

  9. Graduate Studies in Mathematics - Wikipedia

    en.wikipedia.org/wiki/Graduate_Studies_in...

    24 Number Theory: Algebraic Numbers and Functions, Helmut Koch (2000, ISBN 978-0-8218-2054-4) 25 Dirac Operators in Riemannian Geometry , Thomas Friedrich (2000, ISBN 978-0-8218-2055-1 ) 26 An Introduction to Symplectic Geometry , Rolf Berndt (2001, ISBN 978-0-8218-2056-8 )