Glossary of algebraic topology explained

This is a glossary of properties and concepts in algebraic topology in mathematics.

See also: glossary of topology, list of algebraic topology topics, glossary of category theory, glossary of differential geometry and topology, Timeline of manifolds.

• Convention: Throughout the article, I denotes the unit interval, Sⁿ the n-sphere and Dⁿ the n-disk. Also, throughout the article, spaces are assumed to be reasonable; this can be taken to mean for example, a space is a CW complex or compactly generated weakly Hausdorff space. Similarly, no attempt is made to be definitive about the definition of a spectrum. A simplicial set is not thought of as a space; i.e., we generally distinguish between simplicial sets and their geometric realizations.
• Inclusion criterion: As there is no glossary of homological algebra in Wikipedia right now, this glossary also includes a few concepts in homological algebra (e.g., chain homotopy); some concepts in geometric topology and differential topology are also fair game. On the other hand, the items that appear in glossary of topology are generally omitted. Abstract homotopy theory and motivic homotopy theory are also outside the scope. Glossary of category theory covers (or will cover) concepts in theory of model categories. See the glossary of symplectic geometry for the topics in symplectic topology such as quantization.

References

• Book: Adams, J.F. . Frank Adams

. Stable Homotopy and Generalised Homology . 1974 . University of Chicago Press . 978-0-226-00524-9 . Chicago Lectures in Mathematics . Frank Adams.

• Book: Adams, J.F. . Infinite Loop Spaces . 1978 . Princeton University Press . 0-691-08206-5.
• Book: Borel . Armand . Intersection Cohomology . 21 May 2009 . Springer Science & Business Media . 978-0-8176-4765-0 .
  • • Web site: James F. . Davis . Paul . Kirk . Lecture Notes in Algebraic Topology.
• Book: Fulton, William . Algebraic Topology: A First Course . 2013 . Springer . 978-1-4612-4180-5 .
• Web site: Allen . Hatcher . Algebraic topology.
• Book: Hess, Kathryn . Kathryn Hess . math/0604626 . Rational homotopy theory: a brief introduction . 10.1090/conm/436/08409 . 978-0-8218-3814-3 . 2355774 . 175–202 . American Mathematical Society . Providence, Rhode Island . Contemporary Mathematics . Interactions between homotopy theory and algebra . 436 . 2007. 2024-05-12 .
• Web site: algebraic topology . Fall 2010 . Lectures delivered by Michael Hopkins and Notes by Akhil Mathew, Harvard.
• Web site: Lurie . J. . Jacob Lurie . Algebraic K-Theory and Manifold Topology . Math 281 . Harvard University . 2015.
• Web site: Lurie . J. . Chromatic Homotopy Theory . 252x . Harvard University . 2011.
• Web site: May . J. . A Concise Course in Algebraic Topology.
• Web site: May . J. . K. . Ponto . More concise algebraic topology: localization, completion, and model categories.
• Web site: May . Sigurdsson . Parametrized homotopy theory. (despite the title, it contains a significant amount of general results.)
• Rudyak . Yuli B. . Piecewise linear structures on topological manifolds . 23 December 2014 . math/0105047 .
• Web site: Dennis Sullivan . Dennis . Sullivan . Geometric Topology. the 1970 MIT notes
• Book: Whitehead, George William . George W. Whitehead

. George W. Whitehead. Elements of homotopy theory. 3rd. Graduate Texts in Mathematics. 61. 1978. Springer-Verlag . 978-0-387-90336-1. xxi+744. 0516508 .

• Web site: Kirsten Graham . Wickelgren . Kirsten Wickelgren. 8803 Stable Homotopy Theory.

Further reading

• José I. Burgos Gil, The Regulators of Beilinson and Borel
• Lectures on groups of homotopy spheres by JP Levine
• B. I. Dundas, M. Levine, P. A. Østvær, O. Röndigs, and V. Voevodsky. Motivic homotopy theory. Universitext. Springer-Verlag, Berlin, 2007. Lectures from the Summer School held in Nordfjordeid, August 2002. http://www.math.ias.edu/vladimir/sites/math.ias.edu.vladimir/files/Nordfjordeid_lectures_published.pdf

External links

• Algebraic Topology: A guide to literature