nLab
algebraic topology

Idea

Algebraic topology refers to the application of methods of algebra to problems in topology. More specifically, the method of algebraic topology is to assign homeomorphism/homotopy-invariants to topological spaces, or more systematically, to the construction and applications of functors from some category of topological objects (e.g. Hausdorff spaces, topological fibre bundles) to some algebraic category (e.g. abelian groups, modules over the Steenrod algebra). Landing in an algebraic category aids to the computability, but typically loses some information (say getting from a topological spaces with a continuum or more points to rather discrete algebraic structures).

The idea of functorial invariants

The basic idea of the functorial method for the problem of existence of morphisms is the following: If $F:A\to B$ is a functor (we present here a general statement, but in the above context $A$ is a category of topological objects and $B$ some category of algebraic objects) and $d:D\to A$ a diagram in $A$ then $F\circ d$ is a diagram in $B$. If one can fill certain additional arrow $f$ in the diagram $d$ making the extended diagram commutative, then $F(f)$ is a morphism between the corresponding vertices in $B$ extending $F\circ d$ to a commutative diagram. Thus if we prove that there is no morphism extending $F\circ d$ then there was no morphism extending $d$ in the first place. Therefore, the functorial method is very suitable to prove negative existence for morphisms. Sometimes, however, there is a theorem showing that some set of invariants completely characterizes a problem hence being able to show positive existence or uniqueness for maps or spaces. For the uniqueness for morphisms, it is enough to show that $F$ is faithful and that there is at most one solution for the existence problem in the target category. Faithful functors in this context are rare, but it is sufficient for $F$ to be faithful on some subcategory $A_p$ of $A$ containing at least all morphisms which are the possible candidates for the solution of the particular existence problem for morphisms.

Overview of methods

The archetypical example is the classification of surfaces via their Euler characteristic. But as this example already shows, algebraic topology tends to be less about topological spaces themselves as rather about the homotopy types which they present. Therefore the topological invariants in question are typically homotopy invariants of spaces with some exceptions, like the shape invariants for spaces with bad local behaviour.

Hence modern algebraic topology is to a large extent the application of algebraic methods to homotopy theory.

A general and powerful such method is the assignment of homology and cohomology groups to topological spaces, such that these abelian groups depend only on the homotopy type. The simplest such are ordinary homology and ordinary cohomology groups, given by singular simplicial complexes. This way algebraic topology makes use of tools of homological algebra.

The axiomatization of the properties of such cohomology group assignments is what led to the formulation of the trinity of concepts of category, functor and natural transformations, and algebraic topology has come to make intensive use of category theory.

In particular this leads to the formulation of generalized (Eilenberg-Steenrod) cohomology theories which detect more information about classes of homotopy types. By the Brown representability theorem such are represented by spectra (generalizing chain complexes), hence stable homotopy types, and this way algebraic topology comes to use and be about stable homotopy theory.

Still finer invariants of homotopy types are detected by further refinements of these “algebraic” structures, for instance to multiplicative cohomology theories, to equivariant homotopy theory/equivariant stable homotopy theory and so forth. The construction and analysis of these requires the intimate combination of algebra and homotopy theory to higher category theory and higher algebra, notably embodied in the universal higher algebra of operads.

The central tool for breaking down all this higher algebraic data into computable pieces are spectral sequences, which are maybe the main heavy-lifting workhorses of algebraic topology.

Related entries

• topology, differential topology
• homotopy theory, shape theory, nonabelian algebraic topology, rational homotopy theory
• homotopy lifting property, Hurewicz fibration, Hurewicz connection, Serre fibration
• homotopy extension property, Hurewicz cofibration, deformation retract
• suspension, loop space, mapping cylinder, mapping cone, mapping cocylinder
• cohomology, spectrum, Brown representability theorem
• fundamental group, fundamental groupoid
• homotopy group, Eckmann-Hilton duality, H-space, Whitehead product
• topological K-theory, complex cobordism, elliptic cohomology, tmf
• CW complex, CW approximation, simplicial complex, simplicial set
• model category, model structure on topological spaces, homotopy category
• fibration sequence, cofibration sequence
• Freudenthal suspension theorem, Whitehead theorem

References

The following lists basic references on homotopy theory, algebraic topology and some $(\infty,1)$-category theory and homotopy type theory, but see these entries for more pointers.

Historical article at the origin of all these subjects:

• Henri Poincaré, Analysis Situs, Journal de l’École Polytechnique. (2). 1: 1–123 (1895) (gallica:12148/bpt6k4337198/f7, Engl: pdf, pdf)

Topological homotopy theory

Textbook accounts of homotopy theory of topological spaces (i.e. via “point-set topology”):

• Peter J. Hilton, An introduction to homotopy theory, Cambridge University Press 1953 (doi:10.1017/CBO9780511666278)

• Sze-Tsen Hu, Homotopy Theory, Academic Press 1959 (pdf)

• Robert E. Mosher, Martin C. Tangora, Cohomology operations and applications in homotopy theory, Harper & Row, 1968, reprinted by Dover 2008 GoogleBooks

• Tammo tom Dieck, Klaus Heiner Kamps, Dieter Puppe, Homotopietheorie, Lecture Notes in Mathematics 157 Springer 1970 (doi:10.1007/BFb0059721)

• George W. Whitehead, Elements of Homotopy Theory, Springer 1978 (doi:10.1007/978-1-4612-6318-0)

• Renzo A. Piccinini, Lectures on Homotopy Theory, Mathematics Studies 171, North Holland 1992 (ISBN:978-0-444-89238-6)

• Glen Bredon, Chapter VII of: Topology and Geometry, Graduate texts in mathematics 139, Springer 1993 (doi:10.1007/978-1-4757-6848-0, pdf)

• Hans-Joachim Baues, Homotopy types, in Ioan Mackenzie James (ed.) Handbook of Algebraic Topology, North Holland, 1995 (ISBN:9780080532981, doi:10.1016/B978-0-444-81779-2.X5000-7)

• Marcelo Aguilar, Samuel Gitler, Carlos Prieto, Algebraic topology from a homotopical viewpoint, Springer Science & Business Media, 2008 (doi:10.1007/b97586)

• Jeffrey Strom, Modern classical homotopy theory, Graduate Studies in Mathematics Vol. 127, American Mathematical Society, 2011 (doi:10.1090/gsm/127])

• Dai Tamaki, Fiber Bundles and Homotopy, World Scientific 2021 (doi:10.1142/12308)

  (motivated from classifying spaces for principal bundles/fiber bundles)

Algebraic topology

On algebraic topology:

Textbooks:

• Samuel Eilenberg, Norman Steenrod, Foundations of Algebraic Topology, Princeton University Press 1952 (pdf, ISBN:9780691653297)

• Edwin Spanier, Algebraic topology, Springer 1966 (doi:10.1007/978-1-4684-9322-1)

• Robert Switzer, Algebraic Topology - Homotopy and Homology, Die Grundlehren der Mathematischen Wissenschaften in Einzeldarstellungen, Vol. 212, Springer-Verlag, New York, N. Y., 1975 (doi:10.1007/978-3-642-61923-6)

• Raoul Bott, Loring Tu, Differential Forms in Algebraic Topology, Graduate Texts in Mathematics 82, Springer 1982 (doi:10.1007/978-1-4757-3951-0)

  (with focus on differential forms, differential topology)

• Glen Bredon, Topology and Geometry, Graduate texts in mathematics 139, Springer 1993 (doi:10.1007/978-1-4757-6848-0, pdf)

• Albrecht Dold, Lectures on Algebraic Topology, Springer 1995 (doi:10.1007/978-3-642-67821-9, pdf)

• Peter May, A concise course in algebraic topology, University of Chicago Press 1999 (ISBN: 9780226511832, pdf)

• Tammo tom Dieck, Topologie, De Gruyter 2000 (doi:10.1515/9783110802542)

• Peter May, Kate Ponto, More concise algebraic topology, University of Chicago Press (2012) (ISBN:9780226511795, pdf)

• Dai Tamaki, Akira Kono, Generalized Cohomology, Translations of Mathematical Monographs, American Mathematical Society, 2006 (ISBN: 978-0-8218-3514-2)

• Tammo tom Dieck, Algebraic topology, European Mathematical Society, Zürich (2008) (doi:10.4171/048, pdf)

• Allen Hatcher, Algebraic Topology, Cambridge University Press 2002 (ISBN:9780521795401, webpage)

Lecture notes:

• Michael Hopkins (notes by Akhil Mathew), algebraic topology – Lectures (pdf)

• Friedhelm Waldhausen, Algebraische Topologie I (pdf) , II (pdf), III (pdf) (web)

• James F. Davis and Paul Kirk, Lecture notes in algebraic topology (pdf)

• Gereon Quick, Advanced algebraic topology, 2014

Survey of various subjects in algebraic topology:

• Ioan Mackenzie James, Handbook of Algebraic Topology 1995

Survey with relation to differential topology:

• Sergei Novikov, Topology I – General survey, in: Encyclopedia of Mathematical Sciences Vol. 12, Springer 1986 (doi:10.1007/978-3-662-10579-5, pdf)

• Jean Dieudonné, A History of Algebraic and Differential Topology, 1900 - 1960, Modern Birkhäuser Classics 2009 (ISBN:978-0-8176-4907-4)

Some interactive 3D demos:

• Neil Strickland, Interactive pages for Algebraic Topology, web site

Further pointers:

• a thread on AlgTop literature at MathOverflow

Abstract homotopy theory

On localization at weak equivalences to homotopy categories:

• Edgar Brown, Abstract homotopy theory, Trans. AMS 119 no. 1 (1965) (doi:10.1090/S0002-9947-1965-0182970-6)

On localization via calculus of fractions:

• Pierre Gabriel, Michel Zisman, Calculus of fractions and homotopy theory, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 35. Springer, New York (1967)

On localization via model category-theory:

• Daniel Quillen, Homotopical algebra, Lecture Notes in Mathematics 43, Berlin, New York, 1967

• Mark Hovey, Model Categories, Mathematical Surveys and Monographs, Volume 63, AMS (1999) (ISBN:978-0-8218-4361-1, doi:10.1090/surv/063, pdf, Google books)

• Philip Hirschhorn, Model Categories and Their Localizations, AMS Math. Survey and Monographs Vol 99 (2002) (ISBN:978-0-8218-4917-0, pdf toc, pdf)

• William G. Dwyer, Philip S. Hirschhorn, Daniel M. Kan, Jeffrey H. Smith, Homotopy Limits Functors on Model Categories and Homotopical Categories, AMS, Mathematical Surveys and Monographs 113 (2004).

See also:

• Klaus Heiner Kamps, Tim Porter, Abstract Homotopy and Simple Homotopy Theory, World Scientific 1997 (doi:10.1142/2215, GoogleBooks)

• Haynes Miller (ed.), Handbook of Homotopy Theory, 2019

Lecture notes:

• William Dwyer, Homotopy theory and classifying spaces, Copenhagen, June 2008 (pdf, pdf)

• Jesper Michael Møller, Homotopy theory for beginners, 2015 (pdf, pdf)

• Urs Schreiber, Introduction to Homotopy Theory (2016)

• Yuri Ximenes Martins, Introduction to Abstract Homotopy Theory (arXiv:2008.05302)

Introduction, from category theory to (mostly abstract, simplicial) homotopy theory:

• Emily Riehl, Categorical Homotopy Theory, Cambridge University Press, 2014 (pdf, doi:10.1017/CBO9781107261457)

• Birgit Richter, From categories to homotopy theory, Cambridge Studies in Advanced Mathematics 188, Cambridge University Press 2020 (doi:10.1017/9781108855891, book webpage, pdf)

• Urs Schreiber, geometry of physics – categories and toposes

See also:

• William Dwyer, Philip Hirschhorn, Daniel Kan, Jeff Smith, Homotopy Limit Functors on Model Categories and Homotopical Categories, volume 113 of Mathematical Surveys and Monographs, American Mathematical Society (2004) (there exists this pdf copy of what seems to be a preliminary version of this book)

• Zhen Lin Low, Notes on homotopical algebra, 2015

• Brian Munson, Ismar Volic, Cubical homotopy theory, Cambridge University Press, 2015 (pdf, doi:10.1017/CBO9781139343329)

  (with emphasis on cubical objects such as in n-excisive functors and Goodwillie calculus)

Simplicial homotopy theory

On simplicial homotopy theory:

• Peter May, Simplicial objects in algebraic topology, University of Chicago Press 1967 (ISBN:9780226511818, djvu, pdf)

• Edward B. Curtis, Simplicial homotopy theory, Advances in Mathematics 6 (1971) 107–209 (doi:10.1016/0001-8708(71)90015-6, MR279808)

• Paul Goerss, J. F. Jardine, Section V.4 of: Simplicial homotopy theory, Progress in Mathematics, Birkhäuser (1999) Modern Birkhäuser Classics (2009) (doi:10.1007/978-3-0346-0189-4, webpage)

• André Joyal, Myles Tierney Notes on simplicial homotopy theory, Lecture at Advanced Course on Simplicial Methods in Higher Categories, CRM 2008 (pdf)

• André Joyal, Myles Tierney, An introduction to simplicial homotopy theory, 2009 (web, pdf)

• Paul Goerss, Kirsten Schemmerhorn, Model categories and simplicial methods, Notes from lectures given at the University of Chicago, August 2004, in: Interactions between Homotopy Theory and Algebra, Contemporary Mathematics 436, AMS 2007(arXiv:math.AT/0609537, doi:10.1090/conm/436)

• Francis Sergeraert, Introduction to Combinatorial Homotopy Theory, 2008 (pdf, pdf)

Basic $(\infty,1)$-category theory

On (∞,1)-category theory and (∞,1)-topos theory:

• André Joyal, The theory of quasicategories and its applications lectures at Advanced Course on Simplicial Methods in Higher Categories, CRM 2008 (pdf, pdf)

• André Joyal, Notes on Logoi, 2008 (pdf, pdf)

• Jacob Lurie, Higher Topos Theory

• Denis-Charles Cisinski, Higher category theory and homotopical algebra (pdf)

Basic homotopy type theory

On homotopy type theory:

Exposition:

• Mike Shulman, The logic of space, chapter in Gabriel Catren, Mathieu Anel (eds.), New Spaces for Mathematics and Physics (arXiv:1703.03007)

Textbook accounts:

• Univalent Foundations Project, Homotopy Type Theory – Univalent Foundations of Mathematics, 2013 (web, pdf)

• Egbert Rijke, Introduction to Homotopy Type Theory (2019) (web, pdf, GitHub)

Outlook

Indications of open questions and possible future directions in algebraic topology and (stable) homotopy theory:

• Mark Hovey, Algebraic Topology Problem List

• Tyler Lawson, The future, Talbot lectures 2013 (pdf)

• Problems in homotopy theory (wiki)

More regarding the sociology of the field (such as its folklore results):

• Clark Barwick, The future of homotopy theory, 2017 (pdf, pdf)