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.
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edit- *
- The base point of a based space.
- For an unbased space X, X+ is the based space obtained by adjoining a disjoint base point.
A
edit- 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 duality
- Alexander trick
- The Alexander trick produces a section of the restriction map , Top denoting a homeomorphism group; namely, the section is given by sending a homeomorphism to the homeomorphism
- .
- Analysis Situs
- approximate fibration
- 1. An approximate fibration, a generalization of a fibration and a projection in a locally trivial bundle.
- 2. A manifold approximate fibration is a proper approximate fibration between manifolds.
- aspherical space
- Aspherical space
- assembly map
- Atiyah
- 1. Michael Atiyah.
- 2. Atiyah duality.
- 3. The Atiyah–Hirzebruch spectral sequence.
B
edit- bar construction
- based space
- A pair (X, x0) consisting of a space X and a point x0 in X.
- Betti number
- Bing–Borsuk conjecture
- See Bing–Borsuk conjecture.
- Bockstein homomorphism
- Borel
- Borel conjecture.
- Borel–Moore homology
- Borsuk's theorem
- Bott
- 1. Raoul Bott.
- 2. The Bott periodicity theorem for unitary groups say: .
- 3. The Bott periodicity theorem for orthogonal groups say: .
- Brouwer fixed-point theorem
- The Brouwer fixed-point theorem says that any map has a fixed point.
C
edit- cap product
- Casson
- Casson invariant.
- Čech cohomology
- cellular
- 1. A map ƒ:X→Y between CW complexes is cellular if for all n.
- 2. The cellular approximation theorem says that every map between CW complexes is homotopic to a cellular map between them.
- 3. The cellular homology is the (canonical) homology of a CW complex. Note it applies to CW complexes and not to spaces in general. A cellular homology is highly computable; it is especially useful for spaces with natural cell decompositions like projective spaces or Grassmannian.
- chain homotopy
- Given chain maps between chain complexes of modules, a chain homotopy s from f to g is a sequence of module homomorphisms satisfying . It is also called a homotopy operator.
- chain map
- A chain map between chain complexes of modules is a sequence of module homomorphisms that commutes with the differentials; i.e., .
- chain homotopy equivalence
- A chain map that is an isomorphism up to chain homotopy; that is, if ƒ:C→D is a chain map, then it is a chain homotopy equivalence if there is a chain map g:D→C such that gƒ and ƒg are chain homotopic to the identity homomorphisms on C and D, respectively.
- change of fiber
- The change of fiber of a fibration p is a homotopy equivalence, up to homotopy, between the fibers of p induced by a path in the base.
- character variety
- The character variety[2] of a group π and an algebraic group G (e.g., a reductive complex Lie group) is the geometric invariant theory quotient by G:
- .
- is the set of isomorphism classes of G-coverings.
- In particular, if G is abelian, then the left-hand side is (cf. nonabelian cohomology.)
- such that (1) X0 is discrete and (2) Xn is obtained from Xn-1 by attaching n-cells.
D
edit- deck transformation
- Another term for an automorphism of a covering.
- deformation retract
- A subspace is called a deformation retract of X if there is a homotopy such that is the identity, and is the identity (i.e., is a retract of in the sense in category theory). It is called a strong deformation retract if, in addition, satisfies the requirement that is the identity. For example, a homotopy exhibits that the origin is a strong deformation retract of an open ball B centered at the origin.
- Deligne–Beilinson cohomology
- Deligne–Beilinson cohomology
- delooping
- degeneracy cycle
- degree
- de Rham
- 1. de Rham cohomology, the cohomology of complex of differential forms.
- 2. The de Rham theorem gives an explicit isomorphism between the de Rham cohomology and the singular cohomology.
- Dold
- The Dold–Thom theorem.
E
edit- Eckmann–Hilton argument
- The Eckmann–Hilton argument.
- Eckmann–Hilton duality
- Eilenberg–MacLane spaces
- Given an abelian group π, the Eilenberg–MacLane spaces are characterized by
- .
- is an isomorphism.
F
edit- factorization homology
- fiber-homotopy equivalence
- Given D→B, E→B, a map ƒ:D→E over B is a fiber-homotopy equivalence if it is invertible up to homotopy over B. The basic fact is that if D→B, E→B are fibrations, then a homotopy equivalence from D to E is a fiber-homotopy equivalence.
- fiber sequence
- The fiber sequence of a map is the sequence where is the homotopy fiber of f; i.e., the pullback of the path space fibration along f.
- fiber square
- fiber square
- fibration
- A map p:E → B is a fibration if for any given homotopy and a map such that , there exists a homotopy such that . (The above property is called the homotopy lifting property.) A covering map is a basic example of a fibration.
- fibration sequence
- One says is a fibration sequence to mean that p is a fibration and that F is homotopy equivalent to the homotopy fiber of p, with some understanding of base points.
- finitely dominated
- fundamental class
- fundamental group
- The fundamental group of a space X with base point x0 is the group of homotopy classes of loops at x0. It is precisely the first homotopy group of (X, x0) and is thus denoted by .
- fundamental groupoid
- The fundamental groupoid of a space X is the category whose objects are the points of X and whose morphisms x → y are the homotopy classes of paths from x to y; thus, the set of all morphisms from an object x0 to itself is, by definition, the fundament group .
- framed
- A framed manifold is a manifold with a framing.
- free
- Synonymous with unbased. For example, the free path space of a space X refers to the space of all maps from I to X; i.e., while the path space of a based space X consists of such map that preserve the base point (i.e., 0 goes to the base point of X).
- Freudenthal suspension theorem
- For a nondegenerately based space X, the Freudenthal suspension theorem says: if X is (n-1)-connected, then the suspension homomorphism
- Fulton–MacPherson compactification
- The Fulton–MacPherson compactification of the configuration space of n distinct labeled points in a compact complex manifold is a natural smooth compactification introduced by Fulton and MacPherson.
G
edit- G-fibration
- A G-fibration with some topological monoid G. An example is Moore's path space fibration.
- G-space
- A G-space is a space together with an action of a group G (usually satisfying some conditions).
- Γ-space
- generalized cohomology theory
- A generalized cohomology theory is a contravariant functor from the category of pairs of spaces to the category of abelian groups that satisfies all of the Eilenberg–Steenrod axioms except the dimension axiom.
- geometrization conjecture
- geometrization conjecture
- genus
- germ
- germ
- group completion
- grouplike
- An H-space X is said to be group-like or grouplike if is a group; i.e., X satisfies the group axioms up to homotopy.
- Gysin sequence
H
edit- Hauptvermutung
- 1. Hauptvermutung, a German for main conjecture, is short for die Hauptvermutung der kombinatorischen Topologie (the main conjecture of combinatorial topology). It asks whether two simplicial complexes are isomorphic if homeomorphic. It was disproved by Milnor in 1961.
- 2. There are some variants; for example, one can ask whether two PL manifolds are PL-isomorphic if homeomorphic (which is also false).
- h-cobordism
- h-cobordism.
- Hilton–Milnor theorem
- The Hilton–Milnor theorem.
- Hirzebruch
- Hirzebruch signature theorem.
- H-space
- An H-space is a based space that is a unital magma up to homotopy.
- homologus
- Two cycles are homologus if they belong to the same homology class.
- homology sphere
- A homology sphere is a manifold having the homology type of a sphere.
- homotopy category
- Let C be a subcategory of the category of all spaces. Then the homotopy category of C is the category whose class of objects is the same as the class of objects of C but the set of morphisms from an object x to an object y is the set of the homotopy classes of morphisms from x to y in C. For example, a map is a homotopy equivalence if and only if it is an isomorphism in the homotopy category.
- homotopy colimit
- A homotopy colimit is a homotopically-correct version of colimit.
- homotopy over a space B
- A homotopy ht such that for each fixed t, ht is a map over B.
- homotopy equivalence
- 1. A map ƒ:X→Y is a homotopy equivalence if it is invertible up to homotopy; that is, there exists a map g: Y→X such that g ∘ ƒ is homotopic to th identity map on X and ƒ ∘ g is homotopic to the identity map on Y.
- 2. Two spaces are said to be homotopy equivalent if there is a homotopy equivalence between the two. For example, by definition, a space is contractible if it is homotopy equivalent to a point space.
- homotopy excision theorem
- The homotopy excision theorem is a substitute for the failure of excision for homotopy groups.
- homotopy fiber
- The homotopy fiber of a based map ƒ:X→Y, denoted by Fƒ, is the pullback of along f.
- homotopy fiber product
- A fiber product is a particular kind of a limit. Replacing this limit lim with a homotopy limit holim yields a homotopy fiber product.
- homotopy group
- 1. For a based space X, let , the set of homotopy classes of based maps. Then is the set of path-connected components of X, is the fundamental group of X and are the (higher) n-th homotopy groups of X.
- 2. For based spaces , the relative homotopy group is defined as of the space of paths that all start at the base point of X and end somewhere in A. Equivalently, it is the of the homotopy fiber of .
- 3. If E is a spectrum, then
- 4. If X is a based space, then the stable k-th homotopy group of X is . In other words, it is the k-th homotopy group of the suspension spectrum of X.
- homotopy pullback
- A homotopy pullback is a special case of a homotopy limit that is a homotopically-correct pullback.
- homotopy quotient
- If G is a Lie group acting on a manifold X, then the quotient space is called the homotopy quotient (or Borel construction) of X by G, where EG is the universal bundle of G.
- homotopy spectral sequence
- homotopy sphere
- A homotopy sphere is a manifold having the homotopy type of a sphere.
- Hopf
- 1. Heinz Hopf.
- 2. Hopf invariant.
- 3. The Hopf index theorem.
- 4. Hopf construction.
- Hurewicz
- The Hurewicz theorem establishes a relationship between homotopy groups and homology groups.
I
edit- infinite loop space
- infinite loop space machine
- Infinite loop space machine.
- infinite mapping telescope
- intersection
- intersection pairing.
- intersection homology, a substitute for an ordinary (singular) homology for a singular space.
- intersection cohomology
- integration along the fiber
- See integration along the fiber.
- invariance of domain
- invariance of domain.
- isotopy
J
edit- J-homomorphism
- See J-homomorphism.
- join
- The join of based spaces X, Y is
K
edit- k-invariant
- Kan complex
- See Kan complex.
- Kirby–Siebenmann
- Kirby–Siebenmann classification.
- Kervaire invariant
- The Kervaire invariant.
- Koszul duality
- Koszul duality.
- Kuiper
- Kuiper's theorem says that the general linear group of an infinite-dimensional Hilbert space is contractible.
- Künneth formula
L
edit- Lazard ring
- The Lazard ring L is the (huge) commutative ring together with the formal group law ƒ that is universal among all the formal group laws in the sense that any formal group law g over a commutative ring R is obtained via a ring homomorphism L → R mapping ƒ to g. According to Quillen's theorem, it is also the coefficient ring of the complex bordism MU. The Spec of L is called the moduli space of formal group laws.
- Lefschetz
- 1. Solomon Lefschetz
- 2. The Lefschetz fixed-point theorem says: given a finite simplicial complex K and its geometric realization X, if a map has no fixed point, then the Lefschetz number of f; that is,
- 3. The Lefschetz hyperplane theorem.
- lens space
- The lens space is the quotient space where is the group of p-th roots of unity acting on the unit sphere by .
- Leray spectral sequence
- L2
- The L2-cohomology of a Riemannian or Kähler manifold is the cohomology of the complexes of differential forms with square-integrable coefficients (coefficients for forms not cohomology).
- local coefficient
- 1. A module over the group ring for some based space B; in other words, an abelian group together with a homomorphism .
- 2. The local coefficient system over a based space B with an abelian group A is a fiber bundle over B with discrete fiber A. If B admits a universal covering , then this meaning coincides with that of 1. in the sense: every local coefficient system over B can be given as the associated bundle .
- local invariant
- Local invariant cycle theorem.
- local sphere
- The localization of a sphere at some prime number
- local system
- local system.
- localization
- locally constant sheaf
- A locally constant sheaf on a space X is a sheaf such that each point of X has an open neighborhood on which the sheaf is constant.
- loop space
- The loop space of a based space X is the space of all loops starting and ending at the base point of X.
M
edit- Madsen–Weiss theorem
- mapping
- 1.
The mapping cone of a map ƒ:X→Y is obtained by gluing the cone over X to Y. - 2. The mapping cylinder of a map ƒ:X→Y is . Note: .
- 3. The reduced versions of the above are obtained by using reduced cone and reduced cylinder.
- 4. The mapping path space Pp of a map p:E→B is the pullback of along p. If p is fibration, then the natural map E→Pp is a fiber-homotopy equivalence; thus, one can replace E by the mapping path space without changing the homotopy type of the fiber. A mapping path space is also called a mapping cocylinder.
- 5. As a set, the mapping space from a space X to a space Y is the set of all continuous maps from X to Y. It is topologized in such a way the mapping space is a space; tha is, an object in the category of spaces used in algebraic topology; e.g., the category of compactly generated weak Hausdorff spaces. This topology may or may not be compact-open topology.
- Mayer–Vietoris sequence
- microbundle
- microbundle
- model category
- A presentation of an ∞-category.[4] See also model category.
- Moore
- 1. Moore space
- 2. Moore path space.
- multiplicative
- A generalized cohomology theory E is multiplicative if E*(X) is a graded ring. For example, the ordinary cohomology theory and the complex K-theory are multiplicative (in fact, cohomology theories defined by E∞-rings are multiplicative.)
N
edit- n-cell
- Another term for an n-disk.
- n-connected
- A based space X is n-connected if for all integers q ≤ n. For example, "1-connected" is the same thing as "simply connected".
- n-equivalent
- NDR-pair
- A pair of spaces is said to be an NDR-pair (=neighborhood deformation retract pair) if there is a map and a homotopy such that , , and .
If A is a closed subspace of X, then the pair is an NDR-pair if and only if is a cofibration. - nilpotent
- 1. nilpotent space; for example, a simply connected space is nilpotent.
- 2. The nilpotent theorem.
- nonabelian
- 1. nonabelian cohomology
- 2. nonabelian algebraic topology
- normalized
- Given a simplicial group G, the normalized chain complex NG of G is given by with the n-th differential given by ; intuitively, one throws out degenerate chains.[5] It is also called the Moore complex.
O
edit- obstruction cocycle
- obstruction theory
- Obstruction theory is the collection of constructions and calculations indicating when some map on a submanifold (subcomplex) can or cannot be extended to the full manifold. These typically involve the Postnikov tower, killing homotopy groups, obstruction cocycles, etc.
- of finite type
- A CW complex is of finite type if there are only finitely many cells in each dimension.
- operad
- The portmanteau of “operations” and “monad”. See operad.
- orbibundle
- orbibundle.
- orbit category
- orientation
- 1. The orientation covering (or orientation double cover) of a manifold is a two-sheeted covering so that each fiber over x corresponds to two different ways of orienting a neighborhood of x.
- 2. An orientation of a manifold is a section of an orientation covering; i.e., a consistent choice of a point in each fiber.
- 3. An orientation character (also called the first Stiefel–Whitney class) is a group homomorphism that corresponds to an orientation covering of a manifold X (cf. #covering.)
- 4. See also orientation of a vector bundle as well as orientation sheaf.
P
edit- pair
- 1. A pair of spaces is a space X together with a subspace .
- 2. A map of pairs is a map such that .
- p-adic homotopy theory
- The p-adic homotopy theory.
- parallelizable
- path class
- An equivalence class of paths (two paths are equivalent if they are homotopic to each other).
- path lifting
- A path lifting function for a map p: E → B is a section of where is the mapping path space of p. For example, a covering is a fibration with a unique path lifting function. By formal consideration, a map is a fibration if and only if there is a path lifting function for it.
- path space
- The path space of a based space X is , the space of based maps, where the base point of I is 0. Put in another way, it is the (set-theoretic) fiber of over the base point of X. The projection is called the path space fibration, whose fiber over the base point of X is the loop space . See also mapping path space.
- perverse
- A perverse sheaf.
- phantom map
- phantom map
- piecewise algebraic space
- piecewise algebraic space, the notion introduced by Kontsevich and Soibelman.
- PL
- 1. PL short for piecewise linear.
- 2. A PL manifold is a topological manifold with a maximal PL atlas where a PL atlas is an atlas in which the transition maps are PL.
- 3. A PL space is a space with a locally finite simplicial triangulation.
- Poincaré
- 1. Henri Poincaré.
- 2. The Poincaré duality theorem says: given a manifold M of dimension n and an abelian group A, there is a natural isomorphism
- .
- where are homotopy cofiber and homotopy fiber of f.
- ;
- that is X and B are glued together along A through f. The map f is usually called the attaching map.
- An important example is when B = Dn, A = Sn-1; in that case, forming such a pushout is called attaching an n-cell (meaning an n-disk) to X.
Q
edit- quasi-fibration
- A quasi-fibration is a map such that the fibers are homotopy equivalent to each other.
- Quillen
- 1. Daniel Quillen
- 2. Quillen’s theorem says that is the Lazard ring.
R
edit- rational
- 1. The rational homotopy theory.
- 2. The rationalization of a space X is, roughly, the localization of X at zero. More precisely, X0 together with j: X → X0 is a rationalization of X if the map induced by j is an isomorphism of vector spaces and .
- 3. The rational homotopy type of X is the weak homotopy type of X0.
- regulator
- 1. Borel regulator.
- 2. Beilinson regulator.
- Reidemeister
- Reidemeister torsion.
- reduced
- The reduced suspension of a based space X is the smash product . It is related to the loop functor by where is the loop space.
- retract
- 1. A retract of a map f is a map r such that is the identity (in other words, f is a section of r).
- 2. A subspace is called a retract if the inclusion map admits a retract (see #deformation retract).
- ring spectrum
- A ring spectrum is a spectrum satisfying the ring axioms, either on nose or up to homotopy. For example, a complex K-theory is a ring spectrum.
- Rokhlin
- Rokhlin invariant.
S
edit- Samelson product
- Serre
- 1. Jean-Pierre Serre.
- 2. Serre class.
- 3. Serre spectral sequence.
- simple
- simple-homotopy equivalence
- A map ƒ:X→Y between finite simplicial complexes (e.g., manifolds) is a simple-homotopy equivalence if it is homotopic to a composition of finitely many elementary expansions and elementary collapses. A homotopy equivalence is a simple-homotopy equivalence if and only if its Whitehead torsion vanishes.
- simplicial approximation
- See simplicial approximation theorem.
- simplicial complex
- See simplicial complex; the basic example is a triangulation of a manifold.
- simplicial homology
- A simplicial homology is the (canonical) homology of a simplicial complex. Note it applies to simplicial complexes and not to spaces; cf. #singular homology.
- signature invariant
- singular
- 1. Given a space X and an abelian group π, the singular homology group of X with coefficients in π is
- 2. The singular simplices functor is the functor from the category of all spaces to the category of simplicial sets, that is the right adjoint to the geometric realization functor.
- 3. The singular simplicial complex of a space X is the normalized chain complex of the singular simplex of X.
- slant product
- small object argument
- smash product
- The smash product of based spaces X, Y is . It is characterized by the adjoint relation
- .
T
edit- Tate
- Tate sphere
- telescope
- Thom
- 1. René Thom.
- 2. If E is a vector bundle on a paracompact space X, then the Thom space of E is obtained by first replacing each fiber by its compactification and then collapsing the base X.
- 3. The Thom isomorphism says: for each orientable vector bundle E of rank n on a manifold X, a choice of an orientation (the Thom class of E) induces an isomorphism
- .
U
edit- universal coefficient
- The universal coefficient theorem.
- up to homotopy
- A statement holds in the homotopy category as opposed to the category of spaces.
V
edit- V-manifold
- An old term for an orbifold.
- van Kampen
- The van Kampen theorem says: if a space X is path-connected and if x0 is a point in X, then
- Verdier
- Verdier duality.
W
edit- Waldhausen S-construction
- Waldhausen S-construction.
- Wall's finiteness obstruction
- weak equivalence
- A map ƒ:X→Y of based spaces is a weak equivalence if for each q, the induced map is bijective.
- wedge
- For based spaces X, Y, the wedge product of X and Y is the coproduct of X and Y; concretely, it is obtained by taking their disjoint union and then identifying the respective base points.
- well pointed
- A based space is well pointed (or non-degenerately based) if the inclusion of the base point is a cofibration.
- Whitehead
- 1. J. H. C. Whitehead.
- 2. Whitehead's theorem says that for CW complexes, the homotopy equivalence is the same thing as the weak equivalence.
- 3. Whitehead group.
- 4. Whitehead product.
- winding number
- 1. winding number.
Notes
edit- ^ Let r, s denote the restriction and the section. For each f in , define . Then .
- ^ Despite the name, it may not be an algebraic variety in the strict sense; for example, it may not be irreducible. Also, without some finiteness assumption on G, it is only a scheme.
- ^ Hatcher, Ch. 4. H.
- ^ How to think about model categories?
- ^ "Moore complex in nLab".
- ^ "Singular simplicial complex in nLab".
- ^ "Differential topology - Thom's first isotopy lemma".
References
edit- Adams, J.F. (1974). Stable Homotopy and Generalised Homology. Chicago Lectures in Mathematics. University of Chicago Press. ISBN 978-0-226-00524-9.
- Adams, J.F. (1978). Infinite Loop Spaces. Princeton University Press. ISBN 0-691-08206-5.
- Borel, Armand (21 May 2009). Intersection Cohomology. Springer Science & Business Media. ISBN 978-0-8176-4765-0.
- Bott, Raoul; Tu, Loring (1982), Differential Forms in Algebraic Topology, Springer, ISBN 0-387-90613-4
- Bousfield, A. K.; Kan, D. M. (1987), Homotopy Limits, Completions and Localizations, Lecture Notes in Mathematics, vol. 304, Springer, ISBN 9783540061052
- Davis, James F.; Kirk, Paul. "Lecture Notes in Algebraic Topology" (PDF).
- Fulton, William (2013). Algebraic Topology: A First Course. Springer. ISBN 978-1-4612-4180-5.
- Hatcher, Allen. "Algebraic topology".
- Hess, Kathryn (2007). "Rational homotopy theory: a brief introduction". Interactions between homotopy theory and algebra. Contemporary Mathematics. Vol. 436. Providence, Rhode Island: American Mathematical Society. pp. 175–202. arXiv:math/0604626. doi:10.1090/conm/436/08409 (inactive 2024-05-12). ISBN 978-0-8218-3814-3. MR 2355774.
{{cite book}}
: CS1 maint: DOI inactive as of May 2024 (link) - "algebraic topology" (PDF). Fall 2010. Lectures delivered by Michael Hopkins and Notes by Akhil Mathew, Harvard.
- Lurie, J. (2015). "Algebraic K-Theory and Manifold Topology". Math 281. Harvard University.
- Lurie, J. (2011). "Chromatic Homotopy Theory". 252x. Harvard University.
- May, J. "A Concise Course in Algebraic Topology" (PDF).
- May, J.; Ponto, K. "More concise algebraic topology: localization, completion, and model categories" (PDF).
- May; Sigurdsson. "Parametrized homotopy theory" (PDF). (despite the title, it contains a significant amount of general results.)
- Rudyak, Yuli B. (23 December 2014). "Piecewise linear structures on topological manifolds". arXiv:math/0105047.
- Sullivan, Dennis. "Geometric Topology" (PDF). the 1970 MIT notes
- Whitehead, George William (1978). Elements of homotopy theory. Graduate Texts in Mathematics. Vol. 61 (3rd ed.). Springer-Verlag. pp. xxi+744. ISBN 978-0-387-90336-1. MR 0516508.
- Wickelgren, Kirsten Graham. "8803 Stable Homotopy Theory".
Further reading
edit- 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. [1]
External links
edit- Algebraic Topology: A guide to literature Archived 2017-12-17 at the Wayback Machine