glossary of algebraic topology

{{glossary}}

{{term|*}}

{{defn|1=The base point of a based space.}}

{{term|$X\_+$}}

{{defn|1=For an unbased space X, X₊ is the based space obtained by adjoining a disjoint base point.}}

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{{term|absolute neighborhood retract}}

{{term|abstract}}

{{defn|no=1|Abstract homotopy theory}}

{{term|Adams}}

{{defn|no=1|1=John Frank Adams.}}

{{defn|no=2|1=The Adams spectral sequence.}}

{{defn|no=3|1=The Adams conjecture.}}

{{defn|no=4|1=The Adams e-invariant.}}

{{defn|no=5|1=The Adams operations.}}

{{term|Alexander duality}}

{{defn|1=Alexander duality}}

{{term|Alexander trick}}

{{defn|The Alexander trick produces a section of the restriction map $\backslash operatorname\{Top\}(D^\{n+1\})\; \backslash to\; \backslash operatorname\{Top\}(S^n)$, Top denoting a homeomorphism group; namely, the section is given by sending a homeomorphism $f:\; S^n\; \backslash to\; S^n$ to the homeomorphism

:$\backslash widetilde\{f\}:\; D^\{n+1\}\; \backslash to\; D^\{n+1\},\; \backslash ,\; 0\; \backslash mapsto\; 0,\; 0\; \backslash ne\; x\; \backslash mapsto\; |x|f(x/|x|)$.

This section is in fact a homotopy inverse.Let r, s denote the restriction and the section. For each f in $\backslash operatorname\{Top\}(D^\{n+1\})$, define $h\_t(f)(x)\; =\; tf(x/t),\; |x|\; \backslash le\; t,\; h\_t(f)(x)\; =\; |x|f(x/|x|),\; |x|\; >\; t$. Then $h\_t:\; s\; \backslash circ\; r\; \backslash sim\; \backslash operatorname\{id\}$.}}

{{term|Analysis Situs}}

{{term|approximate fibration}}

{{defn|no=1|1=An approximate fibration, a generalization of a fibration and a projection in a locally trivial bundle.}}

{{defn|no=2|1=A manifold approximate fibration is a proper approximate fibration between manifolds.}}

{{term|aspherical space}}

{{defn|1=Aspherical space}}

{{term|assembly map}}

{{term|Atiyah}}

{{defn|no=1|1=Michael Atiyah.}}

{{defn|no=2|1=Atiyah duality.}}

{{defn|no=3|1=The Atiyah–Hirzebruch spectral sequence.}}

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{{term|bar construction}}

{{term|based space}}

{{defn|1=A pair (X, x₀) consisting of a space X and a point x₀ in X.}}

{{term|Betti number}}

{{defn|1=See Betti number.}}

{{term|Bing–Borsuk conjecture}}

{{defn|1=See Bing–Borsuk conjecture.}}

{{term|Bockstein homomorphism}}

{{term|Borel}}

{{defn|1=Borel conjecture.}}

{{term|Borel–Moore homology}}

{{term|Borsuk's theorem}}

{{term|Bott}}

{{defn|no=1|Raoul Bott.}}

{{defn|no=2|The Bott periodicity theorem for unitary groups say: $\backslash pi\_q\; U\; =\; \backslash pi\_\{q+2\}\; U,\; q\; \backslash ge\; 0$.}}

{{defn|no=3|The Bott periodicity theorem for orthogonal groups say: $\backslash pi\_q\; O\; =\; \backslash pi\_\{q+8\}\; O,\; q\; \backslash ge\; 0$.}}

{{term|Brouwer fixed-point theorem}}

{{defn|1=The Brouwer fixed-point theorem says that any map $f:\; D^n\; \backslash to\; D^n$ has a fixed point.}}

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{{term|cap product}}

{{term|Casson}}

{{defn|Casson invariant.}}

{{term|Čech cohomology}}

{{term|cellular}}

{{defn|no=1|A map ƒ:X→Y between CW complexes is cellular if $f(X^n)\; \backslash subset\; Y^n$ for all n.}}

{{defn|no=2|The cellular approximation theorem says that every map between CW complexes is homotopic to a cellular map between them.}}

{{defn|no=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.}}

{{term|chain homotopy}}

{{defn|1=Given chain maps $f,\; g:\; (C,\; d\_C)\; \backslash to\; (D,\; d\_D)$ between chain complexes of modules, a chain homotopy s from f to g is a sequence of module homomorphisms $s\_i:\; C\_i\; \backslash to\; D\_\{i+1\}$ satisfying $f\_i\; -\; g\_i\; =\; d\_D\; \backslash circ\; s\_i\; +\; s\_\{i-1\}\; \backslash circ\; d\_C$. It is also called a homotopy operator.}}

{{term|chain map}}

{{defn|1=A chain map $f:\; (C,\; d\_C)\; \backslash to\; (D,\; d\_D)$ between chain complexes of modules is a sequence of module homomorphisms $f\_i:\; C\_i\; \backslash to\; D\_i$ that commutes with the differentials; i.e., $d\_D\; \backslash circ\; f\_i\; =\; f\_\{i-1\}\; \backslash circ\; d\_C$.}}

{{term|chain homotopy equivalence}}

{{defn|1=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.}}

{{term|change of fiber}}

{{defn|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.}}

{{term|character variety}}

{{defn|1=The character varietyDespite 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. of a group π and an algebraic group G (e.g., a reductive complex Lie group) is the geometric invariant theory quotient by G:

:$\backslash mathcal\{X\}(\backslash pi,\; G)\; =\; \backslash operatorname\{Hom\}(\backslash pi,\; G)/\; \backslash !\; /G$.}}

{{term|characteristic class}}

{{defn|1=Let Vect(X) be the set of isomorphism classes of vector bundles on X. We can view $X\; \backslash mapsto\; \backslash operatorname\{Vect\}(X)$ as a contravariant functor from Top to Set by sending a map ƒ:X → Y to the pullback ƒ^* along it. Then a characteristic class is a natural transformation from Vect to the cohomology functor H^*. Explicitly, to each vector bundle E we assign a cohomology class, say, c(E). The assignment is natural in the sense that ƒ^*c(E) = c(ƒ^*E).}}

{{term|chromatic homotopy theory}}

{{defn|1=chromatic homotopy theory.}}

{{term|class}}

{{defn|no=1|Chern class.}}

{{defn|no=2|Stiefel–Whitney class.}}

{{term|classifying space}}

{{defn|1=Loosely speaking, a classifying space is a space representing some contravariant functor defined on the category of spaces; for example, $BU$ is the classifying space in the sense $[-,\; BU]$ is the functor $X\; \backslash mapsto\; \backslash operatorname\{Vect\}^\{\backslash mathbb\{R\}\}(X)$ that sends a space to the set of isomorphism classes of real vector bundles on the space.}}

{{term|clutching}}

{{term|cobar spectral sequence}}

{{term|cobordism}}

{{defn|no=1|1=See cobordism.}}

{{defn|no=2|1=A cobordism ring is a ring whose elements are cobordism classes.}}

{{defn|no=3|See also h-cobordism theorem, s-cobordism theorem.}}

{{term|coefficient ring}}

{{defn|1=If E is a ring spectrum, then the coefficient ring of it is the ring $\backslash pi\_*\; E$.}}

{{term|cofiber sequence}}

{{defn|1=A cofiber sequence is any sequence that is equivalent to the sequence $X\; \backslash overset\{f\}\backslash to\; Y\; \backslash to\; C\_f$ for some ƒ where $C\_f$ is the reduced mapping cone of ƒ (called the cofiber of ƒ).}}

{{term|cofibrant approximation}}

{{term|cofibration}}

{{defn|1=A map $i:\; A\; \backslash to\; B$ is a cofibration if it satisfies the property: given $h\_0:\; B\; \backslash to\; X$ and homotopy $g\_t:\; A\; \backslash to\; X$ such that $g\_0=h\_0\backslash circ\; i$, there is a homotopy $h\_t:\; B\; \backslash to\; X$ such that $h\_t\; \backslash circ\; i\; =\; g\_t$.{{harvnb|Hatcher|loc=Ch. 4. H.}} A cofibration is injective and is a homeomorphism onto its image.}}

{{term|coherent homotopy}}

{{term|coherency}}

{{defn|1=See coherency (homotopy theory)}}

{{term|cohomotopy group}}

{{defn|1=For a based space X, the set of homotopy classes $[X,\; S^n]$ is called the n-th cohomotopy group of X.}}

{{term|cohomology operation}}

{{term|collapse}}

{{defn|1=An informal phrase but usually means taking a quotient; e.g., a cone is obtained by collapsing the top (or bottom) of a cylinder.}}

{{term|completion}}

{{term|complex bordism}}

{{term|complex-oriented}}

{{defn|1=A multiplicative cohomology theory E is complex-oriented if the restriction map E²(CP^∞) → E²(CP¹) is surjective.}}

{{term|concordant}}

{{term|cone}}

{{defn|1=The cone over a space X is $CX\; =\; X\; \backslash times\; I\; /\; X\; \backslash times\; \backslash \{0\backslash \}$. The reduced cone is obtained from the reduced cylinder $X\; \backslash wedge\; I\_+$ by collapsing the top.}}

{{term|connective}}

{{defn|A spectrum E is connective if $\backslash pi\_q\; E\; =\; 0$ for all negative integers q.}}

{{term|configuration space}}

{{term|1=constant}}

{{defn|1=A constant sheaf on a space X is a sheaf $\backslash mathcal\{F\}$ on X such that for some set A and some map $A\; \backslash to\; \backslash mathcal\{F\}(X)$, the natural map $A\; \backslash to\; \backslash mathcal\{F\}(X)\; \backslash to\; \backslash mathcal\{F\}\_x$ is bijective for any x in X.}}

{{term|1=continuous}}

{{defn|1=Continuous cohomology.}}

{{term|contractible space}}

{{defn|1=A space is contractible if the identity map on the space is homotopic to the constant map.}}

{{term|covering}}

{{defn|no=1|A map p: Y → X is a covering or a covering map if each point of x has a neighborhood N that is evenly covered by p; this means that the pre-image of N is a disjoint union of open sets, each of which maps to N homeomorphically.}}

{{defn|no=2|It is n-sheeted if each fiber p⁻¹(x) has exactly n elements.}}

{{defn|no=3|It is universal if Y is simply connected.}}

{{defn|no=4|A morphism of a covering is a map over X. In particular, an automorphism of a covering p:Y→X (also called a deck transformation) is a map Y→Y over X that has inverse; i.e., a homeomorphism over X.}}

{{defn|no=5|A G-covering is a covering arising from a group action on a space X by a group G, the covering map being the quotient map from X to the orbit space X/G. The notion is used to state the universal property: if X admits a universal covering (in particular connected), then

::$\backslash operatorname\{Hom\}(\backslash pi\_1(X,\; x\_0),\; G)$ is the set of isomorphism classes of G-coverings.

:In particular, if G is abelian, then the left-hand side is $\backslash operatorname\{Hom\}(\backslash pi\_1(X,\; x\_0),\; G)\; =\; \backslash operatorname\{H\}^1(X;\; G)$ (cf. nonabelian cohomology.)}}

{{term|cup product}}

{{term|CW complex}}

{{defn|1=A CW complex is a space X equipped with a CW structure; i.e., a filtration

::$X^0\; \backslash subset\; X^1\; \backslash subset\; X^2\; \backslash subset\; \backslash cdots\; \backslash subset\; X$

:such that (1) X⁰ is discrete and (2) Xⁿ is obtained from X^n-1 by attaching n-cells.}}

{{term|cyclic homology}}

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{{term|deck transformation}}

{{defn|Another term for an automorphism of a covering.}}

{{term|deformation retract}}

{{defn|A subspace $A\; \backslash subset\; X$ is called a deformation retract of X if there is a homotopy $h\_t\; :\; X\; \backslash to\; X$ such that $h\_0$ is the identity, $h\_1(X)\; \backslash subset\; A$ and $\{h\_1\}|\_A$ is the identity (i.e., $h\_1$ is a retract of $A\; \backslash hookrightarrow\; X$ in the sense in category theory). It is called a strong deformation retract if, in addition, $h\_t$ satisfies the requirement that $\{h\_t\}|\_A$ is the identity. For example, a homotopy $h\_t\; :\; B\; \backslash to\; B,\; \backslash ,\; x\; \backslash mapsto\; (1-t)x$ exhibits that the origin is a strong deformation retract of an open ball B centered at the origin.}}

{{term| Deligne–Beilinson cohomology}}

{{defn|1=Deligne–Beilinson cohomology}}

{{term|delooping}}

{{term|degeneracy cycle}}

{{term|degree}}

{{term|de Rham}}

{{defn|no=1|1=de Rham cohomology, the cohomology of complex of differential forms.}}

{{defn|no=2|1=The de Rham theorem gives an explicit isomorphism between the de Rham cohomology and the singular cohomology.}}

{{term|Dold}}

{{defn|1=The Dold–Thom theorem.}}

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{{term|Eckmann–Hilton argument}}

{{defn|1=The Eckmann–Hilton argument.}}

{{term|Eckmann–Hilton duality}}

{{term|Eilenberg–MacLane spaces}}

{{defn|1=Given an abelian group π, the Eilenberg–MacLane spaces $K(\backslash pi,\; n)$ are characterized by

:$\backslash pi\_q\; K(\backslash pi,\; n)=$

\begin{cases}

\pi & \text{if } q = n \\

0 & \text{otherwise}

\end{cases}.}}

{{term|Eilenberg–Steenrod axioms}}

{{defn|1=The Eilenberg–Steenrod axioms are the set of axioms that any cohomology theory (singular, cellular, etc.) must satisfy. Weakening the axioms (namely dropping the dimension axiom) leads to a generalized cohomology theory.}}

{{term|Eilenberg–Zilber theorem}}

{{term|elliptic}}

{{defn|elliptic cohomology.}}

{{term|1=E_n-algebra|2=E_n-algebra}}

{{term|equivariant algebraic topology}}

{{defn|1=Equivariant algebraic topoloy is the study of spaces with (continuous) group action.}}

{{term|etale}}

{{defn|1=étale homotopy.}}

{{term|exact}}

{{defn|1=A sequence of pointed sets $X\; \backslash overset\{f\}\backslash to\; Y\; \backslash overset\{g\}\backslash to\; Z$ is exact if the image of f coincides with the pre-image of the chosen point of Z.}}

{{term|excision}}

{{defn|1=The excision axiom for homology says: if $U\; \backslash subset\; X$ and $\backslash overline\{U\}\; \backslash subset\; \backslash operatorname\{int\}(A)$, then for each q,

::$\backslash operatorname\{H\}\_q(X-U,A-U)\; \backslash to\; \backslash operatorname\{H\}\_q(X,A)$

:is an isomorphism.}}

{{term|excisive pair/triad}}

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{{term|factorization homology}}

{{term|fiber-homotopy equivalence}}

{{defn|1=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.}}

{{term|fiber sequence}}

{{defn|1=The fiber sequence of a map $f:\; X\; \backslash to\; Y$ is the sequence $F\_f\; \backslash overset\{p\}\; \backslash to\; X\; \backslash overset\{f\}\backslash to\; Y$ where $F\_f\; \backslash overset\{p\}\; \backslash to\; X$ is the homotopy fiber of f; i.e., the pullback of the path space fibration $PY\; \backslash to\; Y$ along f.}}

{{term|fiber square}}

{{defn|1=fiber square}}

{{term|fibration}}

{{defn|1=A map p:E → B is a fibration if for any given homotopy $g\_t:\; X\; \backslash to\; B$ and a map $h\_0:\; X\; \backslash to\; E$ such that $p\; \backslash circ\; h\_0\; =\; g\_0$, there exists a homotopy $h\_t:\; X\; \backslash to\; E$ such that $p\; \backslash circ\; h\_t\; =\; g\_t$. (The above property is called the homotopy lifting property.) A covering map is a basic example of a fibration.}}

{{term|fibration sequence}}

{{defn|1=One says $F\; \backslash to\; X\; \backslash overset\{p\}\backslash to\; B$ 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.}}

{{term|finitely dominated}}

{{term|fundamental class}}

{{term|fundamental group}}

{{defn|1=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 $\backslash pi\_1(X,\; x\_0)$.}}

{{term|fundamental groupoid}}

{{defn|1=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 $\backslash pi\_1(X,\; x\_0)$.}}

{{term|framed}}

{{defn|1=A framed manifold is a manifold with a framing.}}

{{term|free}}

{{defn|1=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., $X^I$ 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).}}

{{term|Freudenthal suspension theorem}}

{{defn|1=For a nondegenerately based space X, the Freudenthal suspension theorem says: if X is (n-1)-connected, then the suspension homomorphism

:$\backslash pi\_q\; X\; \backslash to\; \backslash pi\_\{q+1\}\; \backslash Sigma\; X$

is bijective for q < 2n - 1 and is surjective if q = 2n - 1.}}

{{term|Fulton–MacPherson compactification}}

{{defn|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.}}

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{{term|G-fibration}}

{{defn|1=A G-fibration with some topological monoid G. An example is Moore's path space fibration.}}

{{term|G-space}}

{{defn|1=A G-space is a space together with an action of a group G (usually satisfying some conditions).}}

{{term|Γ-space}}

{{term|generalized cohomology theory}}

{{defn|1=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.}}

{{term|geometrization conjecture}}

{{defn|1=geometrization conjecture}}

{{term|genus}}

{{term|germ}}

{{defn|1=germ}}

{{term|group completion}}

{{term|grouplike}}

{{defn|1=An H-space X is said to be group-like or grouplike if $\backslash pi\_0\; X$ is a group; i.e., X satisfies the group axioms up to homotopy.}}

{{term|Gysin sequence}}

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{{term|Hauptvermutung}}

{{defn|no=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.}}

{{defn|no=2|There are some variants; for example, one can ask whether two PL manifolds are PL-isomorphic if homeomorphic (which is also false).}}

{{term|h-cobordism}}

{{defn|1=h-cobordism.}}

{{term|Hilton–Milnor theorem}}

{{defn|1=The Hilton–Milnor theorem.}}

{{term|Hirzebruch}}

{{defn|Hirzebruch signature theorem.}}

{{term|H-space}}

{{defn|1=An H-space is a based space that is a unital magma up to homotopy.}}

{{term|homologus}}

{{defn|1=Two cycles are homologus if they belong to the same homology class.}}

{{term|homology sphere}}

{{defn|A homology sphere is a manifold having the homology type of a sphere.}}

{{term|homotopy category}}

{{defn|1=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.}}

{{term|homotopy colimit}}

{{defn|A homotopy colimit is a homotopically-correct version of colimit.}}

{{term|homotopy over a space B}}

{{defn|A homotopy h_t such that for each fixed t, h_t is a map over B.}}

{{term|homotopy equivalence}}

{{defn|no=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.}}

{{defn|no=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.}}

{{term|homotopy excision theorem}}

{{defn|1=The homotopy excision theorem is a substitute for the failure of excision for homotopy groups.}}

{{term|homotopy fiber}}

{{defn|1=The homotopy fiber of a based map ƒ:X→Y, denoted by Fƒ, is the pullback of $PY\; \backslash to\; Y,\; \backslash ,\; \backslash chi\; \backslash mapsto\; \backslash chi(1)$ along f.}}

{{term|homotopy fiber product}}

{{defn|1=A fiber product is a particular kind of a limit. Replacing this limit lim with a homotopy limit holim yields a homotopy fiber product.}}

{{term|homotopy group}}

{{defn|no=1|For a based space X, let $\backslash pi\_n\; X\; =\; [S^n,\; X]$, the set of homotopy classes of based maps. Then $\backslash pi\_0\; X$ is the set of path-connected components of X, $\backslash pi\_1\; X$ is the fundamental group of X and $\backslash pi\_n\; X,\; \backslash ,\; n\; \backslash ge\; 2$ are the (higher) n-th homotopy groups of X.}}

{{defn|no=2|For based spaces $A\; \backslash subset\; X$, the relative homotopy group $\backslash pi\_n(X,\; A)$ is defined as $\backslash pi\_\{n-1\}$ of the space of paths that all start at the base point of X and end somewhere in A. Equivalently, it is the $\backslash pi\_\{n-1\}$ of the homotopy fiber of $A\; \backslash hookrightarrow\; X$.}}

{{defn|no=3|If E is a spectrum, then $\backslash pi\_k\; E\; =\; \backslash varinjlim\_\{n\}\; \backslash pi\_\{k\; +\; n\}\; E\_n.$}}

{{defn|no=4|If X is a based space, then the stable k-th homotopy group of X is $\backslash pi\_k^s\; X\; =\; \backslash varinjlim\_\{n\}\; \backslash pi\_\{k\; +\; n\}\; \backslash Sigma^n\; X$. In other words, it is the k-th homotopy group of the suspension spectrum of X.}}

{{term|homotopy pullback}}

{{defn|1=A homotopy pullback is a special case of a homotopy limit that is a homotopically-correct pullback.}}

{{term|homotopy quotient}}

{{defn|1=If G is a Lie group acting on a manifold X, then the quotient space $(EG\; \backslash times\; X)/G$ is called the homotopy quotient (or Borel construction) of X by G, where EG is the universal bundle of G.}}

{{term|homotopy spectral sequence}}

{{term|homotopy sphere}}

{{defn|A homotopy sphere is a manifold having the homotopy type of a sphere.}}

{{term|Hopf}}

{{defn|no=1|1=Heinz Hopf.}}

{{defn|no=2|1=Hopf invariant.}}

{{defn|no=3|1=The Hopf index theorem.}}

{{defn|no=4|1=Hopf construction.}}

{{term|Hurewicz}}

{{defn|1=The Hurewicz theorem establishes a relationship between homotopy groups and homology groups.}}

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{{term|infinite loop space}}

{{term|infinite loop space machine}}

{{defn|1=Infinite loop space machine.}}

{{term|infinite mapping telescope}}

{{term|intersection}}

{{defn|1=intersection pairing.}}

{{defn|1=intersection homology, a substitute for an ordinary (singular) homology for a singular space.}}

{{defn|1=intersection cohomology}}

{{term|integration along the fiber}}

{{defn|See integration along the fiber.}}

{{term|invariance of domain}}

{{defn|invariance of domain.}}

{{term|isotopy}}

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{{term|J-homomorphism}}

{{defn|See J-homomorphism.}}

{{term|join}}

{{defn|The join of based spaces X, Y is $X\; \backslash star\; Y\; =\; \backslash Sigma(X\backslash wedge\; Y).$}}

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{{term|k-invariant}}

{{term|Kan complex}}

{{defn|See Kan complex.}}

{{term|Kirby–Siebenmann}}

{{defn|Kirby–Siebenmann classification.}}

{{term|Kervaire invariant}}

{{defn|1=The Kervaire invariant.}}

{{term|Koszul duality}}

{{defn|1=Koszul duality.}}

{{term|Kuiper}}

{{defn|Kuiper's theorem says that the general linear group of an infinite-dimensional Hilbert space is contractible.}}

{{term|Künneth formula}}

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{{term|Lazard ring}}

{{defn|1=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.}}

{{term|Lefschetz}}

{{defn|no=1|1=Solomon Lefschetz}}

{{defn|no=2|1=The Lefschetz fixed-point theorem says: given a finite simplicial complex K and its geometric realization X, if a map $f:\; X\; \backslash to\; X$ has no fixed point, then the Lefschetz number of f; that is,

:$\backslash sum\_0^\backslash infty\; (-1)^q\; \backslash operatorname\{tr\}(f\_*:\; \backslash operatorname\{H\}\_q(X)\; \backslash to\; \backslash operatorname\{H\}\_q(X))$

is zero. For example, it implies the Brouwer fixed-point theorem since the Lefschetz number of $f:\; D^n\; \backslash to\; D^n$ is, as higher homologies vanish, one.}}

{{defn|no=3|1=The Lefschetz hyperplane theorem.}}

{{term|lens space}}

{{defn|1=The lens space is the quotient space $\backslash \{\; z\; \backslash in\; \backslash mathbb\{C\}^n\; |\; |z|\; =\; 1\; \backslash \}/\; \backslash mu\_p$ where $\backslash mu\_p$ is the group of p-th roots of unity acting on the unit sphere by $\backslash zeta\; \backslash cdot\; (z\_1,\; \backslash dots,\; z\_n)\; =\; (\backslash zeta\; z\_1,\; \backslash dots,\; \backslash zeta\; z\_n)$.}}

{{term|Leray spectral sequence}}

{{term|content=L²}}

{{defn|1=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).}}

{{term|local coefficient}}

{{defn|no=1|1=A module over the group ring $\backslash mathbb\{Z\}[\backslash pi\_1\; B]$ for some based space B; in other words, an abelian group together with a homomorphism $\backslash pi\_1\; B\; \backslash to\; \backslash operatorname\{Aut\}(A)$.}}

{{defn|no=2|1=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 $\backslash widetilde\{B\}$, then this meaning coincides with that of 1. in the sense: every local coefficient system over B can be given as the associated bundle $\backslash widetilde\{B\}\; \backslash times\_\{\backslash pi\_1\; B\}\; A$.}}

{{term|local invariant}}

{{defn|Local invariant cycle theorem.}}

{{term|local sphere}}

{{defn|1=The localization of a sphere at some prime number}}

{{term|local system}}

{{defn|1=local system.}}

{{term|localization}}

{{term|locally constant sheaf}}

{{defn|1=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.}}

{{term|loop space}}

{{defn|The loop space $\backslash Omega\; X$ of a based space X is the space of all loops starting and ending at the base point of X.}}

{{glossary end}}

{{glossary}}

{{term|Madsen–Weiss theorem}}

{{term|mapping}}

{{defn|no=1|1=File:Mapping cone.svgThe mapping cone (or cofiber) of a map ƒ:X→Y is $C\_f\; =\; Y\; \backslash cup\_f\; CX$.}}

{{defn|no=2|1=The mapping cylinder of a map ƒ:X→Y is $M\_f\; =\; Y\; \backslash cup\_f\; (X\; \backslash times\; I)$. Note: $C\_f\; =\; M\_f/(X\; \backslash times\; \backslash \{0\backslash \})$.}}

{{defn|no=3|1=The reduced versions of the above are obtained by using reduced cone and reduced cylinder.}}

{{defn|no=4|1=The mapping path space P_p of a map p:E→B is the pullback of $B^I\; \backslash to\; B$ along p. If p is fibration, then the natural map E→P_p 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.}}

{{defn|no=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.}}

{{term|Mayer–Vietoris sequence}}

{{term|microbundle}}

{{defn|microbundle}}

{{term|model category}}

{{defn|1=A presentation of an ∞-category.[https://mathoverflow.net/q/2185 How to think about model categories?] See also model category.}}

{{term|Moore}}

{{defn|no=1|1=Moore space}}

{{defn|no=2|1=Moore path space.}}

{{term|multiplicative}}

{{defn|1=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.) }}

{{glossary end}}

{{glossary}}

{{term|n-cell}}

{{defn|1=Another term for an n-disk.}}

{{term|n-connected}}

{{defn|1=A based space X is n-connected if $\backslash pi\_q\; X\; =\; 0$ for all integers q ≤ n. For example, "1-connected" is the same thing as "simply connected".}}

{{term|n-equivalent}}

{{term|NDR-pair}}

{{defn|1=A pair of spaces $A\; \backslash subset\; X$ is said to be an NDR-pair (=neighborhood deformation retract pair) if there is a map $u:\; X\; \backslash to\; I$ and a homotopy $h\_t:\; X\; \backslash to\; X$ such that $A\; =u^\{-1\}(0)$, $h\_0\; =\; \backslash operatorname\{id\}\_X$, $h\_t|\_A\; =\; \backslash operatorname\{id\}\_A$ and .

If A is a closed subspace of X, then the pair $A\; \backslash subset\; X$ is an NDR-pair if and only if $A\; \backslash hookrightarrow\; X$ is a cofibration.

{{term|nilpotent}}

{{defn|no=1|1=nilpotent space; for example, a simply connected space is nilpotent.}}

{{defn|no=2|1=The nilpotent theorem.}}

{{term|nonabelian}}

{{defn|no=1|1=nonabelian cohomology}}

{{defn|no=2|1=nonabelian algebraic topology}}

{{term|normalized}}

{{defn|1=Given a simplicial group G, the normalized chain complex NG of G is given by $(NG)\_n\; =\; \backslash cap\_1^\{\backslash infty\}\; \backslash operatorname\{ker\}d\_i^n$ with the n-th differential given by $d^n\_0$; intuitively, one throws out degenerate chains.{{Cite web|url=https://ncatlab.org/nlab/show/Moore+complex|title = Moore complex in nLab}} It is also called the Moore complex.}}

{{glossary end}}

{{glossary}}

{{term|obstruction cocycle}}

{{term|obstruction theory}}

{{defn|1=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.}}

{{term|of finite type}}

{{defn|1=A CW complex is of finite type if there are only finitely many cells in each dimension.}}

{{term|operad}}

{{defn|1=The portmanteau of “operations” and “monad”. See operad.}}

{{term|orbibundle}}

{{defn|1=orbibundle.}}

{{term|orbit category}}

{{term|orientation}}

{{defn|no=1|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.}}

{{defn|no=2|1=An orientation of a manifold is a section of an orientation covering; i.e., a consistent choice of a point in each fiber.}}

{{defn|no=3|1=An orientation character (also called the first Stiefel–Whitney class) is a group homomorphism $\backslash pi\_1(X,\; x\_0)\; \backslash to\; \backslash \{\backslash pm\; 1\backslash \}$ that corresponds to an orientation covering of a manifold X (cf. #covering.)}}

{{defn|no=4|1=See also orientation of a vector bundle as well as orientation sheaf.}}

{{glossary end}}

{{glossary}}

{{term|pair}}

{{defn|no=1|1=A pair $(X,\; A)$ of spaces is a space X together with a subspace $A\; \backslash subset\; X$.}}

{{defn|no=2|1=A map of pairs $(X,\; A)\; \backslash to\; (Y,\; B)$ is a map $X\; \backslash to\; Y$ such that $f(A)\; \backslash subset\; B$.}}

{{term|p-adic homotopy theory}}

{{defn|1=The p-adic homotopy theory.}}

{{term|parallelizable}}

{{term|path class}}

{{defn|An equivalence class of paths (two paths are equivalent if they are homotopic to each other).}}

{{term|path lifting}}

{{defn|1=A path lifting function for a map p: E → B is a section of $E^I\; \backslash to\; P\_p$ where $P\_p$ 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.}}

{{term|path space}}

{{defn|1=The path space of a based space X is $PX\; =\; \backslash operatorname\{Map\}(I,\; X)$, the space of based maps, where the base point of I is 0. Put in another way, it is the (set-theoretic) fiber of $X^I\; \backslash to\; X,\; \backslash ,\; \backslash chi\; \backslash mapsto\; \backslash chi(0)$ over the base point of X. The projection $PX\; \backslash to\; X,\; \backslash ,\; \backslash chi\; \backslash mapsto\; \backslash chi(1)$ is called the path space fibration, whose fiber over the base point of X is the loop space $\backslash Omega\; X$. See also mapping path space.}}

{{term|perverse}}

{{defn|1=A perverse sheaf.}}

{{term|phantom map}}

{{defn|phantom map}}

{{term|piecewise algebraic space}}

{{defn|piecewise algebraic space, the notion introduced by Kontsevich and Soibelman.}}

{{term|PL}}

{{defn|no=1|1=PL short for piecewise linear.}}

{{defn|no=2|1=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.}}

{{defn|no=3|1=A PL space is a space with a locally finite simplicial triangulation.}}

{{term|Poincaré}}

{{defn|no=1|1=Henri Poincaré.}}

{{defn|no=2|1=The Poincaré duality theorem says: given a manifold M of dimension n and an abelian group A, there is a natural isomorphism

:$\backslash operatorname\{H\}^*\_c(M;\; A)\; \backslash simeq\; \backslash operatorname\{H\}\_\{n\; -\; *\}(M;\; A)$.}}

{{defn|no=3|1=Poincaré conjecture}}

{{defn|no=4|1=Poincaré lemma states the higher de Rham cohomology of a contractible smooth manifold vanishes.}}

{{defn|no=5|1=Poincaré homology sphere.}}

{{term|Pontrjagin–Thom construction}}

{{term|Postnikov system}}

{{defn|1=A Postnikov system is a sequence of fibrations, such that all preceding manifolds have vanishing homotopy groups below a given dimension.}}

{{term|principal fibration}}

{{defn|1=Usually synonymous with G-fibration.}}

{{term|prime decomposition}}

{{term|profinite}}

{{defn|1=profinite homotopy theory; it studies profinite spaces.}}

{{term|properly discontinuous}}

{{defn|Not particularly a precise term. But it could mean, for example, that G is discrete and each point of the G-space has a neighborhood V such that for each g in G that is not the identity element, gV intersects V at finitely many points.}}

{{term|pseudomanifold}}

{{defn|pseudomanifold}}

{{term|pullback}}

{{defn|1=Given a map p:E→B, the pullback of p along ƒ:X→B is the space $f^*E\; =\; \backslash \{\; (e,\; x)\; \backslash in\; E\; \backslash times\; X\; |\; p(e)\; =\; f(x)\; \backslash \}$ (succinctly it is the equalizer of p and f). It is a space over X through a projection.}}

{{term|Puppe sequence}}

{{defn|1=The Puppe sequence refers ro either of the sequences

::$X\; \backslash overset\{f\}\backslash to\; Y\; \backslash to\; C\_f\; \backslash to\; \backslash Sigma\; X\; \backslash to\; \backslash Sigma\; Y\; \backslash to\; \backslash cdots,$

::$\backslash cdots\; \backslash to\; \backslash Omega\; X\; \backslash to\; \backslash Omega\; Y\; \backslash to\; F\_f\; \backslash to\; X\; \backslash overset\{f\}\backslash to\; Y$

:where $C\_f,\; F\_f$ are homotopy cofiber and homotopy fiber of f.}}

{{term|pushout}}

{{defn|1=Given $A\; \backslash subset\; B$ and a map $f:\; A\; \backslash to\; X$, the pushout of X and B along f is

::$X\; \backslash cup\_f\; B\; =\; X\; \backslash sqcup\; B/(a\; \backslash sim\; f(a))$;

: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 = S^n-1; in that case, forming such a pushout is called attaching an n-cell (meaning an n-disk) to X.}}

{{glossary end}}

{{glossary}}

{{term|quasi-fibration}}

{{defn|1=A quasi-fibration is a map such that the fibers are homotopy equivalent to each other.}}

{{term|Quillen}}

{{defn|no=1|1=Daniel Quillen}}

{{defn|no=2|1=Quillen’s theorem says that $\backslash pi\_*\; MU$ is the Lazard ring.}}

{{glossary end}}

{{glossary}}

{{term|rational}}

{{defn|no=1|1=The rational homotopy theory.}}

{{defn|no=2|1=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 $\backslash pi\_*\; X\; \backslash otimes\; \backslash mathbb\{Q\}\; \backslash to\; \backslash pi\_*\; X\_0\; \backslash otimes\; \backslash mathbb\{Q\}$ induced by j is an isomorphism of vector spaces and $\backslash pi\_*\; X\_0\; \backslash otimes\; \backslash mathbb\{Q\}\; \backslash simeq\; \backslash pi\_*\; X\_0$.}}

{{defn|no=3|1=The rational homotopy type of X is the weak homotopy type of X₀.}}

{{term|regulator}}

{{defn|no=1|Borel regulator.}}

{{defn|no=2|Beilinson regulator.}}

{{term|Reidemeister}}

{{defn|1=Reidemeister torsion.}}

{{term|reduced}}

{{defn|1=The reduced suspension of a based space X is the smash product $\backslash Sigma\; X\; =\; X\; \backslash wedge\; S^1$. It is related to the loop functor by $\backslash operatorname\{Map\}(\backslash Sigma\; X,\; Y)\; =\; \backslash operatorname\{Map\}(X,\; \backslash Omega\; Y)$ where $\backslash Omega\; Y\; =\; \backslash operatorname\{Map\}(S^1,\; Y)$ is the loop space.}}

{{term|retract}}

{{defn|no=1|1=A retract of a map f is a map r such that $r\; \backslash circ\; f$ is the identity (in other words, f is a section of r).}}

{{defn|no=2|1=A subspace $A\; \backslash subset\; X$ is called a retract if the inclusion map $A\; \backslash hookrightarrow\; X$ admits a retract (see #deformation retract).}}

{{term|ring spectrum}}

{{defn|1=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.}}

{{term|Rokhlin}}

{{defn|Rokhlin invariant.}}

{{glossary end}}

{{glossary}}

{{term|Samelson product}}

{{term|Serre}}

{{defn|no=1|1=Jean-Pierre Serre.}}

{{defn|no=2|1=Serre class.}}

{{defn|no=3|1=Serre spectral sequence.}}

{{term|simple}}

{{term|simple-homotopy equivalence}}

{{defn|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.}}

{{term|simplicial approximation}}

{{defn|1=See simplicial approximation theorem.}}

{{term|simplicial complex}}

{{defn|1=See simplicial complex; the basic example is a triangulation of a manifold.}}

{{term|simplicial homology}}

{{defn|1=A simplicial homology is the (canonical) homology of a simplicial complex. Note it applies to simplicial complexes and not to spaces; cf. #singular homology.}}

{{term|signature invariant}}

{{term|singular}}

{{defn|no=1|1=Given a space X and an abelian group π, the singular homology group of X with coefficients in π is

:$\backslash operatorname\{H\}\_*(X;\; \backslash pi)\; =\; \backslash operatorname\{H\}\_*(C\_*(X)\; \backslash otimes\; \backslash pi)$

where $C\_*(X)$ is the singular chain complex of X; i.e., the n-th degree piece is the free abelian group generated by all the maps $\backslash triangle^n\; \backslash to\; X$ from the standard n-simplex to X. A singular homology is a special case of a simplicial homology; indeed, for each space X, there is the singular simplicial complex of X {{Cite web|url=http://ncatlab.org/nlab/show/singular+simplicial+complex|title = Singular simplicial complex in nLab}} whose homology is the singular homology of X.}}

{{defn|no=2|The singular simplices functor is the functor $\backslash mathbf\{Top\}\; \backslash to\; s\backslash mathbf\{Set\}$ from the category of all spaces to the category of simplicial sets, that is the right adjoint to the geometric realization functor.}}

{{defn|no=3|The singular simplicial complex of a space X is the normalized chain complex of the singular simplex of X.}}

{{term|slant product}}

{{term|small object argument}}

{{term|smash product}}

{{defn|1=The smash product of based spaces X, Y is $X\; \backslash wedge\; Y\; =\; X\; \backslash times\; Y\; /\; X\; \backslash vee\; Y$. It is characterized by the adjoint relation

:$\backslash operatorname\{Map\}(X\; \backslash wedge\; Y,\; Z)\; =\; \backslash operatorname\{Map\}(X,\; \backslash operatorname\{Map\}(Y,\; Z))$.}}

{{term|Spanier–Whitehead}}

{{defn|1=The Spanier–Whitehead duality.}}

{{term|spectrum}}

{{defn|1=Roughly a sequence of spaces together with the maps (called the structure maps) between the consecutive terms; see spectrum (topology).}}

{{term|sphere bundle}}

{{defn|1=A sphere bundle is a fiber bundle whose fibers are spheres.}}

{{term|sphere spectrum}}

{{defn|The sphere spectrum is a spectrum consisting of a sequence of spheres $S^0,\; S^1,\; S^2,\; S^3,\; \backslash dots$ together with the maps between the spheres given by suspensions. In short, it is the suspension spectrum of $S^0$.}}

{{term|Spivak}}

{{defn|Spivak normal fibration}}

{{term|stable homotopy group}}

{{defn|1=See #homotopy group.}}

{{term|Steenrod homology}}

{{defn|1=Steenrod homology.}}

{{term|Steenrod operation}}

{{term|Sullivan}}

{{defn|no=1|1=Dennis Sullivan.}}

{{defn|no=2|1=The Sullivan conjecture.}}

{{defn|no=3|1={{citation| title=Infinitesimal computations in topology|journal=Publications Mathématiques de l'IHÉS|year=1977|volume=47|pages=269–331|url=http://www.numdam.org/item?id=PMIHES_1977__47__269_0|last1=Sullivan|first1=Dennis|doi=10.1007/BF02684341|s2cid=42019745}} - introduces rational homotopy theory (along with Quillen's paper).}}

{{defn|no=4|The Sullivan algebra in the rational homotopy theory.}}

{{term|suspension spectrum}}

{{defn|The suspension spectrum of a based space X is the spectrum given by $X\_n\; =\; \backslash Sigma^n\; X$.}}

{{term|stratified}}

{{defn|no=1|A stratified space is a topological space with a stratification.}}

{{defn|no=2|A stratified Morse theory is a Morse theory done on a stratified space.}}

{{term|symmetric spectrum}}

{{defn|1=See symmetric spectrum.}}

{{term|symplectic topology}}

{{defn|1=symplectic topology.}}

{{glossary end}}

{{glossary}}

{{term|Tate}}

{{defn|Tate sphere}}

{{term|telescope}}

{{term|Thom}}

{{defn|no=1|1=René Thom.}}

{{defn|no=2|1=If E is a vector bundle on a paracompact space X, then the Thom space $\backslash text\{Th\}(E)$ of E is obtained by first replacing each fiber by its compactification and then collapsing the base X.}}

{{defn|no=3|1=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

:$\backslash widetilde\{\backslash operatorname\{H\}\}^\{*+n\}(\backslash text\{Th\}(E);\; \backslash mathbb\{Z\})\backslash simeq\; \backslash operatorname\{H\}^*(X;\; \backslash mathbb\{Z\})$.}}

{{defn|no=4|1=Thom's first and second isotopy lemmas.{{cite web | url=https://mathoverflow.net/questions/304298/thoms-first-isotopy-lemma | title=Differential topology - Thom's first isotopy lemma }}}}

{{defn|no=5|1=A Thom mapping originally called a mapping "sans éclatement"}}

{{term|topological chiral homology}}

{{term|transfer}}

{{term|transgression}}

{{term|triangulation}}

{{defn|triangulation.}}

{{glossary end}}

{{glossary}}

{{term|universal coefficient}}

{{defn|1=The universal coefficient theorem.}}

{{term|up to homotopy}}

{{defn|1=A statement holds in the homotopy category as opposed to the category of spaces.}}

{{glossary end}}

{{glossary}}

{{term|V-manifold}}

{{defn|An old term for an orbifold.}}

{{term|van Kampen}}

{{defn|1=The van Kampen theorem says: if a space X is path-connected and if x₀ is a point in X, then

:$\backslash pi\_1(X,\; x\_0)\; =\; \backslash varinjlim\; \backslash pi\_1(U,\; x\_0)$

where the colimit runs over some open cover of X consisting of path-connected open subsets containing x₀ such that the cover is closed under finite intersections.}}

{{term|Verdier}}

{{defn|Verdier duality.}}

{{glossary end}}

{{glossary}}

{{term|Waldhausen S-construction}}

{{defn|1=Waldhausen S-construction.}}

{{term|Wall's finiteness obstruction}}

{{term|weak equivalence}}

{{defn|1=A map ƒ:X→Y of based spaces is a weak equivalence if for each q, the induced map $f\_*:\; \backslash pi\_q\; X\; \backslash to\; \backslash pi\_q\; Y$ is bijective.}}

{{term|wedge}}

{{defn|1=For based spaces X, Y, the wedge product $X\; \backslash wedge\; Y$ 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.}}

{{term|well pointed}}

{{defn|1=A based space is well pointed (or non-degenerately based) if the inclusion of the base point is a cofibration.}}

{{term|Whitehead}}

{{defn|no=1|J. H. C. Whitehead.}}

{{defn|no=2|Whitehead's theorem says that for CW complexes, the homotopy equivalence is the same thing as the weak equivalence.}}

{{defn|no=3|Whitehead group.}}

{{defn|no=4|Whitehead product.}}

{{term|winding number}}

{{defn|no=1|winding number.}}

{{glossary end}}

{{reflist}}

• {{cite book |first=J.F. |last=Adams |title=Stable Homotopy and Generalised Homology |url=https://books.google.com/books?id=6vG13YQcPnYC |year=1974 |publisher=University of Chicago Press |isbn=978-0-226-00524-9 |series=Chicago Lectures in Mathematics |authorlink=Frank Adams}}
• {{cite book |first=J.F. |last=Adams |title=Infinite Loop Spaces |url=https://books.google.com/books?id=e2rYkg9lGnsC |date=1978 |publisher=Princeton University Press |isbn=0-691-08206-5}}
• {{cite book |last1=Borel |first1=Armand |title=Intersection Cohomology |date=21 May 2009 |publisher=Springer Science & Business Media |isbn=978-0-8176-4765-0 |url=https://books.google.com/books?id=0c5HAAAAQBAJ&dq=borel+intersection+cohomology&pg=PA1}}
• {{citation | last1 = Bott | first1 = Raoul | authorlink = Raoul Bott | last2=Tu |first2= Loring | title = Differential Forms in Algebraic Topology | year = 1982 | publisher = Springer | isbn = 0-387-90613-4}}
• {{citation|title=Homotopy Limits, Completions and Localizations|volume=304|series=Lecture Notes in Mathematics|first1=A. K.|last1=Bousfield|first2=D. M.|last2=Kan|publisher=Springer|year=1987|isbn=9783540061052}}
• {{cite web |first1=James F. |last1=Davis |first2=Paul |last2=Kirk |url=http://www.maths.ed.ac.uk/~aar/papers/davkir.pdf |title=Lecture Notes in Algebraic Topology}}
• {{cite book |first=William |last=Fulton |title=Algebraic Topology: A First Course |url=https://books.google.com/books?id=8-OPBAAAQBAJ |date=2013 |publisher=Springer |isbn=978-1-4612-4180-5 }}
• {{cite web |first=Allen |last=Hatcher |url=http://pi.math.cornell.edu/~hatcher/AT/ATpage.html |title=Algebraic topology}}
• {{cite book

| last = Hess | first = Kathryn | author-link = Kathryn Hess

| arxiv = math/0604626

| contribution = Rational homotopy theory: a brief introduction

| doi = 10.1090/conm/436/08409

| isbn = 978-0-8218-3814-3

| mr = 2355774

| pages = 175–202

| publisher = American Mathematical Society | location = Providence, Rhode Island

| series = Contemporary Mathematics

| title = Interactions between homotopy theory and algebra

| volume = 436

| year = 2007}}

• {{cite web |url=https://math.berkeley.edu/~amathew/ATnotes.pdf |title=algebraic topology |date=Fall 2010 }} Lectures delivered by Michael Hopkins and Notes by Akhil Mathew, Harvard.
• {{cite web |last=Lurie |first=J. |authorlink=Jacob Lurie |url=http://www.math.harvard.edu/~lurie/281.html |title=Algebraic K-Theory and Manifold Topology |work=Math 281 |publisher=Harvard University |year=2015}}
• {{cite web |last=Lurie |first=J. |url=http://www.math.harvard.edu/~lurie/252x.html |title=Chromatic Homotopy Theory |work=252x |publisher=Harvard University |year=2011}}
• {{cite web |last=May |first=J. |url=http://www.math.uchicago.edu/~may/CONCISE/ConciseRevised.pdf |title=A Concise Course in Algebraic Topology}}
• {{cite web |last1=May |first1=J. |first2=K. |last2=Ponto |url=http://www.math.uchicago.edu/~may/TEAK/KateBookFinal.pdf |title=More concise algebraic topology: localization, completion, and model categories}}
• {{cite web |last1=May |last2=Sigurdsson |url=http://www.math.uchicago.edu/~may/EXTHEORY/MaySig.pdf |title=Parametrized homotopy theory}} (despite the title, it contains a significant amount of general results.)
• {{cite arXiv |last1=Rudyak |first1=Yuli B. |title=Piecewise linear structures on topological manifolds |date=23 December 2014 |eprint=math/0105047 }}
• {{cite web |authorlink=Dennis Sullivan |first=Dennis |last=Sullivan |url=http://www.maths.ed.ac.uk/~aar/books/gtop.pdf |title=Geometric Topology}} the 1970 MIT notes
• {{cite book|first=George William |last=Whitehead |authorlink=George W. Whitehead|title=Elements of homotopy theory|url=https://books.google.com/books?id=wlrvAAAAMAAJ |edition=3rd|series=Graduate Texts in Mathematics|volume=61|year=1978|publisher=Springer-Verlag |isbn=978-0-387-90336-1|pages=xxi+744|mr=0516508 }}
• {{cite web |first=Kirsten Graham |last=Wickelgren |authorlink=Kirsten Wickelgren|url=http://people.math.gatech.edu/~kwickelgren3/8803_Stable/ |title=8803 Stable Homotopy Theory}}

• José I. Burgos Gil, [https://www.icmat.es/miembros/burgos/files/brbr.pdf The Regulators of Beilinson and Borel]
• [https://math.mit.edu/juvitop/old/notes/2014_Fall/JPLEVINElectures.pdf 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]

• [http://pantodon.shinshu-u.ac.jp/topology/literature/ Algebraic Topology: A guide to literature] {{Webarchive|url=https://web.archive.org/web/20171217014101/http://pantodon.shinshu-u.ac.jp/topology/literature/ |date=2017-12-17 }}

Category:Algebraic topology

Algebraic topology

Category:Wikipedia glossaries using description lists