Glossary of algebraic topology

absolute neighborhood retract

abstract
1.  Abstract homotopy theory

Adams
1.  <a href="/facts/John_Frank_Adams/fxgpN9Xk">John Frank Adams</a>.
2.  The <a href="/facts/Adams_spectral_sequence/XPqI80V4">Adams spectral sequence</a>.
3.  The <a href="/facts/Adams_conjecture/AhQh93y8">Adams conjecture</a>.
4.  The <a href="/facts/Adams_e-invariant/AhQh93y8">Adams e-invariant</a>.
5.  The <a href="/facts/Adams_operations/HpWVycRE">Adams operations</a>.

<a href="/facts/Alexander_duality/ugNX3fH1">Alexander duality</a>
<a href="/facts/Alexander_duality/ugNX3fH1">Alexander duality</a>

<a href="/facts/Alexander_trick/fTsPL4aT">Alexander trick</a>
The <a href="/facts/Alexander_trick/fTsPL4aT">Alexander trick</a> produces a section of the restriction map 
 
 
 
 Top
 ⁡
 (
 
 D
 
 n
 +
 1
 
 
 )
 →
 Top
 ⁡
 (
 
 S
 
 n
 
 
 )
 
 
 {\displaystyle \operatorname {Top} (D^{n+1})\to \operatorname {Top} (S^{n})}
 
, Top denoting a <a href="/facts/Homeomorphism_group/oGNAzgWq">homeomorphism group</a>; namely, the section is given by sending a homeomorphism 
 
 
 
 f
 :
 
 S
 
 n
 
 
 →
 
 S
 
 n
 
 
 
 
 {\displaystyle f:S^{n}\to S^{n}}
 
 to the homeomorphism

f
 ~
 
 
 
 :
 
 D
 
 n
 +
 1
 
 
 →
 
 D
 
 n
 +
 1
 
 
 ,
 
 0
 ↦
 0
 ,
 0
 ≠
 x
 ↦
 
 |
 
 x
 
 |
 
 f
 (
 x
 
 /
 
 
 |
 
 x
 
 |
 
 )
 
 
 {\displaystyle {\widetilde {f}}:D^{n+1}\to D^{n+1},\,0\mapsto 0,0\neq x\mapsto |x|f(x/|x|)}
 
.
This section is in fact a homotopy inverse.<a class="footnote-ref" id="fnref:1" href="#fn:1">1</a>

Analysis Situs

<a href="/facts/Approximate_fibration/GSDDSpgX">approximate fibration</a>
1.  An <a href="/facts/Approximate_fibration/GSDDSpgX">approximate fibration</a>, a generalization of a fibration and a projection in a locally trivial bundle.
2.  A <a href="/facts/Manifold_approximate_fibration/GSDDSpgX">manifold approximate fibration</a> is a proper approximate fibration between manifolds.

aspherical space
<a href="/facts/Aspherical_space/M43BNIGF">Aspherical space</a>

assembly map

Atiyah
1.  <a href="/facts/Michael_Atiyah/RQVJ9ATE">Michael Atiyah</a>.
2.  Atiyah duality.
3.  The <a href="/facts/Atiyah%25E2%2580%2593Hirzebruch_spectral_sequence/2cwxuanD">Atiyah–Hirzebruch spectral sequence</a>.

bar construction

based space
A pair (X, x0) consisting of a space X and a point x0 in X.

Betti number
See <a href="/facts/Betti_number/r42Y8k6f">Betti number</a>.
Bing–Borsuk conjecture
See <a href="/facts/Bing%25E2%2580%2593Borsuk_conjecture/8ybNHTfn">Bing–Borsuk conjecture</a>.

Bockstein homomorphism

Borel
<a href="/facts/Borel_conjecture/hNYGPhKB">Borel conjecture</a>.

Borel–Moore homology

Borsuk's theorem

Bott
1.  <a href="/facts/Raoul_Bott/5JHkhPx8">Raoul Bott</a>.
2.  The <a href="/facts/Bott_periodicity_theorem/GqgJRllU">Bott periodicity theorem</a> for unitary groups say: 
 
 
 
 
 π
 
 q
 
 
 U
 =
 
 π
 
 q
 +
 2
 
 
 U
 ,
 q
 ≥
 0
 
 
 {\displaystyle \pi _{q}U=\pi _{q+2}U,q\geq 0}
 
.
3.  The <a href="/facts/Bott_periodicity_theorem/GqgJRllU">Bott periodicity theorem</a> for orthogonal groups say: 
 
 
 
 
 π
 
 q
 
 
 O
 =
 
 π
 
 q
 +
 8
 
 
 O
 ,
 q
 ≥
 0
 
 
 {\displaystyle \pi _{q}O=\pi _{q+8}O,q\geq 0}
 
.

Brouwer fixed-point theorem
The <a href="/facts/Brouwer_fixed-point_theorem/xP2p03lQ">Brouwer fixed-point theorem</a> says that any map 
 
 
 
 f
 :
 
 D
 
 n
 
 
 →
 
 D
 
 n
 
 
 
 
 {\displaystyle f:D^{n}\to D^{n}}
 
 has a fixed point.

cap product

Casson
<a href="/facts/Casson_invariant/uRs2DNPp">Casson invariant</a>.

Čech cohomology

cellular
1.  A map ƒ:X→Y between CW complexes is <a href="/facts/Cellular_map/KYLg2rwL">cellular</a> if 
 
 
 
 f
 (
 
 X
 
 n
 
 
 )
 ⊂
 
 Y
 
 n
 
 
 
 
 {\displaystyle f(X^{n})\subset Y^{n}}
 
 for all n.
2.  The <a href="/facts/Cellular_approximation_theorem/KYLg2rwL">cellular approximation theorem</a> says that every map between CW complexes is homotopic to a <a href="/facts/Cellular_map/KYLg2rwL">cellular map</a> between them.
3.  The <a href="/facts/Cellular_homology/B4BfgxtM">cellular homology</a> 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 
 
 
 
 f
 ,
 g
 :
 (
 C
 ,
 
 d
 
 C
 
 
 )
 →
 (
 D
 ,
 
 d
 
 D
 
 
 )
 
 
 {\displaystyle f,g:(C,d_{C})\to (D,d_{D})}
 
 between chain complexes of modules, a <a href="/facts/Chain_homotopy/FrGBMJxd">chain homotopy</a> s from f to g is a sequence of module homomorphisms 
 
 
 
 
 s
 
 i
 
 
 :
 
 C
 
 i
 
 
 →
 
 D
 
 i
 +
 1
 
 
 
 
 {\displaystyle s_{i}:C_{i}\to D_{i+1}}
 
 satisfying 
 
 
 
 
 f
 
 i
 
 
 −
 
 g
 
 i
 
 
 =
 
 d
 
 D
 
 
 ∘
 
 s
 
 i
 
 
 +
 
 s
 
 i
 −
 1
 
 
 ∘
 
 d
 
 C
 
 
 
 
 {\displaystyle f_{i}-g_{i}=d_{D}\circ s_{i}+s_{i-1}\circ d_{C}}
 
. It is also called a <a href="/facts/Homotopy_operator/FrGBMJxd">homotopy operator</a>.

chain map
A chain map 
  
    
      
        f
        :
        (
        C
        ,
        
          d
          
            C
          
        
        )
        →
        (
        D
        ,
        
          d
          
            D
          
        
        )
      
    
    {\displaystyle f:(C,d_{C})\to (D,d_{D})}
  
 between chain complexes of modules is a sequence of module homomorphisms 
  
    
      
        
          f
          
            i
          
        
        :
        
          C
          
            i
          
        
        →
        
          D
          
            i
          
        
      
    
    {\displaystyle f_{i}:C_{i}\to D_{i}}
  
 that commutes with the differentials; i.e., 
  
    
      
        
          d
          
            D
          
        
        ∘
        
          f
          
            i
          
        
        =
        
          f
          
            i
            −
            1
          
        
        ∘
        
          d
          
            C
          
        
      
    
    {\displaystyle d_{D}\circ f_{i}=f_{i-1}\circ d_{C}}
  
.

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 <a href="/facts/Change_of_fiber/qCbwHlVL">change of fiber</a> 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 <a href="/facts/Character_variety/XIRJ6koa">character variety</a><a class="footnote-ref" id="fnref:2" href="#fn:2">2</a> of a group π and an algebraic group G (e.g., a reductive complex Lie group) is the <a href="/facts/Geometric_invariant_theory_quotient/BGS7ASbx">geometric invariant theory quotient</a> by G:

X
          
        
        (
        π
        ,
        G
        )
        =
        Hom
        ⁡
        (
        π
        ,
        G
        )
        
          /
        
        
        
          /
        
        G
      
    
    {\displaystyle {\mathcal {X}}(\pi ,G)=\operatorname {Hom} (\pi ,G)/\!/G}
  
.

characteristic class
Let Vect(X) be the set of isomorphism classes of vector bundles on X. We can view 
 
 
 
 X
 ↦
 Vect
 ⁡
 (
 X
 )
 
 
 {\displaystyle X\mapsto \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 <a href="/facts/Characteristic_class/A6NV9tUu">characteristic class</a> is a <a href="/facts/Natural_transformation/xwPumTHt">natural transformation</a> 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).

chromatic homotopy theory
<a href="/facts/Chromatic_homotopy_theory/gtcvY7LU">chromatic homotopy theory</a>.

class
1.  <a href="/facts/Chern_class/Tw2WD5hM">Chern class</a>.
2.  <a href="/facts/Stiefel%25E2%2580%2593Whitney_class/H3f972We">Stiefel–Whitney class</a>.

classifying space
Loosely speaking, a <a href="/facts/Classifying_space/gGIAmhtL">classifying space</a> is a space representing some contravariant functor defined on the category of spaces; for example, 
 
 
 
 B
 U
 
 
 {\displaystyle BU}
 
 is the classifying space in the sense 
 
 
 
 [
 −
 ,
 B
 U
 ]
 
 
 {\displaystyle [-,BU]}
 
 is the functor 
 
 
 
 X
 ↦
 
 Vect
 
 
 R
 
 
 
 ⁡
 (
 X
 )
 
 
 {\displaystyle X\mapsto \operatorname {Vect} ^{\mathbb {R} }(X)}
 
 that sends a space to the set of isomorphism classes of real vector bundles on the space.

clutching

cobar spectral sequence

cobordism
1.  See <a href="/facts/Cobordism/ctN2FSTB">cobordism</a>.
2.  A <a href="/facts/Cobordism_ring/rwKC7qme">cobordism ring</a> is a ring whose elements are cobordism classes.
3.  See also <a href="/facts/H-cobordism_theorem/gh4z1VVW">h-cobordism theorem</a>, <a href="/facts/S-cobordism_theorem/gh4z1VVW">s-cobordism theorem</a>.

coefficient ring
If E is a ring spectrum, then the coefficient ring of it is the ring 
 
 
 
 
 π
 
 ∗
 
 
 E
 
 
 {\displaystyle \pi _{*}E}
 
.

cofiber sequence
A cofiber sequence is any sequence that is equivalent to the sequence 
  
    
      
        X
        
          
            →
            f
          
        
        Y
        →
        
          C
          
            f
          
        
      
    
    {\displaystyle X{\overset {f}{\to }}Y\to C_{f}}
  
 for some ƒ where 
  
    
      
        
          C
          
            f
          
        
      
    
    {\displaystyle C_{f}}
  
 is the reduced mapping cone of ƒ (called the cofiber of ƒ).

cofibrant approximation

cofibration
A map 
 
 
 
 i
 :
 A
 →
 B
 
 
 {\displaystyle i:A\to B}
 
 is a <a href="/facts/Cofibration/NCWE6Cbt">cofibration</a> if it satisfies the property: given 
 
 
 
 
 h
 
 0
 
 
 :
 B
 →
 X
 
 
 {\displaystyle h_{0}:B\to X}
 
 and homotopy 
 
 
 
 
 g
 
 t
 
 
 :
 A
 →
 X
 
 
 {\displaystyle g_{t}:A\to X}
 
 such that 
 
 
 
 
 g
 
 0
 
 
 =
 
 h
 
 0
 
 
 ∘
 i
 
 
 {\displaystyle g_{0}=h_{0}\circ i}
 
, there is a homotopy 
 
 
 
 
 h
 
 t
 
 
 :
 B
 →
 X
 
 
 {\displaystyle h_{t}:B\to X}
 
 such that 
 
 
 
 
 h
 
 t
 
 
 ∘
 i
 =
 
 g
 
 t
 
 
 
 
 {\displaystyle h_{t}\circ i=g_{t}}
 
.<a class="footnote-ref" id="fnref:3" href="#fn:3">3</a> A cofibration is injective and is a homeomorphism onto its image.

coherent homotopy

coherency
See <a href="/facts/Coherency_(homotopy_theory)/ezWC0V7p">coherency (homotopy theory)</a>

cohomotopy group
For a based space X, the set of homotopy classes 
 
 
 
 [
 X
 ,
 
 S
 
 n
 
 
 ]
 
 
 {\displaystyle [X,S^{n}]}
 
 is called the n-th <a href="/facts/Cohomotopy_group/BrUb4TId">cohomotopy group</a> of X.

cohomology operation

collapse
An informal phrase but usually means taking a quotient; e.g., a cone is obtained by collapsing the top (or bottom) of a cylinder.

completion

complex bordism

complex-oriented
A multiplicative cohomology theory E is <a href="/facts/Complex-oriented_cohomology_theory/GQqDCpML">complex-oriented</a> if the restriction map E2(CP∞) → E2(CP1) is surjective.

concordant

cone
The <a href="/facts/Cone_(topology)/nUGIb0uE">cone</a> over a space X is 
 
 
 
 C
 X
 =
 X
 ×
 I
 
 /
 
 X
 ×
 {
 0
 }
 
 
 {\displaystyle CX=X\times I/X\times \{0\}}
 
. The <a href="/facts/Reduced_cone/nUGIb0uE">reduced cone</a> is obtained from the reduced cylinder 
 
 
 
 X
 ∧
 
 I
 
 +
 
 
 
 
 {\displaystyle X\wedge I_{+}}
 
 by collapsing the top.

connective
A spectrum E is <a href="/facts/Connective_spectrum/VvKBoEMT">connective</a> if 
 
 
 
 
 π
 
 q
 
 
 E
 =
 0
 
 
 {\displaystyle \pi _{q}E=0}
 
 for all negative integers q.

configuration space

constant
A <a href="/facts/Constant_sheaf/RfrbWp81">constant sheaf</a> on a space X is a sheaf 
 
 
 
 
 
 F
 
 
 
 
 {\displaystyle {\mathcal {F}}}
 
 on X such that for some set A and some map 
 
 
 
 A
 →
 
 
 F
 
 
 (
 X
 )
 
 
 {\displaystyle A\to {\mathcal {F}}(X)}
 
, the natural map 
 
 
 
 A
 →
 
 
 F
 
 
 (
 X
 )
 →
 
 
 
 F
 
 
 
 x
 
 
 
 
 {\displaystyle A\to {\mathcal {F}}(X)\to {\mathcal {F}}_{x}}
 
 is bijective for any x in X.

continuous
Continuous cohomology.

contractible space
A space is <a href="/facts/Contractible_space/REXYj8uS">contractible</a> if the identity map on the space is homotopic to the constant map.

covering
1.  A map p: Y → X is a <a href="/facts/Covering_space/bDRGasl7">covering</a> 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.
2.  It is n-sheeted if each fiber p−1(x) has exactly n elements.
3.  It is <a href="/facts/Universal_covering/bDRGasl7">universal</a> if Y is simply connected.
4.  A morphism of a covering is a map over X. In particular, an automorphism of a covering p:Y→X (also called a <a href="/facts/Deck_transformation/bDRGasl7">deck transformation</a>) is a map Y→Y over X that has inverse; i.e., a homeomorphism over X.
5.  A G-covering is a covering arising from a <a href="/facts/Group_action_(mathematics)/Vh9VtUUw">group action</a> on a space X by a group G, the covering map being the quotient map from X to the <a href="/facts/Orbit_space/Vh9VtUUw">orbit space</a> X/G. The notion is used to state the universal property: if X admits a universal covering (in particular connected), then

Hom
 ⁡
 (
 
 π
 
 1
 
 
 (
 X
 ,
 
 x
 
 0
 
 
 )
 ,
 G
 )
 
 
 {\displaystyle \operatorname {Hom} (\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 
 
 
 
 Hom
 ⁡
 (
 
 π
 
 1
 
 
 (
 X
 ,
 
 x
 
 0
 
 
 )
 ,
 G
 )
 =
 
 H
 
 1
 
 
 ⁡
 (
 X
 ;
 G
 )
 
 
 {\displaystyle \operatorname {Hom} (\pi _{1}(X,x_{0}),G)=\operatorname {H} ^{1}(X;G)}
 
 (cf. <a href="/facts/Nonabelian_cohomology/FvaB41Cy">nonabelian cohomology</a>.)

cup product

CW complex
A <a href="/facts/CW_complex/rN7buLMo">CW complex</a> is a space X equipped with a CW structure; i.e., a filtration

X
          
            0
          
        
        ⊂
        
          X
          
            1
          
        
        ⊂
        
          X
          
            2
          
        
        ⊂
        ⋯
        ⊂
        X
      
    
    {\displaystyle X^{0}\subset X^{1}\subset X^{2}\subset \cdots \subset X}

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

cyclic homology

deck transformation
Another term for an automorphism of a covering.

deformation retract
A subspace 
 
 
 
 A
 ⊂
 X
 
 
 {\displaystyle A\subset X}
 
 is called a <a href="/facts/Deformation_retract/477Gm67f">deformation retract</a> of X if there is a homotopy 
 
 
 
 
 h
 
 t
 
 
 :
 X
 →
 X
 
 
 {\displaystyle h_{t}:X\to X}
 
 such that 
 
 
 
 
 h
 
 0
 
 
 
 
 {\displaystyle h_{0}}
 
 is the identity, 
 
 
 
 
 h
 
 1
 
 
 (
 X
 )
 ⊂
 A
 
 
 {\displaystyle h_{1}(X)\subset A}
 
 and 
 
 
 
 
 
 h
 
 1
 
 
 
 
 
 |
 
 
 A
 
 
 
 
 {\displaystyle {h_{1}}|_{A}}
 
 is the identity (i.e., 
 
 
 
 
 h
 
 1
 
 
 
 
 {\displaystyle h_{1}}
 
 is a <a href="/facts/Retract_(category_theory)/Ee1wexPw">retract</a> of 
 
 
 
 A
 ↪
 X
 
 
 {\displaystyle A\hookrightarrow X}
 
 in the sense in category theory). It is called a <a href="/facts/Strong_deformation_retract/477Gm67f">strong deformation retract</a> if, in addition, 
 
 
 
 
 h
 
 t
 
 
 
 
 {\displaystyle h_{t}}
 
 satisfies the requirement that 
 
 
 
 
 
 h
 
 t
 
 
 
 
 
 |
 
 
 A
 
 
 
 
 {\displaystyle {h_{t}}|_{A}}
 
 is the identity. For example, a homotopy 
 
 
 
 
 h
 
 t
 
 
 :
 B
 →
 B
 ,
 
 x
 ↦
 (
 1
 −
 t
 )
 x
 
 
 {\displaystyle h_{t}:B\to B,\,x\mapsto (1-t)x}
 
 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.  <a href="/facts/De_Rham_cohomology/4qGKeJq0">de Rham cohomology</a>, the cohomology of complex of differential forms.
2.  The <a href="/facts/De_Rham_theorem/4qGKeJq0">de Rham theorem</a> gives an explicit isomorphism between the de Rham cohomology and the singular cohomology.

Dold
The <a href="/facts/Dold%25E2%2580%2593Thom_theorem/pSe6z5Hf">Dold–Thom theorem</a>.

Eckmann–Hilton argument
The <a href="/facts/Eckmann%25E2%2580%2593Hilton_argument/5VRxTYAW">Eckmann–Hilton argument</a>.

Eckmann–Hilton duality

Eilenberg–MacLane spaces
Given an abelian group π, the <a href="/facts/Eilenberg%25E2%2580%2593MacLane_spaces/mrux6oMJ">Eilenberg–MacLane spaces</a> 
 
 
 
 K
 (
 π
 ,
 n
 )
 
 
 {\displaystyle K(\pi ,n)}
 
 are characterized by

π
          
            q
          
        
        K
        (
        π
        ,
        n
        )
        =
        
          
            {
            
              
                
                  π
                
                
                  
                    if 
                  
                  q
                  =
                  n
                
              
              
                
                  0
                
                
                  
                    otherwise
                  
                
              
            
            
          
        
      
    
    {\displaystyle \pi _{q}K(\pi ,n)={\begin{cases}\pi &{\text{if }}q=n\\0&{\text{otherwise}}\end{cases}}}
  
.

Eilenberg–Steenrod axioms
The <a href="/facts/Eilenberg%25E2%2580%2593Steenrod_axioms/2c42f4He">Eilenberg–Steenrod axioms</a> 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 <a href="/facts/Generalized_cohomology_theory/J7FJzTNh">generalized cohomology theory</a>.

Eilenberg–Zilber theorem

elliptic
<a href="/facts/Elliptic_cohomology/Y0nSafzr">elliptic cohomology</a>.

En-algebra

equivariant algebraic topology
<a href="/facts/Equivariant_algebraic_topoloy/01HreDnt">Equivariant algebraic topoloy</a> is the study of spaces with <a href="/facts/Continuous_group_action/jB9QBrbx">(continuous) group action</a>.

etale
<a href="/facts/%25C3%2589tale_homotopy/QN598o1X">étale homotopy</a>.

exact
A sequence of pointed sets 
 
 
 
 X
 
 
 →
 f
 
 
 Y
 
 
 →
 g
 
 
 Z
 
 
 {\displaystyle X{\overset {f}{\to }}Y{\overset {g}{\to }}Z}
 
 is <a href="/facts/Exact_sequence/leUJnwJb">exact</a> if the image of f coincides with the pre-image of the chosen point of Z.

excision
The <a href="/facts/Excision_theorem/GCH1wz2c">excision</a> axiom for homology says: if 
 
 
 
 U
 ⊂
 X
 
 
 {\displaystyle U\subset X}
 
 and 
 
 
 
 
 
 U
 ¯
 
 
 ⊂
 int
 ⁡
 (
 A
 )
 
 
 {\displaystyle {\overline {U}}\subset \operatorname {int} (A)}
 
, then for each q,

H
          
            q
          
        
        ⁡
        (
        X
        −
        U
        ,
        A
        −
        U
        )
        →
        
          H
          
            q
          
        
        ⁡
        (
        X
        ,
        A
        )
      
    
    {\displaystyle \operatorname {H} _{q}(X-U,A-U)\to \operatorname {H} _{q}(X,A)}

is an isomorphism.

excisive pair/triad

factorization homology

fiber-homotopy equivalence
Given D→B, E→B, a map ƒ:D→E over B is a <a href="/facts/Fiber-homotopy_equivalence/LESXbm8e">fiber-homotopy equivalence</a> 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 <a href="/facts/Fiber_sequence/qjUjNpGq">fiber sequence</a> of a map 
 
 
 
 f
 :
 X
 →
 Y
 
 
 {\displaystyle f:X\to Y}
 
 is the sequence 
 
 
 
 
 F
 
 f
 
 
 
 
 →
 p
 
 
 X
 
 
 →
 f
 
 
 Y
 
 
 {\displaystyle F_{f}{\overset {p}{\to }}X{\overset {f}{\to }}Y}
 
 where 
 
 
 
 
 F
 
 f
 
 
 
 
 →
 p
 
 
 X
 
 
 {\displaystyle F_{f}{\overset {p}{\to }}X}
 
 is the homotopy fiber of f; i.e., the pullback of the path space fibration 
 
 
 
 P
 Y
 →
 Y
 
 
 {\displaystyle PY\to Y}
 
 along f.

fiber square
<a href="/facts/Fiber_square/SCSzvgxx">fiber square</a>

fibration
A map p:E → B is a <a href="/facts/Fibration/h51Cx7xU">fibration</a> if for any given homotopy 
 
 
 
 
 g
 
 t
 
 
 :
 X
 →
 B
 
 
 {\displaystyle g_{t}:X\to B}
 
 and a map 
 
 
 
 
 h
 
 0
 
 
 :
 X
 →
 E
 
 
 {\displaystyle h_{0}:X\to E}
 
 such that 
 
 
 
 p
 ∘
 
 h
 
 0
 
 
 =
 
 g
 
 0
 
 
 
 
 {\displaystyle p\circ h_{0}=g_{0}}
 
, there exists a homotopy 
 
 
 
 
 h
 
 t
 
 
 :
 X
 →
 E
 
 
 {\displaystyle h_{t}:X\to E}
 
 such that 
 
 
 
 p
 ∘
 
 h
 
 t
 
 
 =
 
 g
 
 t
 
 
 
 
 {\displaystyle p\circ h_{t}=g_{t}}
 
. (The above property is called the <a href="/facts/Homotopy_lifting_property/w1cMx4Ko">homotopy lifting property</a>.) A covering map is a basic example of a fibration.

fibration sequence
One says 
 
 
 
 F
 →
 X
 
 
 →
 p
 
 
 B
 
 
 {\displaystyle F\to X{\overset {p}{\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.

finitely dominated

fundamental class

fundamental group
The <a href="/facts/Fundamental_group/VyYtLqSP">fundamental group</a> 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 
 
 
 
 
 π
 
 1
 
 
 (
 X
 ,
 
 x
 
 0
 
 
 )
 
 
 {\displaystyle \pi _{1}(X,x_{0})}
 
.

fundamental groupoid
The <a href="/facts/Fundamental_groupoid/gt44ExMY">fundamental groupoid</a> 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 
 
 
 
 
 π
 
 1
 
 
 (
 X
 ,
 
 x
 
 0
 
 
 )
 
 
 {\displaystyle \pi _{1}(X,x_{0})}
 
.

framed
A <a href="/facts/Framed_manifold/CQq7nQsJ">framed manifold</a> is a manifold with a framing.

free
Synonymous with unbased. For example, the <a href="/facts/Free_path_space/7JFfGWQx">free path space</a> of a space X refers to the space of all maps from I to X; i.e., 
 
 
 
 
 X
 
 I
 
 
 
 
 {\displaystyle 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).

Freudenthal suspension theorem
For a nondegenerately based space X, the <a href="/facts/Freudenthal_suspension_theorem/qGCDXhgl">Freudenthal suspension theorem</a> says: if X is (n-1)-connected, then the suspension homomorphism

π
          
            q
          
        
        X
        →
        
          π
          
            q
            +
            1
          
        
        Σ
        X
      
    
    {\displaystyle \pi _{q}X\to \pi _{q+1}\Sigma X}

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

Fulton–MacPherson compactification
The <a href="/facts/Fulton%25E2%2580%2593MacPherson_compactification/Te8RASyR">Fulton–MacPherson compactification</a> of the <a href="/facts/Configuration_space_(mathematics)/qBdhCN3z">configuration space</a> of n distinct labeled points in a compact complex manifold is a natural smooth compactification introduced by Fulton and MacPherson.

G-fibration
A <a href="/facts/G-fibration/drT50zxf">G-fibration</a> with some <a href="/facts/Topological_monoid/nJ2XlAuh">topological monoid</a> G. An example is <a href="/facts/Moore%2527s_path_space_fibration/7JFfGWQx">Moore's path space fibration</a>.

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 <a href="/facts/Generalized_cohomology_theory/J7FJzTNh">generalized cohomology theory</a> 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
<a href="/facts/Geometrization_conjecture/FDpqIM7y">geometrization conjecture</a>

genus

germ
<a href="/facts/Germ_(mathematics)/g28mtBu5">germ</a>

group completion 
grouplike
An H-space X is said to be group-like or grouplike if 
 
 
 
 
 π
 
 0
 
 
 X
 
 
 {\displaystyle \pi _{0}X}
 
 is a <a href="/facts/Group_(mathematics)/IQTYqINt">group</a>; i.e., X satisfies the group axioms up to homotopy.

Gysin sequence

Hauptvermutung
1.  <a href="/facts/Hauptvermutung/y7gRbjGy">Hauptvermutung</a>, 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
<a href="/facts/H-cobordism/gh4z1VVW">h-cobordism</a>.

Hilton–Milnor theorem
The <a href="/facts/Hilton%25E2%2580%2593Milnor_theorem/u8BkM3W9">Hilton–Milnor theorem</a>.

Hirzebruch
<a href="/facts/Hirzebruch_signature_theorem/uDcIGALT">Hirzebruch signature theorem</a>.

H-space
An <a href="/facts/H-space/v5FRJX8M">H-space</a> is a based space that is a <a href="/facts/Unital_magma/9g3vp7Wl">unital magma</a> up to homotopy.

homologus
Two cycles are homologus if they belong to the same homology class.

homology sphere
A <a href="/facts/Homology_sphere/8XnrYQ1T">homology sphere</a> 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 <a href="/facts/Homotopy_category/JaEnPakj">homotopy category</a> 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 <a href="/facts/Homotopy_colimit/wYTpWPCV">homotopy colimit</a> 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 <a href="/facts/Homotopy_equivalence/1nWrPxdj">homotopy equivalence</a> 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 <a href="/facts/Point_space/tlvI3oX1">point space</a>.

homotopy excision theorem
The <a href="/facts/Homotopy_excision_theorem/5Cs89DN0">homotopy excision theorem</a> is a substitute for the failure of excision for homotopy groups.

homotopy fiber
The <a href="/facts/Homotopy_fiber/hH4omFFQ">homotopy fiber</a> of a based map ƒ:X→Y, denoted by Fƒ, is the pullback of 
 
 
 
 P
 Y
 →
 Y
 ,
 
 χ
 ↦
 χ
 (
 1
 )
 
 
 {\displaystyle PY\to Y,\,\chi \mapsto \chi (1)}
 
 along f.

homotopy fiber product
A fiber product is a particular kind of a <a href="/facts/Limit_(category_theory)/JoLugLs3">limit</a>. Replacing this limit lim with a <a href="/facts/Homotopy_limit/wYTpWPCV">homotopy limit</a> holim yields a homotopy fiber product.

homotopy group
1.  For a based space X, let 
 
 
 
 
 π
 
 n
 
 
 X
 =
 [
 
 S
 
 n
 
 
 ,
 X
 ]
 
 
 {\displaystyle \pi _{n}X=[S^{n},X]}
 
, the set of homotopy classes of based maps. Then 
 
 
 
 
 π
 
 0
 
 
 X
 
 
 {\displaystyle \pi _{0}X}
 
 is the set of path-connected components of X, 
 
 
 
 
 π
 
 1
 
 
 X
 
 
 {\displaystyle \pi _{1}X}
 
 is the fundamental group of X and 
 
 
 
 
 π
 
 n
 
 
 X
 ,
 
 n
 ≥
 2
 
 
 {\displaystyle \pi _{n}X,\,n\geq 2}
 
 are the (higher) n-th <a href="/facts/Homotopy_group/xjogL95Y">homotopy groups</a> of X.
2.  For based spaces 
 
 
 
 A
 ⊂
 X
 
 
 {\displaystyle A\subset X}
 
, the <a href="/facts/Relative_homotopy_group/xjogL95Y">relative homotopy group</a> 
 
 
 
 
 π
 
 n
 
 
 (
 X
 ,
 A
 )
 
 
 {\displaystyle \pi _{n}(X,A)}
 
 is defined as 
 
 
 
 
 π
 
 n
 −
 1
 
 
 
 
 {\displaystyle \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 
 
 
 
 
 π
 
 n
 −
 1
 
 
 
 
 {\displaystyle \pi _{n-1}}
 
 of the homotopy fiber of 
 
 
 
 A
 ↪
 X
 
 
 {\displaystyle A\hookrightarrow X}
 
.
3.  If E is a spectrum, then 
 
 
 
 
 π
 
 k
 
 
 E
 =
 
 
 
 lim
 →
 
 
 
 n
 
 
 ⁡
 
 π
 
 k
 +
 n
 
 
 
 E
 
 n
 
 
 .
 
 
 {\displaystyle \pi _{k}E=\varinjlim _{n}\pi _{k+n}E_{n}.}

4.  If X is a based space, then the <a href="/facts/Stable_homotopy_group/wnedyf7K">stable k-th homotopy group</a> of X is 
 
 
 
 
 π
 
 k
 
 
 s
 
 
 X
 =
 
 
 
 lim
 →
 
 
 
 n
 
 
 ⁡
 
 π
 
 k
 +
 n
 
 
 
 Σ
 
 n
 
 
 X
 
 
 {\displaystyle \pi _{k}^{s}X=\varinjlim _{n}\pi _{k+n}\Sigma ^{n}X}
 
. In other words, it is the k-th homotopy group of the suspension spectrum of X.

homotopy pullback
A <a href="/facts/Homotopy_pullback/wYTpWPCV">homotopy pullback</a> is a special case of a homotopy limit that is a homotopically-correct pullback.

homotopy quotient
If G is a <a href="/facts/Lie_group/a9jCQyjF">Lie group</a> acting on a manifold X, then the quotient space 
 
 
 
 (
 E
 G
 ×
 X
 )
 
 /
 
 G
 
 
 {\displaystyle (EG\times X)/G}
 
 is called the <a href="/facts/Homotopy_quotient/N6zTy9fn">homotopy quotient</a> (or Borel construction) of X by G, where EG is the universal bundle of G.

homotopy spectral sequence

homotopy sphere
A <a href="/facts/Homotopy_sphere/17cB1EkD">homotopy sphere</a> is a manifold having the homotopy type of a sphere.

Hopf
1.  <a href="/facts/Heinz_Hopf/PMks10NV">Heinz Hopf</a>.
2.  <a href="/facts/Hopf_invariant/6MCJDeDC">Hopf invariant</a>.
3.  The <a href="/facts/Hopf_index_theorem/IV1aG75X">Hopf index theorem</a>.
4.  <a href="/facts/Hopf_construction/N9j4XSEf">Hopf construction</a>.

Hurewicz
The <a href="/facts/Hurewicz_theorem/rHClbqTd">Hurewicz theorem</a> establishes a relationship between homotopy groups and homology groups.

infinite loop space

infinite loop space machine
<a href="/facts/Infinite_loop_space_machine/3tM95ubb">Infinite loop space machine</a>.

infinite mapping telescope

intersection
intersection pairing.
<a href="/facts/Intersection_homology/XnBz8gmT">intersection homology</a>, a substitute for an ordinary (singular) homology for a singular space.
<a href="/facts/Intersection_cohomology/XnBz8gmT">intersection cohomology</a>

integration along the fiber
See <a href="/facts/Integration_along_the_fiber/BPpfJ8HD">integration along the fiber</a>.

invariance of domain
<a href="/facts/Invariance_of_domain/n1uFhWRY">invariance of domain</a>.

isotopy

J-homomorphism
See <a href="/facts/J-homomorphism/AhQh93y8">J-homomorphism</a>.

join
The <a href="/facts/Join_(topology)/81UQka6h">join</a> of based spaces X, Y is 
 
 
 
 X
 ⋆
 Y
 =
 Σ
 (
 X
 ∧
 Y
 )
 .
 
 
 {\displaystyle X\star Y=\Sigma (X\wedge Y).}

k-invariant

Kan complex
See <a href="/facts/Kan_complex/TWxx0fWK">Kan complex</a>.

Kirby–Siebenmann
Kirby–Siebenmann classification.

Kervaire invariant
The <a href="/facts/Kervaire_invariant/vQRoPxz1">Kervaire invariant</a>.

Koszul duality
<a href="/facts/Koszul_duality/rFGMWpCp">Koszul duality</a>.

Kuiper
<a href="/facts/Kuiper%2527s_theorem/1nDafxlY">Kuiper's theorem</a> says that the general linear group of an infinite-dimensional Hilbert space is contractible.

Künneth formula

Lazard ring
The <a href="/facts/Lazard_ring/VY8LzJCo">Lazard ring</a> L is the (huge) commutative ring together with the <a href="/facts/Formal_group_law/xUnVSCC5">formal group law</a> 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 <a href="/facts/Complex_bordism/tAkFKIL1">complex bordism</a> MU. The <a href="/facts/Spectrum_of_a_ring/2k2nDE5w">Spec</a> of L is called the <a href="/facts/Moduli_space_of_formal_group_laws/gQLS8XYz">moduli space of formal group laws</a>.

Lefschetz
1.  <a href="/facts/Solomon_Lefschetz/Ut2VFwGr">Solomon Lefschetz</a>
2.  The <a href="/facts/Lefschetz_fixed-point_theorem/TEjN2DQ8">Lefschetz fixed-point theorem</a> says: given a finite simplicial complex K and its geometric realization X, if a map 
 
 
 
 f
 :
 X
 →
 X
 
 
 {\displaystyle f:X\to X}
 
 has no fixed point, then the Lefschetz number of f; that is,

∑
          
            0
          
          
            ∞
          
        
        (
        −
        1
        
          )
          
            q
          
        
        tr
        ⁡
        (
        
          f
          
            ∗
          
        
        :
        
          H
          
            q
          
        
        ⁡
        (
        X
        )
        →
        
          H
          
            q
          
        
        ⁡
        (
        X
        )
        )
      
    
    {\displaystyle \sum _{0}^{\infty }(-1)^{q}\operatorname {tr} (f_{*}:\operatorname {H} _{q}(X)\to \operatorname {H} _{q}(X))}

is zero. For example, it implies the <a href="/facts/Brouwer_fixed-point_theorem/xP2p03lQ">Brouwer fixed-point theorem</a> since the Lefschetz number of 
 
 
 
 f
 :
 
 D
 
 n
 
 
 →
 
 D
 
 n
 
 
 
 
 {\displaystyle f:D^{n}\to D^{n}}
 
 is, as higher homologies vanish, one.
3.  The <a href="/facts/Lefschetz_hyperplane_theorem/GzH8FaHG">Lefschetz hyperplane theorem</a>.

lens space
The <a href="/facts/Lens_space/5hQ8dgA5">lens space</a> is the quotient space 
 
 
 
 {
 z
 ∈
 
 
 C
 
 
 n
 
 
 
 |
 
 
 |
 
 z
 
 |
 
 =
 1
 }
 
 /
 
 
 μ
 
 p
 
 
 
 
 {\displaystyle \{z\in \mathbb {C} ^{n}||z|=1\}/\mu _{p}}
 
 where 
 
 
 
 
 μ
 
 p
 
 
 
 
 {\displaystyle \mu _{p}}
 
 is the group of p-th roots of unity acting on the unit sphere by 
 
 
 
 ζ
 ⋅
 (
 
 z
 
 1
 
 
 ,
 …
 ,
 
 z
 
 n
 
 
 )
 =
 (
 ζ
 
 z
 
 1
 
 
 ,
 …
 ,
 ζ
 
 z
 
 n
 
 
 )
 
 
 {\displaystyle \zeta \cdot (z_{1},\dots ,z_{n})=(\zeta z_{1},\dots ,\zeta z_{n})}
 
.

Leray spectral sequence

L2
The <a href="/facts/L2-cohomology/PntXCWzS">L2-cohomology</a> of a <a href="/facts/Riemannian_manifold/5KYnmEgF">Riemannian</a> or <a href="/facts/K%25C3%25A4hler_manifold/JFPM6Vxg">Kähler manifold</a> is the cohomology of the complexes of differential forms with <a href="/facts/Square-integrable/IASblcJJ">square-integrable</a> coefficients (coefficients for forms not cohomology).

local coefficient
1.  A module over the <a href="/facts/Group_ring/TJlkt5Te">group ring</a> 
 
 
 
 
 Z
 
 [
 
 π
 
 1
 
 
 B
 ]
 
 
 {\displaystyle \mathbb {Z} [\pi _{1}B]}
 
 for some based space B; in other words, an abelian group together with a homomorphism 
 
 
 
 
 π
 
 1
 
 
 B
 →
 Aut
 ⁡
 (
 A
 )
 
 
 {\displaystyle \pi _{1}B\to \operatorname {Aut} (A)}
 
.
2.  The <a href="/facts/Local_coefficient_system/URIcDzNk">local coefficient system</a> 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 
 
 
 
 
 
 
 B
 ~
 
 
 
 
 
 {\displaystyle {\widetilde {B}}}
 
, then this meaning coincides with that of 1. in the sense: every local coefficient system over B can be given as the <a href="/facts/Associated_bundle/7JfKDcBJ">associated bundle</a> 
 
 
 
 
 
 
 B
 ~
 
 
 
 
 ×
 
 
 π
 
 1
 
 
 B
 
 
 A
 
 
 {\displaystyle {\widetilde {B}}\times _{\pi _{1}B}A}
 
.

local invariant
<a href="/facts/Local_invariant_cycle_theorem/UwBdAKlc">Local invariant cycle theorem</a>.

local sphere
The localization of a sphere at some prime number

local system
<a href="/facts/Local_system/URIcDzNk">local system</a>.

localization

locally constant sheaf
A <a href="/facts/Locally_constant_sheaf/mW64LDW9">locally constant sheaf</a> on a space X is a sheaf such that each point of X has an open neighborhood on which the sheaf is <a href="/facts/Constant_sheaf/RfrbWp81">constant</a>.

loop space
The <a href="/facts/Loop_space/S5xkdCkx">loop space</a> 
 
 
 
 Ω
 X
 
 
 {\displaystyle \Omega X}
 
 of a based space X is the space of all loops starting and ending at the base point of X.

Madsen–Weiss theorem

mapping
1.  The <a href="/facts/Mapping_cone_(topology)/a32m1wAa">mapping cone</a> (or cofiber) of a map ƒ:X→Y is 
 
 
 
 
 C
 
 f
 
 
 =
 Y
 
 ∪
 
 f
 
 
 C
 X
 
 
 {\displaystyle C_{f}=Y\cup _{f}CX}
 
.
2.  The <a href="/facts/Mapping_cylinder/kB1yUpLb">mapping cylinder</a> of a map ƒ:X→Y is 
 
 
 
 
 M
 
 f
 
 
 =
 Y
 
 ∪
 
 f
 
 
 (
 X
 ×
 I
 )
 
 
 {\displaystyle M_{f}=Y\cup _{f}(X\times I)}
 
. Note: 
 
 
 
 
 C
 
 f
 
 
 =
 
 M
 
 f
 
 
 
 /
 
 (
 X
 ×
 {
 0
 }
 )
 
 
 {\displaystyle C_{f}=M_{f}/(X\times \{0\})}
 
.
3.  The reduced versions of the above are obtained by using reduced cone and reduced cylinder.
4.  The <a href="/facts/Mapping_path_space/7JFfGWQx">mapping path space</a> Pp of a map p:E→B is the pullback of 
 
 
 
 
 B
 
 I
 
 
 →
 B
 
 
 {\displaystyle B^{I}\to B}
 
 along p. If p is fibration, then the natural map E→Pp is a <a href="/facts/Fiber-homotopy_equivalence/LESXbm8e">fiber-homotopy equivalence</a>; 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 <a href="/facts/Mapping_space/4C6EEjPD">mapping space</a> 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 <a href="/facts/Category_of_compactly_generated_weak_Hausdorff_spaces/1Td2gkzn">category of compactly generated weak Hausdorff spaces</a>. This topology may or may not be compact-open topology.

Mayer–Vietoris sequence

microbundle
<a href="/facts/Microbundle/W8JCPpKt">microbundle</a>

model category
A presentation of an <a href="/facts/%25E2%2588%259E-category/YgCfQmMD">∞-category</a>.<a class="footnote-ref" id="fnref:4" href="#fn:4">4</a> See also <a href="/facts/Model_category/KcxFUgY8">model category</a>.

Moore
1.  Moore space
2.  <a href="/facts/Moore_path_space/7JFfGWQx">Moore path space</a>.

multiplicative
A <a href="/facts/Generalized_cohomology_theory/J7FJzTNh">generalized cohomology theory</a> E is multiplicative if E*(X) is a <a href="/facts/Graded_ring/AIUjkSaP">graded ring</a>. For example, the ordinary cohomology theory and the complex K-theory are multiplicative (in fact, cohomology theories defined by <a href="/facts/E-infinity_ring/GC2bU7L7">E∞-rings</a> are multiplicative.)

n-cell
Another term for an n-disk.

n-connected
A based space X is <a href="/facts/N-connected/dbCgWgu3">n-connected</a> if 
 
 
 
 
 π
 
 q
 
 
 X
 =
 0
 
 
 {\displaystyle \pi _{q}X=0}
 
 for all integers q ≤ n. For example, "1-connected" is the same thing as "<a href="/facts/Simply_connected/xFlta63J">simply connected</a>".

n-equivalent

NDR-pair
A pair of spaces 
 
 
 
 A
 ⊂
 X
 
 
 {\displaystyle A\subset X}
 
 is said to be an <a href="/facts/NDR-pair/477Gm67f">NDR-pair</a> (=neighborhood deformation retract pair) if there is a map 
 
 
 
 u
 :
 X
 →
 I
 
 
 {\displaystyle u:X\to I}
 
 and a homotopy 
 
 
 
 
 h
 
 t
 
 
 :
 X
 →
 X
 
 
 {\displaystyle h_{t}:X\to X}
 
 such that 
 
 
 
 A
 =
 
 u
 
 −
 1
 
 
 (
 0
 )
 
 
 {\displaystyle A=u^{-1}(0)}
 
, 
 
 
 
 
 h
 
 0
 
 
 =
 
 id
 
 X
 
 
 
 
 {\displaystyle h_{0}=\operatorname {id} _{X}}
 
, 
 
 
 
 
 h
 
 t
 
 
 
 
 |
 
 
 A
 
 
 =
 
 id
 
 A
 
 
 
 
 {\displaystyle h_{t}|_{A}=\operatorname {id} _{A}}
 
 and 
 
 
 
 
 h
 
 1
 
 
 (
 {
 x
 
 |
 
 u
 (
 x
 )
 <
 1
 }
 )
 ⊂
 A
 
 
 {\displaystyle h_{1}(\{x|u(x)<1\})\subset A}
 
.

If A is a closed subspace of X, then the pair 
 
 
 
 A
 ⊂
 X
 
 
 {\displaystyle A\subset X}
 
 is an NDR-pair if and only if 
 
 
 
 A
 ↪
 X
 
 
 {\displaystyle A\hookrightarrow X}
 
 is a <a href="/facts/Cofibration/NCWE6Cbt">cofibration</a>.
nilpotent
1.  <a href="/facts/Nilpotent_space/TJj0XSVQ">nilpotent space</a>; for example, a simply connected space is nilpotent.
2.  The nilpotent theorem.

nonabelian
1.  <a href="/facts/Nonabelian_cohomology/FvaB41Cy">nonabelian cohomology</a>
2.  <a href="/facts/Nonabelian_algebraic_topology/gS8XxoW5">nonabelian algebraic topology</a>

normalized
Given a <a href="/facts/Simplicial_group/YWY6cWoR">simplicial group</a> G, the normalized chain complex NG of G is given by 
 
 
 
 (
 N
 G
 
 )
 
 n
 
 
 =
 
 ∩
 
 1
 
 
 ∞
 
 
 ker
 ⁡
 
 d
 
 i
 
 
 n
 
 
 
 
 {\displaystyle (NG)_{n}=\cap _{1}^{\infty }\operatorname {ker} d_{i}^{n}}
 
 with the n-th differential given by 
 
 
 
 
 d
 
 0
 
 
 n
 
 
 
 
 {\displaystyle d_{0}^{n}}
 
; intuitively, one throws out degenerate chains.<a class="footnote-ref" id="fnref:5" href="#fn:5">5</a> It is also called the Moore complex.

obstruction cocycle

obstruction theory
<a href="/facts/Obstruction_theory/h5T62I8z">Obstruction theory</a> 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 <a href="/facts/Postnikov_tower/wdHYnz7F">Postnikov tower</a>, 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 <a href="/facts/Operad/RfUyH38R">operad</a>.

orbibundle
orbibundle.

orbit category

orientation
1.  The <a href="/facts/Orientation_covering/1SEQM2Pt">orientation covering</a> (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 <a href="/facts/Orientation_of_a_manifold/1SEQM2Pt">orientation of a manifold</a> is a section of an orientation covering; i.e., a consistent choice of a point in each fiber.
3.  An <a href="/facts/Orientation_character/EcnXcu43">orientation character</a> (also called the first <a href="/facts/Stiefel%25E2%2580%2593Whitney_class/H3f972We">Stiefel–Whitney class</a>) is a group homomorphism 
 
 
 
 
 π
 
 1
 
 
 (
 X
 ,
 
 x
 
 0
 
 
 )
 →
 {
 ±
 1
 }
 
 
 {\displaystyle \pi _{1}(X,x_{0})\to \{\pm 1\}}
 
 that corresponds to an orientation covering of a manifold X (cf. #covering.)
4.  See also <a href="/facts/Orientation_of_a_vector_bundle/FXAzI2UD">orientation of a vector bundle</a> as well as <a href="/facts/Orientation_sheaf/cThhN158">orientation sheaf</a>.

pair
1.  A pair 
 
 
 
 (
 X
 ,
 A
 )
 
 
 {\displaystyle (X,A)}
 
 of spaces is a space X together with a subspace 
 
 
 
 A
 ⊂
 X
 
 
 {\displaystyle A\subset X}
 
.
2.  A map of pairs 
 
 
 
 (
 X
 ,
 A
 )
 →
 (
 Y
 ,
 B
 )
 
 
 {\displaystyle (X,A)\to (Y,B)}
 
 is a map 
 
 
 
 X
 →
 Y
 
 
 {\displaystyle X\to Y}
 
 such that 
 
 
 
 f
 (
 A
 )
 ⊂
 B
 
 
 {\displaystyle f(A)\subset B}
 
.

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 
 
 
 
 
 E
 
 I
 
 
 →
 
 P
 
 p
 
 
 
 
 {\displaystyle E^{I}\to P_{p}}
 
 where 
 
 
 
 
 P
 
 p
 
 
 
 
 {\displaystyle P_{p}}
 
 is the <a href="/facts/Mapping_path_space/7JFfGWQx">mapping path space</a> 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 <a href="/facts/Path_space_(algebraic_topology)/XeS9AWD2">path space</a> of a based space X is 
 
 
 
 P
 X
 =
 Map
 ⁡
 (
 I
 ,
 X
 )
 
 
 {\displaystyle PX=\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
 
 
 →
 X
 ,
 
 χ
 ↦
 χ
 (
 0
 )
 
 
 {\displaystyle X^{I}\to X,\,\chi \mapsto \chi (0)}
 
 over the base point of X. The projection 
 
 
 
 P
 X
 →
 X
 ,
 
 χ
 ↦
 χ
 (
 1
 )
 
 
 {\displaystyle PX\to X,\,\chi \mapsto \chi (1)}
 
 is called the <a href="/facts/Path_space_fibration/7JFfGWQx">path space fibration</a>, whose fiber over the base point of X is the loop space 
 
 
 
 Ω
 X
 
 
 {\displaystyle \Omega X}
 
. See also <a href="/facts/Mapping_path_space/7JFfGWQx">mapping path space</a>.

perverse
A <a href="/facts/Perverse_sheaf/HyvX2c0Q">perverse sheaf</a>.

phantom map
<a href="/facts/Phantom_map/1tcLp1GB">phantom map</a>

piecewise algebraic space
<a href="/facts/Piecewise_algebraic_space/M3Vepckc">piecewise algebraic space</a>, the notion introduced by Kontsevich and Soibelman.

PL
1.  PL short for piecewise linear.
2.  A <a href="/facts/PL_manifold/dq3fTkPT">PL manifold</a> 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.  <a href="/facts/Henri_Poincar%25C3%25A9/AazL2j4X">Henri Poincaré</a>.
2.  The <a href="/facts/Poincar%25C3%25A9_duality_theorem/lwNjUCIa">Poincaré duality theorem</a> says: given a manifold M of dimension n and an abelian group A, there is a natural isomorphism

H
 
 c
 
 
 ∗
 
 
 ⁡
 (
 M
 ;
 A
 )
 ≃
 
 H
 
 n
 −
 ∗
 
 
 ⁡
 (
 M
 ;
 A
 )
 
 
 {\displaystyle \operatorname {H} _{c}^{*}(M;A)\simeq \operatorname {H} _{n-*}(M;A)}
 
.
3.  <a href="/facts/Poincar%25C3%25A9_conjecture/mxk2Pju3">Poincaré conjecture</a>
4.  Poincaré lemma states the higher de Rham cohomology of a contractible smooth manifold vanishes.
5.  <a href="/facts/Poincar%25C3%25A9_homology_sphere/8XnrYQ1T">Poincaré homology sphere</a>.

Pontrjagin–Thom construction

Postnikov system
A <a href="/facts/Postnikov_system/wdHYnz7F">Postnikov system</a> is a sequence of fibrations, such that all preceding manifolds have vanishing <a href="/facts/Homotopy_group/xjogL95Y">homotopy groups</a> below a given dimension.

principal fibration
Usually synonymous with <a href="/facts/G-fibration/drT50zxf">G-fibration</a>.

prime decomposition

profinite
profinite homotopy theory; it studies <a href="/facts/Profinite_space/wqW5xc48">profinite spaces</a>.

properly discontinuous
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.

pseudomanifold
<a href="/facts/Pseudomanifold/7spVTmma">pseudomanifold</a>

pullback
Given a map p:E→B, the pullback of p along ƒ:X→B is the space 
 
 
 
 
 f
 
 ∗
 
 
 E
 =
 {
 (
 e
 ,
 x
 )
 ∈
 E
 ×
 X
 
 |
 
 p
 (
 e
 )
 =
 f
 (
 x
 )
 }
 
 
 {\displaystyle f^{*}E=\{(e,x)\in E\times X|p(e)=f(x)\}}
 
 (succinctly it is the <a href="/facts/Equalizer_(mathematics)/4eD7pB8y">equalizer</a> of p and f). It is a space over X through a projection.

Puppe sequence
The <a href="/facts/Puppe_sequence/qjUjNpGq">Puppe sequence</a> refers ro either of the sequences

X
        
          
            →
            f
          
        
        Y
        →
        
          C
          
            f
          
        
        →
        Σ
        X
        →
        Σ
        Y
        →
        ⋯
        ,
      
    
    {\displaystyle X{\overset {f}{\to }}Y\to C_{f}\to \Sigma X\to \Sigma Y\to \cdots ,}

⋯
        →
        Ω
        X
        →
        Ω
        Y
        →
        
          F
          
            f
          
        
        →
        X
        
          
            →
            f
          
        
        Y
      
    
    {\displaystyle \cdots \to \Omega X\to \Omega Y\to F_{f}\to X{\overset {f}{\to }}Y}

where 
 
 
 
 
 C
 
 f
 
 
 ,
 
 F
 
 f
 
 
 
 
 {\displaystyle C_{f},F_{f}}
 
 are homotopy cofiber and homotopy fiber of f.

pushout
Given 
 
 
 
 A
 ⊂
 B
 
 
 {\displaystyle A\subset B}
 
 and a map 
 
 
 
 f
 :
 A
 →
 X
 
 
 {\displaystyle f:A\to X}
 
, the <a href="/facts/Pushout_(topology)/qxczNFKG">pushout</a> of X and B along f is

X
 
 ∪
 
 f
 
 
 B
 =
 X
 ⊔
 B
 
 /
 
 (
 a
 ∼
 f
 (
 a
 )
 )
 
 
 {\displaystyle X\cup _{f}B=X\sqcup B/(a\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 = Sn-1; in that case, forming such a pushout is called attaching an n-cell (meaning an n-disk) to X.

quasi-fibration
A <a href="/facts/Quasi-fibration/guu4VBnj">quasi-fibration</a> is a map such that the fibers are homotopy equivalent to each other.

Quillen
1.  <a href="/facts/Daniel_Quillen/E4Nak15V">Daniel Quillen</a>
2.  Quillen’s theorem says that 
 
 
 
 
 π
 
 ∗
 
 
 M
 U
 
 
 {\displaystyle \pi _{*}MU}
 
 is the <a href="/facts/Lazard_ring/VY8LzJCo">Lazard ring</a>.

rational
1.  The <a href="/facts/Rational_homotopy_theory/b91f4L7z">rational homotopy theory</a>.
2.  The <a href="/facts/Rationalization_(algebraic_topology)/b91f4L7z">rationalization</a> of a space X is, roughly, the <a href="/facts/Localization_of_a_space/u2KxMWK1">localization</a> of X at zero. More precisely, X0 together with j: X → X0 is a rationalization of X if the map 
 
 
 
 
 π
 
 ∗
 
 
 X
 ⊗
 
 Q
 
 →
 
 π
 
 ∗
 
 
 
 X
 
 0
 
 
 ⊗
 
 Q
 
 
 
 {\displaystyle \pi _{*}X\otimes \mathbb {Q} \to \pi _{*}X_{0}\otimes \mathbb {Q} }
 
 induced by j is an isomorphism of vector spaces and 
 
 
 
 
 π
 
 ∗
 
 
 
 X
 
 0
 
 
 ⊗
 
 Q
 
 ≃
 
 π
 
 ∗
 
 
 
 X
 
 0
 
 
 
 
 {\displaystyle \pi _{*}X_{0}\otimes \mathbb {Q} \simeq \pi _{*}X_{0}}
 
.
3.  The <a href="/facts/Rational_homotopy_type/b91f4L7z">rational homotopy type</a> of X is the weak homotopy type of X0.

regulator
1.  Borel regulator.
2.  <a href="/facts/Beilinson_regulator/M2zhEDep">Beilinson regulator</a>.

Reidemeister
<a href="/facts/Reidemeister_torsion/a4yHL1Kd">Reidemeister torsion</a>.

reduced
The <a href="/facts/Reduced_suspension/pIWr58dM">reduced suspension</a> of a based space X is the smash product 
 
 
 
 Σ
 X
 =
 X
 ∧
 
 S
 
 1
 
 
 
 
 {\displaystyle \Sigma X=X\wedge S^{1}}
 
. It is related to the <a href="/facts/Loop_functor/S5xkdCkx">loop functor</a> by 
 
 
 
 Map
 ⁡
 (
 Σ
 X
 ,
 Y
 )
 =
 Map
 ⁡
 (
 X
 ,
 Ω
 Y
 )
 
 
 {\displaystyle \operatorname {Map} (\Sigma X,Y)=\operatorname {Map} (X,\Omega Y)}
 
 where 
 
 
 
 Ω
 Y
 =
 Map
 ⁡
 (
 
 S
 
 1
 
 
 ,
 Y
 )
 
 
 {\displaystyle \Omega Y=\operatorname {Map} (S^{1},Y)}
 
 is the loop space.

retract
1.  A <a href="/facts/Retraction_(topology)/477Gm67f">retract</a> of a map f is a map r such that 
 
 
 
 r
 ∘
 f
 
 
 {\displaystyle r\circ f}
 
 is the identity (in other words, f is a section of r).
2.  A subspace 
 
 
 
 A
 ⊂
 X
 
 
 {\displaystyle A\subset X}
 
 is called a retract if the inclusion map 
 
 
 
 A
 ↪
 X
 
 
 {\displaystyle A\hookrightarrow X}
 
 admits a retract (see #deformation retract).

ring spectrum
A <a href="/facts/Ring_spectrum/fB4IaL5T">ring spectrum</a> is a spectrum satisfying the ring axioms, either on nose or up to homotopy. For example, a <a href="/facts/Complex_K-theory/dNCXyK71">complex K-theory</a> is a ring spectrum.

Rokhlin
<a href="/facts/Rokhlin_invariant/lwaxv5lW">Rokhlin invariant</a>.

Samelson product

Serre
1.  <a href="/facts/Jean-Pierre_Serre/1xSr7BCh">Jean-Pierre Serre</a>.
2.  Serre class.
3.  <a href="/facts/Serre_spectral_sequence/9kAf3sYV">Serre spectral sequence</a>.

simple

simple-homotopy equivalence
A map ƒ:X→Y between finite simplicial complexes (e.g., manifolds) is a <a href="/facts/Simple-homotopy_equivalence/50mT3MVr">simple-homotopy equivalence</a> if it is homotopic to a composition of finitely many elementary expansions and <a href="/facts/Elementary_collapse/kXCsPpLk">elementary collapses</a>. A homotopy equivalence is a simple-homotopy equivalence if and only if its <a href="/facts/Whitehead_torsion/fq5THont">Whitehead torsion</a> vanishes.

simplicial approximation
See <a href="/facts/Simplicial_approximation_theorem/rsSYXbqF">simplicial approximation theorem</a>.

simplicial complex
See <a href="/facts/Simplicial_complex/fF2282CF">simplicial complex</a>; the basic example is a triangulation of a manifold.

simplicial homology
A <a href="/facts/Simplicial_homology/Tc3I3IJh">simplicial homology</a> 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 <a href="/facts/Singular_homology_group/6c2AHr85">singular homology group</a> of X with coefficients in π is

H
          
            ∗
          
        
        ⁡
        (
        X
        ;
        π
        )
        =
        
          H
          
            ∗
          
        
        ⁡
        (
        
          C
          
            ∗
          
        
        (
        X
        )
        ⊗
        π
        )
      
    
    {\displaystyle \operatorname {H} _{*}(X;\pi )=\operatorname {H} _{*}(C_{*}(X)\otimes \pi )}

where 
 
 
 
 
 C
 
 ∗
 
 
 (
 X
 )
 
 
 {\displaystyle C_{*}(X)}
 
 is the <a href="/facts/Singular_chain_complex/6c2AHr85">singular chain complex</a> of X; i.e., the n-th degree piece is the free abelian group generated by all the maps 
 
 
 
 
 △
 
 n
 
 
 →
 X
 
 
 {\displaystyle \triangle ^{n}\to X}
 
 from the standard n-simplex to X. A singular homology is a special case of a <a href="/facts/Simplicial_homology/Tc3I3IJh">simplicial homology</a>; indeed, for each space X, there is the <a href="/facts/Singular_simplicial_complex/a2IrPL4j">singular simplicial complex</a> of X <a class="footnote-ref" id="fnref:6" href="#fn:6">6</a> whose homology is the singular homology of X.
2.  The <a href="/facts/Singular_simplices_functor/0FCfNRN8">singular simplices functor</a> is the functor 
 
 
 
 
 T
 o
 p
 
 →
 s
 
 S
 e
 t
 
 
 
 {\displaystyle \mathbf {Top} \to s\mathbf {Set} }
 
 from the category of all spaces to the category of simplicial sets, that is the right adjoint to the <a href="/facts/Geometric_realization_functor/6DaqXMLT">geometric realization functor</a>.
3.  The singular simplicial complex of a space X is the <a href="/facts/Normalized_chain_complex/a2IrPL4j">normalized chain complex</a> of the singular simplex of X.

slant product

small object argument

smash product
The <a href="/facts/Smash_product/3gLz9z6f">smash product</a> of based spaces X, Y is 
 
 
 
 X
 ∧
 Y
 =
 X
 ×
 Y
 
 /
 
 X
 ∨
 Y
 
 
 {\displaystyle X\wedge Y=X\times Y/X\vee Y}
 
. It is characterized by the adjoint relation

Map
        ⁡
        (
        X
        ∧
        Y
        ,
        Z
        )
        =
        Map
        ⁡
        (
        X
        ,
        Map
        ⁡
        (
        Y
        ,
        Z
        )
        )
      
    
    {\displaystyle \operatorname {Map} (X\wedge Y,Z)=\operatorname {Map} (X,\operatorname {Map} (Y,Z))}
  
.

Spanier–Whitehead
The <a href="/facts/Spanier%25E2%2580%2593Whitehead_duality/a3W4ADYQ">Spanier–Whitehead duality</a>.

spectrum
Roughly a sequence of spaces together with the maps (called the structure maps) between the consecutive terms; see <a href="/facts/Spectrum_(topology)/mWho3roy">spectrum (topology)</a>.

sphere bundle
A <a href="/facts/Sphere_bundle/7yxMQ74h">sphere bundle</a> is a fiber bundle whose fibers are spheres.

sphere spectrum
The <a href="/facts/Sphere_spectrum/NB9vNpGj">sphere spectrum</a> is a spectrum consisting of a sequence of spheres 
 
 
 
 
 S
 
 0
 
 
 ,
 
 S
 
 1
 
 
 ,
 
 S
 
 2
 
 
 ,
 
 S
 
 3
 
 
 ,
 …
 
 
 {\displaystyle S^{0},S^{1},S^{2},S^{3},\dots }
 
 together with the maps between the spheres given by suspensions. In short, it is the <a href="/facts/Suspension_spectrum/mWho3roy">suspension spectrum</a> of 
 
 
 
 
 S
 
 0
 
 
 
 
 {\displaystyle S^{0}}
 
.

Spivak
Spivak normal fibration

stable homotopy group
See #homotopy group.

Steenrod homology
<a href="/facts/Steenrod_homology/dkeNrz7h">Steenrod homology</a>.

Steenrod operation

Sullivan
1.  <a href="/facts/Dennis_Sullivan/qb6wrKXr">Dennis Sullivan</a>.
2.  The <a href="/facts/Sullivan_conjecture/2XPHt0zA">Sullivan conjecture</a>.
3.  Sullivan, Dennis (1977), <a href="http://www.numdam.org/item?id=PMIHES_1977__47__269_0">"Infinitesimal computations in topology"</a>, Publications Mathématiques de l'IHÉS, 47: 269–331, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1007%2FBF02684341">10.1007/BF02684341</a>, <a href="/facts/S2CID_(identifier)/ldJsHa2Y">S2CID</a> <a href="https://api.semanticscholar.org/CorpusID:42019745">42019745</a> - introduces <a href="/facts/Rational_homotopy_theory/b91f4L7z">rational homotopy theory</a> (along with Quillen's paper).
4.  The <a href="/facts/Sullivan_algebra/b91f4L7z">Sullivan algebra</a> in the rational homotopy theory.

suspension spectrum
The <a href="/facts/Suspension_spectrum/mWho3roy">suspension spectrum</a> of a based space X is the spectrum given by 
 
 
 
 
 X
 
 n
 
 
 =
 
 Σ
 
 n
 
 
 X
 
 
 {\displaystyle X_{n}=\Sigma ^{n}X}
 
.

stratified
1.  A <a href="/facts/Stratified_space/1Ldd6EgY">stratified space</a> is a topological space with a stratification.
2.  A <a href="/facts/Stratified_Morse_theory/d0BBkSRx">stratified Morse theory</a> is a Morse theory done on a stratified space.

symmetric spectrum
See <a href="/facts/Symmetric_spectrum/krINuHyo">symmetric spectrum</a>.

symplectic topology
<a href="/facts/Symplectic_topology/runXFCea">symplectic topology</a>.

Tate
Tate sphere

telescope

Thom
1.  <a href="/facts/Ren%25C3%25A9_Thom/hdzhWSaC">René Thom</a>.
2.  If E is a vector bundle on a paracompact space X, then the <a href="/facts/Thom_space/5tgFlV5m">Thom space</a> 
 
 
 
 
 Th
 
 (
 E
 )
 
 
 {\displaystyle {\text{Th}}(E)}
 
 of E is obtained by first replacing each fiber by its compactification and then collapsing the base X.
3.  The <a href="/facts/Thom_isomorphism/5tgFlV5m">Thom isomorphism</a> says: for each <a href="/facts/Orientable_vector_bundle/FXAzI2UD">orientable vector bundle</a> E of rank n on a manifold X, a choice of an orientation (the <a href="/facts/Thom_class/5tgFlV5m">Thom class</a> of E) induces an isomorphism

H
 ~
 
 
 
 
 ∗
 +
 n
 
 
 (
 
 Th
 
 (
 E
 )
 ;
 
 Z
 
 )
 ≃
 
 H
 
 ∗
 
 
 ⁡
 (
 X
 ;
 
 Z
 
 )
 
 
 {\displaystyle {\widetilde {\operatorname {H} }}^{*+n}({\text{Th}}(E);\mathbb {Z} )\simeq \operatorname {H} ^{*}(X;\mathbb {Z} )}
 
.
4.  <a href="/facts/Thom%2527s_first_isotopy_lemma/oCYpHU3c">Thom's first</a> and <a href="/facts/Thom%2527s_second_isotopy_lemma/lSJ6j9JE">second</a> isotopy lemmas.<a class="footnote-ref" id="fnref:7" href="#fn:7">7</a>
5.  A <a href="/facts/Thom_mapping/lSJ6j9JE">Thom mapping</a> originally called a mapping "sans éclatement"

topological chiral homology

transfer

transgression

triangulation
<a href="/facts/Triangulation/lpmlmhGd">triangulation</a>.

universal coefficient
The <a href="/facts/Universal_coefficient_theorem/N8ALJkMd">universal coefficient theorem</a>.

up to homotopy
A statement holds in the <a href="/facts/Homotopy_category/JaEnPakj">homotopy category</a> as opposed to the category of spaces.

V-manifold
An old term for an <a href="/facts/Orbifold/KR6dQNzH">orbifold</a>.

van Kampen
The <a href="/facts/Van_Kampen_theorem/B2PbCFwC">van Kampen theorem</a> says: if a space X is path-connected and if x0 is a point in X, then

π
          
            1
          
        
        (
        X
        ,
        
          x
          
            0
          
        
        )
        =
        
          
            lim
            →
          
        
        ⁡
        
          π
          
            1
          
        
        (
        U
        ,
        
          x
          
            0
          
        
        )
      
    
    {\displaystyle \pi _{1}(X,x_{0})=\varinjlim \pi _{1}(U,x_{0})}

where the <a href="/facts/Colimit/JoLugLs3">colimit</a> runs over some open cover of X consisting of path-connected open subsets containing x0 such that the cover is closed under finite intersections.

Verdier
<a href="/facts/Verdier_duality/moQWLPdQ">Verdier duality</a>.

Waldhausen S-construction
<a href="/facts/Waldhausen_S-construction/ys6y0Xtw">Waldhausen S-construction</a>.

Wall's finiteness obstruction

weak equivalence
A map ƒ:X→Y of based spaces is a <a href="/facts/Weak_equivalence_(homotopy_theory)/KUWfV7rc">weak equivalence</a> if for each q, the induced map 
 
 
 
 
 f
 
 ∗
 
 
 :
 
 π
 
 q
 
 
 X
 →
 
 π
 
 q
 
 
 Y
 
 
 {\displaystyle f_{*}:\pi _{q}X\to \pi _{q}Y}
 
 is bijective.

wedge
For based spaces X, Y, the <a href="/facts/Wedge_product_(topology)/s59BzuKT">wedge product</a> 
 
 
 
 X
 ∧
 Y
 
 
 {\displaystyle X\wedge Y}
 
 of X and Y is the <a href="/facts/Coproduct_(category_theory)/0bJ8SCdu">coproduct</a> 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.  <a href="/facts/J._H._C._Whitehead/NhjAGNjR">J. H. C. Whitehead</a>.
2.  <a href="/facts/Whitehead%2527s_theorem/43UKropG">Whitehead's theorem</a> says that for <a href="/facts/CW_complexes/rN7buLMo">CW complexes</a>, the <a href="/facts/Homotopy_equivalence/1nWrPxdj">homotopy equivalence</a> is the same thing as the <a href="/facts/Weak_equivalence_(homotopy_theory)/KUWfV7rc">weak equivalence</a>.
3.  <a href="/facts/Whitehead_torsion/fq5THont">Whitehead group</a>.
4.  <a href="/facts/Whitehead_product/hWSWnUVe">Whitehead product</a>.

winding number
1.  <a href="/facts/Winding_number/kaquyUL2">winding number</a>.

<h2 id="notes">Notes</h2>

<ul><li><a href="/facts/Frank_Adams/fxgpN9Xk">Adams, J.F.</a> (1974). <a href="https://books.google.com/books?id=6vG13YQcPnYC">Stable Homotopy and Generalised Homology</a>. Chicago Lectures in Mathematics. University of Chicago Press. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-226-00524-9.</li>
<li>Adams, J.F. (1978). <a href="https://books.google.com/books?id=e2rYkg9lGnsC">Infinite Loop Spaces</a>. Princeton University Press. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-691-08206-5.</li>
<li>Borel, Armand (21 May 2009). <a href="https://books.google.com/books?id=0c5HAAAAQBAJ&dq=borel+intersection+cohomology&pg=PA1">Intersection Cohomology</a>. Springer Science & Business Media. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-8176-4765-0.</li>
<li><a href="/facts/Raoul_Bott/5JHkhPx8">Bott, Raoul</a>; Tu, Loring (1982), Differential Forms in Algebraic Topology, Springer, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-387-90613-4</li>
<li>Bousfield, A. K.; Kan, D. M. (1987), Homotopy Limits, Completions and Localizations, Lecture Notes in Mathematics, vol. 304, Springer, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 9783540061052</li>
<li>Davis, James F.; Kirk, Paul. <a href="http://www.maths.ed.ac.uk/~aar/papers/davkir.pdf">"Lecture Notes in Algebraic Topology"</a> (PDF).</li>
<li>Fulton, William (2013). <a href="https://books.google.com/books?id=8-OPBAAAQBAJ">Algebraic Topology: A First Course</a>. Springer. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-1-4612-4180-5.</li>
<li>Hatcher, Allen. <a href="http://pi.math.cornell.edu/~hatcher/AT/ATpage.html">"Algebraic topology"</a>.</li>
<li><a href="/facts/Kathryn_Hess/IDKJqEGL">Hess, Kathryn</a> (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. <a href="/facts/ArXiv_(identifier)/H6EtgnBe">arXiv</a>:<a href="https://arxiv.org/abs/math/0604626">math/0604626</a>. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1090%2Fconm%2F436%2F08409">10.1090/conm/436/08409</a>. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-8218-3814-3. <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=2355774">2355774</a>.</li>
<li><a href="https://math.berkeley.edu/~amathew/ATnotes.pdf">"algebraic topology"</a> (PDF). Fall 2010. Lectures delivered by Michael Hopkins and Notes by Akhil Mathew, Harvard.</li>
<li><a href="/facts/Jacob_Lurie/QUNLubLt">Lurie, J.</a> (2015). <a href="http://www.math.harvard.edu/~lurie/281.html">"Algebraic K-Theory and Manifold Topology"</a>. Math 281. Harvard University.</li>
<li>Lurie, J. (2011). <a href="http://www.math.harvard.edu/~lurie/252x.html">"Chromatic Homotopy Theory"</a>. 252x. Harvard University.</li>
<li>May, J. <a href="http://www.math.uchicago.edu/~may/CONCISE/ConciseRevised.pdf">"A Concise Course in Algebraic Topology"</a> (PDF).</li>
<li>May, J.; Ponto, K. <a href="http://www.math.uchicago.edu/~may/TEAK/KateBookFinal.pdf">"More concise algebraic topology: localization, completion, and model categories"</a> (PDF).</li>
<li>May; Sigurdsson. <a href="http://www.math.uchicago.edu/~may/EXTHEORY/MaySig.pdf">"Parametrized homotopy theory"</a> (PDF). (despite the title, it contains a significant amount of general results.)</li>
<li>Rudyak, Yuli B. (23 December 2014). "Piecewise linear structures on topological manifolds". <a href="/facts/ArXiv_(identifier)/H6EtgnBe">arXiv</a>:<a href="https://arxiv.org/abs/math/0105047">math/0105047</a>.</li>
<li><a href="/facts/Dennis_Sullivan/qb6wrKXr">Sullivan, Dennis</a>. <a href="http://www.maths.ed.ac.uk/~aar/books/gtop.pdf">"Geometric Topology"</a> (PDF). the 1970 MIT notes</li>
<li><a href="/facts/George_W._Whitehead/AWRvB3Hr">Whitehead, George William</a> (1978). <a href="https://books.google.com/books?id=wlrvAAAAMAAJ">Elements of homotopy theory</a>. Graduate Texts in Mathematics. Vol. 61 (3rd ed.). Springer-Verlag. pp. xxi+744. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-387-90336-1. <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0516508">0516508</a>.</li>
<li><a href="/facts/Kirsten_Wickelgren/8s8lzvE6">Wickelgren, Kirsten Graham</a>. <a href="http://people.math.gatech.edu/~kwickelgren3/8803_Stable/">"8803 Stable Homotopy Theory"</a>.</li></ul>
<h2 id="further-reading">Further reading</h2>
<ul><li>José I. Burgos Gil, <a href="https://www.icmat.es/miembros/burgos/files/brbr.pdf">The Regulators of Beilinson and Borel</a></li>
<li><a href="https://math.mit.edu/juvitop/old/notes/2014_Fall/JPLEVINElectures.pdf">Lectures on groups of homotopy spheres</a> by JP Levine</li>
<li>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. <a href="http://www.math.ias.edu/vladimir/sites/math.ias.edu.vladimir/files/Nordfjordeid_lectures_published.pdf">[1]</a></li></ul>
<h2 id="external-links">External links</h2>
<ul><li><a href="http://pantodon.shinshu-u.ac.jp/topology/literature/">Algebraic Topology: A guide to literature</a> <a href="https://web.archive.org/web/20171217014101/http://pantodon.shinshu-u.ac.jp/topology/literature/">Archived</a> 2017-12-17 at the <a href="/facts/Wayback_Machine/nmQ3a6JC">Wayback Machine</a></li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1">Let r, s denote the restriction and the section. For each f in 
 
 
 
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 {\displaystyle h_{t}(f)(x)=tf(x/t),|x|\leq t,h_{t}(f)(x)=|x|f(x/|x|),|x|>t}
 
. Then 
 
 
 
 
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 {\displaystyle h_{t}:s\circ r\sim \operatorname {id} }
 
. <a href="#fnref:1" class="footnote-back-ref">↩</a></li>
<li id="fn:2">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. <a href="/wiki/Algebraic_variety" target="_blank">/wiki/Algebraic_variety</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></li>
<li id="fn:3">Hatcher, Ch. 4. H. - Hatcher, Allen. "Algebraic topology". <a href="http://pi.math.cornell.edu/~hatcher/AT/ATpage.html" target="_blank">http://pi.math.cornell.edu/~hatcher/AT/ATpage.html</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></li>
<li id="fn:4">How to think about model categories? <a href="https://mathoverflow.net/q/2185" target="_blank">https://mathoverflow.net/q/2185</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></li>
<li id="fn:5">"Moore complex in nLab". <a href="https://ncatlab.org/nlab/show/Moore+complex" target="_blank">https://ncatlab.org/nlab/show/Moore+complex</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></li>
<li id="fn:6">"Singular simplicial complex in nLab". <a href="http://ncatlab.org/nlab/show/singular+simplicial+complex" target="_blank">http://ncatlab.org/nlab/show/singular+simplicial+complex</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></li>
<li id="fn:7">"Differential topology - Thom's first isotopy lemma". <a href="https://mathoverflow.net/questions/304298/thoms-first-isotopy-lemma" target="_blank">https://mathoverflow.net/questions/304298/thoms-first-isotopy-lemma</a> <a href="#fnref:7" class="footnote-back-ref">↩</a></li>
</ol>

Glossary of algebraic topology open-in-new

Glossary of algebraic topology