nLab
integral cohomology
**
cohomology**
*
cocycle,
coboundary,
coefficient
*
homology
*
chain,
cycle,
boundary
*
characteristic class
*
universal characteristic class
*
secondary characteristic class
*
differential characteristic class
*
fiber sequence/
long exact sequence in cohomology
*
fiber ∞bundle,
principal ∞bundle,
associated ∞bundle,
twisted ∞bundle
*
∞group extension
*
obstruction
### Special and general types ###
*
cochain cohomology
*
ordinary cohomology,
singular cohomology
*
group cohomology,
nonabelian group cohomology,
Lie group cohomology
*
Galois cohomology
*
groupoid cohomology,
nonabelian groupoid cohomology
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generalized (EilenbergSteenrod) cohomology
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cobordism cohomology theory
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integral cohomology
*
Ktheory
*
elliptic cohomology,
tmf
*
taf
*
abelian sheaf cohomology
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Deligne cohomology
*
de Rham cohomology
*
Dolbeault cohomology
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etale cohomology
*
group of units,
Picard group,
Brauer group
*
crystalline cohomology
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syntomic cohomology
*
motivic cohomology
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cohomology of operads
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Hochschild cohomology,
cyclic cohomology
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string topology
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nonabelian cohomology
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principal ∞bundle
*
universal principal ∞bundle,
groupal model for universal principal ∞bundles
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principal bundle,
Atiyah Lie groupoid
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principal 2bundle/
gerbe
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covering ∞bundle/
local system
*
(∞,1)vector bundle /
(∞,n)vector bundle
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quantum anomaly
*
orientation,
Spin structure,
Spin^c structure,
String structure,
Fivebrane structure
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cohomology with constant coefficients /
with a local system of coefficients
*
∞Lie algebra cohomology
*
Lie algebra cohomology,
nonabelian Lie algebra cohomology,
Lie algebra extensions,
GelfandFuks cohomology,
*
bialgebra cohomology
### Special notions
*
Čech cohomology
*
hypercohomology
### Variants ###
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equivariant cohomology
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equivariant homotopy theory
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Bredon cohomology
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twisted cohomology
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twisted bundle
*
twisted Ktheory,
twisted spin structure,
twisted spin^c structure
*
twisted differential cstructures
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twisted differential string structure,
twisted differential fivebrane structure
* differential cohomology
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differential generalized (EilenbergSteenrod) cohomology
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differential cobordism cohomology
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Deligne cohomology
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differential Ktheory
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differential elliptic cohomology
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differential cohomology in a cohesive topos
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ChernWeil theory
*
∞ChernWeil theory
*
relative cohomology
### Extra structure
*
Hodge structure
*
orientation,
in generalized cohomology
### Operations ###
*
cohomology operations
*
cup product
*
connecting homomorphism,
Bockstein homomorphism
*
fiber integration,
transgression
*
cohomology localization
### Theorems
*
universal coefficient theorem
*
Künneth theorem
*
de Rham theorem,
Poincare lemma,
Stokes theorem
*
Hodge theory,
Hodge theorem
nonabelian Hodge theory,
noncommutative Hodge theory
*
Brown representability theorem
*
hypercovering theorem
*
EckmannHiltonFuks duality
Integral cohomology or “ordinary cohomology” is the ordinary version of generalized (EilenbergSteenrod) cohomology, the one that is represented by the EilenbergMacLane spectrum.
Integral cohomology is best known maybe in its incarnation as singular cohomology or Čech cohomology with coefficients in the integers.
Geometric models

integral cohomology in degree 1 classifies complex line bundle;

integral cohomology in degree 2 classifies complex line bundle gerbe / line 2bundles;

integral cohomology in degree $n$ classifies line nbundles .
Revised on December 19, 2009 03:21:02
by
Toby Bartels
(173.60.119.197)