nLab ordinary cohomology

Idea

One says ordinary cohomology for the cohomology theory represented by the Eilenberg-MacLane spectrum $K(\mathbb{A})$ for some abelian group $A$.

This is to distinguish from more general notions of cohomology, such as generalized (Eilenberg-Steenrod) cohomology and nonabelian cohomology.

Properties

Presentation by singular cohomology

Ordinary cohomology is modeled by singular cohomology.

General

• universal coefficient theorem

• Künneth theorem

• Hopf degree theorem

• Hurewicz theorem, Boardman homomorphism

• Eilenberg-Zilber theorem

Chromatic level

chromatic homotopy theory

chromatic level	complex oriented cohomology theory	E-∞ ring/A-∞ ring	real oriented cohomology theory
0	ordinary cohomology	Eilenberg-MacLane spectrum $H \mathbb{Z}$	HZR-theory
	0th Morava K-theory	$K(0)$
1	complex K-theory	complex K-theory spectrum $KU$	KR-theory
	first Morava K-theory	$K(1)$
	first Morava E-theory	$E(1)$
2	elliptic cohomology	elliptic spectrum $Ell_E$
	second Morava K-theory	$K(2)$
	second Morava E-theory	$E(2)$
	algebraic K-theory of KU	$K(KU)$
3 …10	K3 cohomology	K3 spectrum
$n$	$n$th Morava K-theory	$K(n)$
	$n$th Morava E-theory	$E(n)$	BPR-theory
$n+1$	algebraic K-theory applied to chrom. level $n$	$K(E_n)$ (red-shift conjecture)
$\infty$	complex cobordism cohomology	MU	MR-theory

Related concepts

• cohomology in homotopy type theory

• The refinement to equivariant cohomology is equivariant ordinary cohomology

• The refinement to differential cohomology is ordinary differential cohomology.

• ordinary homology

• twisted ordinary cohomology

• affine modality

• comparison theorem (étale cohomology)

References

Early references on (co)homology

The original references on chain homology/cochain cohomology and ordinary cohomology in the form of cellular cohomology:

• Andrei Kolmogoroff, Über die Dualität im Aufbau der kombinatorischen Topologie, Recueil Mathématique 1(43) (1936), 97–102. (mathnet)

A footnote on the first page reads as follows, giving attribution to Alexander 35a, 35b:

Die Resultate dieser Arbeit wurden für den Fall gewöhnlicher Komplexe vom Verfasser im Frühling und im Sommer 1934 erhalten und teilweise an der Internationalen Konferenz für Tensoranalysis (Moskau) im Mai 1934 vorgetragen. Die hier dargestellte allgemeinere Theorie bildete den Gegenstand eines Vortrages, den der Verfasser an der Internationalen Topologischen Konferenz (Moskau, September 1935) hielt; bei letzterer Gelegenheit erfuhr er, dass ein grosser Teil dieser Resultate im Falle von Komplexen indessen von Herrn Alexander erhalten worden ist. Vgl. die inzwischen erschienenen Noten von Herrn Alexander in den «Proceedings of the National Academy of Sciences U.S.A.», 21, (1935), 509—512. Herr Alexander trug über seine Resultate ebenfalls an der Moskauer Topologischen Konferenz vor. Verallgemeinerungen für abgeschlossene Mengen und die Konstruktion eines Homologieringes für Komplexe und abgeschlossene Mengen, über welche der Verfasser ebenso an der Tensorkonferenz 1934 vorgetragen hat, werden in einer weiteren Publikation dargestellt. Diese weitere Begriffsbildungen sind übrigens ebenfalls von Herrn Alexander gefunden und teilweise in den erwähnten Noten publiziert.

• Andrei Kolmogoroff, Homologiering des Komplexes und des lokal-bicompakten Raumes, Recueil Mathématique 1(43) (1936), 701–705. mathnet.

• J. W. Alexander, On the chains of a complex and their duals, Proc. Nat. Acad. Sei. USA, 21(1935), 509–511 (doi:10.1073/pnas.21.8.50)

• J. W. Alexander, On the ring of a compact metric space, Proc. Nat. Acad. Sci. USA, 21 (1935), 511–512 (doi:10.1073/pnas.21.8.511)

• J. W. Alexander, On the connectivity ring of an abstract space, Ann. of Math., 37 (1936), 698–708 (doi:10.2307/1968484, pdf)

The term “cohomology” was introduced by Hassler Whitney in

• Hassler Whitney, On matrices of integers and combinatorial topology. Duke Mathematical Journal 3:1 (1937), 35–45 (doi:10.1215/S0012-7094-37-00304-1)

See also

• Hassler Whitney, On products in a complex, Annals of Math. 39 (1938) 397–432 (doi:10.2307/1968795)

The notion of singular cohomology is due to

• Samuel Eilenberg, Singular homology theory, Annals of Mathematics 45:3 (1944) (doi:10.2307/1969185)

The notion of monadic cohomology via canonical resolutions:

• Michael Barr, Jon Beck, Homology and Standard Constructions, in: Seminar on Triples and Categorical Homology Theory, Lecture Notes in Maths. 80, Springer (1969), Reprints in Theory and Applications of Categories 18 (2008) 186-248 [tac:18, pdf]

• Michael Barr, Cartan-Eilenberg cohomology and triples, J. Pure Applied Algebra 112 3 (1996) 219–238 [doi:10.1016/0022-4049(95)00138-7, pdf, pdf]

• Michael Barr, Algebraic cohomology: the early days, in Galois Theory, Hopf Algebras, and Semiabelian Categories, Fields Institute Communications 43 (2004) 1–26 [doi:10.1090/fic/043, pdf, pdf]

The general abstract perspective on cohomology (subsuming sheaf cohomology, hypercohomology, non-abelian cohomology and indications of Whitehead-generalized cohomology) was essentially established in:

• Kenneth Brown, Abstract homotopy theory and generalized sheaf cohomology (1973)

General

Textbook account:

• Jean Gallier, Jocelyn Quaintance, Homology, Cohomology, and Sheaf Cohomology for Algebraic Topology, Algebraic Geometry, and Differential Geometry, World Scientific (2022) [doi:10.1142/12495, webpage]

See the references at:

• singular cohomology, cellular cohomology

• sheaf cohomology

• Cech cohomology

• integral cohomology

Discussion in homotopy type theory:

• Guillaume Brunerie, Axel Ljungström, Anders Mörtberg, Synthetic Integral Cohomology in Cubical Agda, 30th EACSL Annual Conference on Computer Science Logic (CSL 2022) 216 (2022) $[$doi:10.4230/LIPIcs.CSL.2022.11$]$

  (in cubical Agda)