Cohomology

In mathematics, specifically in homology theory and algebraic topology, cohomology is a general term for a sequence of abelian groups associated to a topological space, often defined from a cochain complex. Cohomology can be viewed as a method of assigning richer algebraic invariants to a space than homology. Some versions of cohomology arise by dualizing the construction of homology. In other words, cochains are functions on the group of chains in homology theory.
From its beginning in topology, this idea became a dominant method in the mathematics of the second half of the twentieth century. From the initial idea of homology as a method of constructing algebraic invariants of topological spaces, the range of applications of homology and cohomology theories has spread throughout geometry and algebra. The terminology tends to hide the fact that cohomology, a contravariant theory, is more natural than homology in many applications. At a basic level, this has to do with functions and pullbacks in geometric situations: given spaces "X" and "Y", and some kind of function "F" on "Y", for any mapping , composition with "f" gives rise to a function on "X". The most important cohomology theories have a product, the cup product, which gives them a ring structure. Because of this feature, cohomology is usually a stronger invariant than homology.
Singular cohomology.
Singular cohomology is a powerful invariant in topology, associating a graded-commutative ring to any topological space. Every continuous map "f": "X" → "Y" determines a homomorphism from the cohomology ring of "Y" to that of "X"; this puts strong restrictions on the possible maps from "X" to "Y". Unlike more subtle invariants such as homotopy groups, the cohomology ring tends to be computable in practice for spaces of interest.
For a topological space "X", the definition of singular cohomology starts with the singular chain complex:
By definition, the singular homology of "X" is the homology of this chain complex (the kernel of one homomorphism modulo the image of the previous one). In more detail, "Ci" is the free abelian group on the set of continuous maps from the standard "i"-simplex to "X" (called "singular "i"-simplices in "X""), and ∂"i" is the "i"th boundary homomorphism. The groups "C""i" are zero for "i" negative.
Now fix an abelian group "A", and replace each group "Ci" by its dual group formula_2 and formula_3 by its dual homomorphism
This has the effect of "reversing all the arrows" of the original complex, leaving a cochain complex
For an integer "i", the "i"th cohomology group of "X" with coefficients in "A" is defined to be ker("di")/im("d""i"−1) and denoted by "H""i"("X", "A"). The group "H""i"("X", "A") is zero for "i" negative. The elements of formula_6 are called singular "i"-cochains with coefficients in "A". (Equivalently, an "i"-cochain on "X" can be identified with a function from the set of singular "i"-simplices in "X" to "A".) Elements of ker("d") and im("d") are called cocycles and coboundaries, respectively, while elements of ker("d")/im("d") = "H""i"("X", "A") are called cohomology classes (because they are equivalence classes of cocycles).
In what follows, the coefficient group "A" is sometimes not written. It is common to take "A" to be a commutative ring "R"; then the cohomology groups are "R"-modules. A standard choice is the ring Z of integers.
Some of the formal properties of cohomology are only minor variants of the properties of homology:
On the other hand, cohomology has a crucial structure that homology does not: for any topological space "X" and commutative ring "R", there is a bilinear map, called the cup product:
defined by an explicit formula on singular cochains. The product of cohomology classes "u" and "v" is written as "u" ∪ "v" or simply as "uv". This product makes the direct sum
into a graded ring, called the cohomology ring of "X". It is graded-commutative in the sense that:
For any continuous map formula_20 the pullback formula_21 is a homomorphism of graded "R"-algebras. It follows that if two spaces are homotopy equivalent, then their cohomology rings are isomorphic.
Here are some of the geometric interpretations of the cup product. In what follows, manifolds are understood to be without boundary, unless stated otherwise. A closed manifold means a compact manifold (without boundary), whereas a closed submanifold "N" of a manifold "M" means a submanifold that is a closed subset of "M", not necessarily compact (although "N" is automatically compact if "M" is).
Very informally, for any topological space "X", elements of formula_25 can be thought of as represented by codimension-"i" subspaces of "X" that can move freely on "X". For example, one way to define an element of formula_25 is to give a continuous map "f" from "X" to a manifold "M" and a closed codimension-"i" submanifold "N" of "M" with an orientation on the normal bundle. Informally, one thinks of the resulting class formula_27 as lying on the subspace formula_28 of "X"; this is justified in that the class formula_29 restricts to zero in the cohomology of the open subset formula_30 The cohomology class formula_29 can move freely on "X" in the sense that "N" could be replaced by any continuous deformation of "N" inside "M".
Examples.
In what follows, cohomology is taken with coefficients in the integers Z, unless stated otherwise. 
The diagonal.
The cup product on cohomology can be viewed as coming from the diagonal map Δ: "X" → "X" × "X", "x" ↦ ("x","x"). Namely, for any spaces "X" and "Y" with cohomology classes "u" ∈ "H""i"("X","R") and "v" ∈ "H""j"("Y","R"), there is an external product (or cross product) cohomology class "u" × "v" ∈ "H""i"+"j"("X" × "Y","R"). The cup product of classes "u" ∈ "H""i"("X","R") and "v" ∈ "H""j"("X","R") can be defined as the pullback of the external product by the diagonal:
Alternatively, the external product can be defined in terms of the cup product. For spaces "X" and "Y", write "f": "X" × "Y" → "X" and "g": "X" × "Y" → "Y" for the two projections. Then the external product of classes "u" ∈ "H""i"("X","R") and "v" ∈ "H""j"("Y","R") is:
Poincaré duality.
Another interpretation of Poincaré duality is that the cohomology ring of a closed oriented manifold is self-dual in a strong sense. Namely, let "X" be a closed connected oriented manifold of dimension "n", and let "F" be a field. Then "H""n"("X","F") is isomorphic to "F", and the product
is a perfect pairing for each integer "i". In particular, the vector spaces "H""i"("X","F") and "H""n"−"i"("X","F") have the same (finite) dimension. Likewise, the product on integral cohomology modulo torsion with values in "H""n"("X",Z) ≅ Z is a perfect pairing over Z.
Characteristic classes.
An oriented real vector bundle "E" of rank "r" over a topological space "X" determines a cohomology class on "X", the Euler class χ("E") ∈ "H""r"("X",Z). Informally, the Euler class is the class of the zero set of a general section of "E". That interpretation can be made more explicit when "E" is a smooth vector bundle over a smooth manifold "X", since then a general smooth section of "X" vanishes on a codimension-"r" submanifold of "X".
There are several other types of characteristic classes for vector bundles that take values in cohomology, including Chern classes, Stiefel–Whitney classes, and Pontryagin classes.
Eilenberg–MacLane spaces.
For each abelian group "A" and natural number "j", there is a space formula_40 whose "j"-th homotopy group is isomorphic to "A" and whose other homotopy groups are zero. Such a space is called an Eilenberg–MacLane space. This space has the remarkable property that it is a classifying space for cohomology: there is a natural element "u" of formula_41, and every cohomology class of degree "j" on every space "X" is the pullback of "u" by some continuous map formula_42. More precisely, pulling back the class "u" gives a bijection
for every space "X" with the homotopy type of a CW complex. Here formula_44 denotes the set of homotopy classes of continuous maps from "X" to "Y".
For example, the space formula_45 (defined up to homotopy equivalence) can be taken to be the circle formula_34. So the description above says that every element of formula_47 is pulled back from the class "u" of a point on formula_34 by some map formula_49.
There is a related description of the first cohomology with coefficients in any abelian group "A", say for a CW complex "X". Namely, formula_50 is in one-to-one correspondence with the set of isomorphism classes of Galois covering spaces of "X" with group "A", also called principal "A"-bundles over "X". For "X" connected, it follows that formula_50 is isomorphic to formula_52, where formula_53 is the fundamental group of "X". For example, formula_54 classifies the double covering spaces of "X", with the element formula_55 corresponding to the trivial double covering, the disjoint union of two copies of "X".
Cap product.
For any topological space "X", the cap product is a bilinear map
for any integers "i" and "j" and any commutative ring "R". The resulting map
makes the singular homology of "X" into a module over the singular cohomology ring of "X".
For "i" = "j", the cap product gives the natural homomorphism
which is an isomorphism for "R" a field.
For example, let "X" be an oriented manifold, not necessarily compact. Then a closed oriented codimension-"i" submanifold "Y" of "X" (not necessarily compact) determines an element of "H""i"("X","R"), and a compact oriented "j"-dimensional submanifold "Z" of "X" determines an element of "H""j"("X","R"). The cap product ["Y"] ∩ ["Z"] ∈ "H""j"−"i"("X","R") can be computed by perturbing "Y" and "Z" to make them intersect transversely and then taking the class of their intersection, which is a compact oriented submanifold of dimension "j" − "i".
A closed oriented manifold "X" of dimension "n" has a fundamental class ["X"] in "H""n"("X","R"). The Poincaré duality isomorphism
is defined by cap product with the fundamental class of "X".
History, to the birth of singular cohomology.
Although cohomology is fundamental to modern algebraic topology, its importance was not seen for some 40 years after the development of homology. The concept of "dual cell structure", which Henri Poincaré used in his proof of his Poincaré duality theorem, contained the germ of the idea of cohomology, but this was not seen until later.
There were various precursors to cohomology. In the mid-1920s, J. W. Alexander and Solomon Lefschetz founded the intersection theory of cycles on manifolds. On a closed oriented "n"-dimensional manifold "M", an "i"-cycle and a "j"-cycle with nonempty intersection will, if in general position, have intersection an ("i" + "j" − "n")-cycle. This leads to a multiplication of homology classes
which in retrospect can be identified with the cup product on the cohomology of "M".
Alexander had by 1930 defined a first notion of a cochain, by thinking of an "i"-cochain on a space "X" as a function on small neighborhoods of the diagonal in "X""i"+1.
In 1931, Georges de Rham related homology and differential forms, proving de Rham's theorem. This result can be stated more simply in terms of cohomology.
In 1934, Lev Pontryagin proved the Pontryagin duality theorem; a result on topological groups. This (in rather special cases) provided an interpretation of Poincaré duality and Alexander duality in terms of group characters.
At a 1935 conference in Moscow, Andrey Kolmogorov and Alexander both introduced cohomology and tried to construct a cohomology product structure.
In 1936, Norman Steenrod constructed Čech cohomology by dualizing Čech homology.
From 1936 to 1938, Hassler Whitney and Eduard Čech developed the cup product (making cohomology into a graded ring) and cap product, and realized that Poincaré duality can be stated in terms of the cap product. Their theory was still limited to finite cell complexes.
In 1944, Samuel Eilenberg overcame the technical limitations, and gave the modern definition of singular homology and cohomology.
In 1945, Eilenberg and Steenrod stated the axioms defining a homology or cohomology theory, discussed below. In their 1952 book, "Foundations of Algebraic Topology", they proved that the existing homology and cohomology theories did indeed satisfy their axioms.
In 1946, Jean Leray defined sheaf cohomology.
In 1948 Edwin Spanier, building on work of Alexander and Kolmogorov, developed Alexander–Spanier cohomology.
Sheaf cohomology.
Sheaf cohomology is a rich generalization of singular cohomology, allowing more general "coefficients" than simply an abelian group. For every sheaf of abelian groups "E" on a topological space "X", one has cohomology groups "H""i"("X","E") for integers "i". In particular, in the case of the constant sheaf on "X" associated to an abelian group "A", the resulting groups "H""i"("X","A") coincide with singular cohomology for "X" a manifold or CW complex (though not for arbitrary spaces "X"). Starting in the 1950s, sheaf cohomology has become a central part of algebraic geometry and complex analysis, partly because of the importance of the sheaf of regular functions or the sheaf of holomorphic functions.
Grothendieck elegantly defined and characterized sheaf cohomology in the language of homological algebra. The essential point is to fix the space "X" and think of sheaf cohomology as a functor from the abelian category of sheaves on "X" to abelian groups. Start with the functor taking a sheaf "E" on "X" to its abelian group of global sections over "X", "E"("X"). This functor is left exact, but not necessarily right exact. Grothendieck defined sheaf cohomology groups to be the right derived functors of the left exact functor "E" ↦ "E"("X").
That definition suggests various generalizations. For example, one can define the cohomology of a topological space "X" with coefficients in any complex of sheaves, earlier called hypercohomology (but usually now just "cohomology"). From that point of view, sheaf cohomology becomes a sequence of functors from the derived category of sheaves on "X" to abelian groups.
In a broad sense of the word, "cohomology" is often used for the right derived functors of a left exact functor on an abelian category, while "homology" is used for the left derived functors of a right exact functor. For example, for a ring "R", the Tor groups Tor"i""R"("M","N") form a "homology theory" in each variable, the left derived functors of the tensor product "M"⊗"R""N" of "R"-modules. Likewise, the Ext groups Ext"i""R"("M","N") can be viewed as a "cohomology theory" in each variable, the right derived functors of the Hom functor Hom"R"("M","N").
Sheaf cohomology can be identified with a type of Ext group. Namely, for a sheaf "E" on a topological space "X", "H""i"("X","E") is isomorphic to Ext"i"(Z"X", "E"), where Z"X" denotes the constant sheaf associated to the integers Z, and Ext is taken in the abelian category of sheaves on "X".
Cohomology of varieties.
There are numerous machines built for computing the cohomology of algebraic varieties. The simplest case being the determination of cohomology for smooth projective varieties over a field of characteristic formula_61. Tools from Hodge theory, called Hodge structures help give computations of cohomology of these types of varieties (with the addition of more refined information). In the simplest case the cohomology of a smooth hypersurface in formula_62 can be determined from the degree of the polynomial alone.
When considering varieties over a finite field, or a field of characteristic formula_63, more powerful tools are required because the classical definitions of homology/cohomology break down. This is because varieties over finite fields will only be a finite set of points. Grothendieck came up with the idea for a Grothendieck topology and used sheaf cohomology over the etale topology to define the cohomology theory for varieties over a finite field. Using the étale topology for a variety over a field of characteristic formula_63 one can construct formula_65-adic cohomology for formula_66. This is defined as
If we have a scheme of finite type
then there is an equality of dimensions for the Betti cohomology of formula_69 and the formula_65-adic cohomology of formula_71 whenever the variety is smooth over both fields. In addition to these cohomology theories there are other cohomology theories called Weil cohomology theories which behave similarly to singular cohomology. There is a conjectured theory of motives which underlie all of the Weil cohomology theories.
Another useful computational tool is the blowup sequence. Given a codimension formula_72 subscheme formula_73 there is a Cartesian square
From this there is an associated long exact sequence
If the subvariety formula_76 is smooth, then the connecting morphisms are all trivial, hence
Axioms and generalized cohomology theories.
There are various ways to define cohomology for topological spaces (such as singular cohomology, Čech cohomology, Alexander–Spanier cohomology or sheaf cohomology). (Here sheaf cohomology is considered only with coefficients in a constant sheaf.) These theories give different answers for some spaces, but there is a large class of spaces on which they all agree. This is most easily understood axiomatically: there is a list of properties known as the Eilenberg–Steenrod axioms, and any two constructions that share those properties will agree at least on all CW complexes. There are versions of the axioms for a homology theory as well as for a cohomology theory. Some theories can be viewed as tools for computing singular cohomology for special topological spaces, such as simplicial cohomology for simplicial complexes, cellular cohomology for CW complexes, and de Rham cohomology for smooth manifolds.
One of the Eilenberg–Steenrod axioms for a cohomology theory is the dimension axiom: if "P" is a single point, then "Hi"("P") = 0 for all "i" ≠ 0. Around 1960, George W. Whitehead observed that it is fruitful to omit the dimension axiom completely: this gives the notion of a generalized homology theory or a generalized cohomology theory, defined below. There are generalized cohomology theories such as K-theory or complex cobordism that give rich information about a topological space, not directly accessible from singular cohomology. (In this context, singular cohomology is often called "ordinary cohomology".)
By definition, a generalized homology theory is a sequence of functors "h""i" (for integers "i") from the category of CW-pairs ("X", "A") (so "X" is a CW complex and "A" is a subcomplex) to the category of abelian groups, together with a natural transformation ∂"i": "h""i"("X", "A") → "h""i"−1("A") called the boundary homomorphism (here "h""i"−1("A") is a shorthand for "h""i"−1("A",∅)). The axioms are:
The axioms for a generalized cohomology theory are obtained by reversing the arrows, roughly speaking. In more detail, a generalized cohomology theory is a sequence of contravariant functors "h""i" (for integers "i") from the category of CW-pairs to the category of abelian groups, together with a natural transformation "d": "h""i"("A") → "h""i"+1("X","A") called the boundary homomorphism (writing "h""i"("A") for "h""i"("A",∅)). The axioms are:
A spectrum determines both a generalized homology theory and a generalized cohomology theory. A fundamental result by Brown, Whitehead, and Adams says that every generalized homology theory comes from a spectrum, and likewise every generalized cohomology theory comes from a spectrum. This generalizes the representability of ordinary cohomology by Eilenberg–MacLane spaces.
A subtle point is that the functor from the stable homotopy category (the homotopy category of spectra) to generalized homology theories on CW-pairs is not an equivalence, although it gives a bijection on isomorphism classes; there are nonzero maps in the stable homotopy category (called phantom maps) that induce the zero map between homology theories on CW-pairs. Likewise, the functor from the stable homotopy category to generalized cohomology theories on CW-pairs is not an equivalence. It is the stable homotopy category, not these other categories, that has good properties such as being triangulated.
If one prefers homology or cohomology theories to be defined on all topological spaces rather than on CW complexes, one standard approach is to include the axiom that every weak homotopy equivalence induces an isomorphism on homology or cohomology. (That is true for singular homology or singular cohomology, but not for sheaf cohomology, for example.) Since every space admits a weak homotopy equivalence from a CW complex, this axiom reduces homology or cohomology theories on all spaces to the corresponding theory on CW complexes.
Some examples of generalized cohomology theories are:
Many of these theories carry richer information than ordinary cohomology, but are harder to compute.
A cohomology theory "E" is said to be multiplicative if formula_95 has the structure of a graded ring for each space "X". In the language of spectra, there are several more precise notions of a ring spectrum, such as an "E"∞ ring spectrum, where the product is commutative and associative in a strong sense.
Other cohomology theories.
Cohomology theories in a broader sense (invariants of other algebraic or geometric structures, rather than of topological spaces) include: