Geometrical arguments with homology were only gradually replaced by rigorous techniques in the early 20th Century. Originally, they used combinatorial topology which assumes that the spaces that are treated are simplicial complexes while the most interesting spaces are manifolds so that artificial triangulations have to be introduced to apply the tools. Combinatorial topology allowed Brouwer to prove the simplicial approximation theorem, which is based on the idea that homology is a functor. Brouwer also proved the Jordan curve theorem, and the invariance of domain.
The transition from combinatorial topology to algebraic topology is attributed to Emmy Noether who said that homology classes are in quotient groups. Emmy Noether, in the period from 1920 onwards, was with her students elaborating the theory of modules for any ring. The ideas of homology and rings were combined to give the idea of homology with coefficients in a ring. The coefficients are the way in which chains are linear combinations of the basic geometric chains traced on the space. Originally, they had been assumed to be integers. Then they could be viewed as integers, real, or complex numbers, or possibly residue classes mod 2. However, with the recent developments by Emmy Noether, they could, for instance, be residue classes mod 3. Then the cycle would be a more complicated geometrical situation where the number of incoming edges at every vertex has to be multiple of 3. The universal coefficient theorem determines all other homology theories through the tensor product.
During the 1930’s, you had the development of cohomology theories. Several research directions came together, and the de Rham cohomology that was implicit in Poincare’s work became the subject of definite theorems. Homology and cohomology are dual theories. The details required working out. Also, singular homology avoided the need for the apparatus of triangulations, although this required using infinitely generated modules.
From 1940 to 1960, the subject of algebraic topology advanced very rapidly. Homology theory was often used as a baseline theory, easy to compute, and in terms of which topologists sought to calculate other functors. The axiomatisation of homology theory by Eilenberg and Steenrod showed that what various homology theories had in common was some exact sequences, such as the Mayer-Vietoris theorem, and the dimension axiom, which calculated the homology of a point. Then the dimension axiom was relaxed to admit cohomology derived from topological K-theory and cobordism. This vastly extended cohomology to include extraordinary cohomology theories, that became standard in homotopy theory. These can be easily characterized for the category of CW complexes. For more general spaces, recourse to sheaf theory brought some extension of homology theories, such as the Borel-Moore theory for locally compact spaces.
Cohomology is a general term for a sequence of abelian groups defined from a cochain complex. Cohomology is defined as the abstract study of cochains, cocycles, and coboundaries. Cohomology can be viewed as a method of assigning algebraic invariants to a topological space that has a more refined algebraic structure than homology. Cohomology arises from the algebraic dualization of homology. Cochains should assign quantities to the chains in homology.
From its beginning in topology, this idea became a dominant method in the mathematics of the second half of the 20th Century. For many applications, cohomology, a contravariant theory, is more natural than homology. This has to do with functors and pullbacks in geometric situations. Given spaces X and Y, and some kind of functor F on Y, for any mapping f : X → Y, composition with f gives rise to a functor F o f on X. Cohomology groups also have natural products, makin calculation easier.
Although cohomology is fundamental to modern algebraic topology, its importance was not realized until 40 years after the development of homology. The concept of dual cell structure, which Henri Poincare used in his proof of the Poincare duality theorem, contained the origin of the idea of cohomology, but this was not realized until later.
There were various precursors to cohomology. In the 1930’s, J. W. Alexander and Lefschetz founded the intersection theory of cycles on manifolds. On an n-dimensional manifold M, a p-cycle and a q-cycle with non-empty intersection, will, if in a general position, have an intersection of a (p + q – n)-cycle. Therefore, you can define a multiplication of homology classes.
Hp(M) x Hq(M) → Hp + q – n(M)
Here is a brief chronology of cohomology.
1. In 1893, Henri Poincare used what was later recognized to be a reference to cohomology to prove the Poincare duality theorem.
2. In 1930, Alexander defined the cochain based on a p-chain on a space X having relevance to small neighborhoods of the diagonal in Xp + 1.
3. In 1931, de Rham related homology and exterior differential forms, proving de Rham’s theorem. This result is now understood to be more naturally interpreted in terms of cohomology.
4. In 1934, Pontrjagin proved the Pontrjagin duality theorem, which is a result on topological groups. This, in special cases, provided an interpretation of Poincare’s duality and Alexander’s duality in terms of group characters.
5. In a 1935 conference in Moscow, Kolmogorov and Alexander both introduced cohomology, and tried to construct a cohomology product structure.
6. In 1936, Steenrod published a paper constructing Cech cohomology by dualizing Cech homology.
7. From 1936 to 1938, Hassler Whitney and Eduard Cech developed the cup product, making cohomology into a graded ring, and the cap product, and realized that Poincare duality can be stated in terms of the cap product. Their theory was still limited to cell complexes.
8. In 1944, Eilenberg overcame technical limitations, and gave the modern definition of singular homology and cohomology.
9. In 1945, Eilenberg and Steenrod stated the axioms defining a homology and cohomology theory.
10. In 1948, Spanier, building on work of Alexander and Kolmogorov, developed Alexander-Spanier cohomology.
11. In 1952, Eilenberg and Steenrod publish “Foundations of Algebraic Topology”, where they prove that the existing homology and cohomology theories satisfied their axioms.
12. In 1959, Jean-Pierre Serre used the analogy of vector bundles with projective modules to create algebraic K-theory.
A cohomology theory is a family of contravariant functors from the category of pairs of topological spaces and continuous functions, or some subcategory thereof, such as the category of CW complexes, to the category of abelian groups and group homomorphisms that satisfies the Eilenberg-Steenrod axioms.
The Eilenberg-Steenrod axioms apply to a sequence of functors Hn from the category of pairs of topological spaces to the category of abelian groups, together with a natural transformation
∂ : Hi(X, A) → Hi – 1(A)
called the boundary map. The Eileen-Steenrod axioms are
1. Homotopy – Homotopic maps induce the same maps in homology.
2. Excision – If (X, A) is a pair and U is contained in the interior of A, then the inclusion map
i : (X – U, A – U) → (X, A)
induces an isomorphism in homology.
3. Let P be the one-point space. Then Hn(P) = 0 for all n ≠ 0.
4. Addivity – If
X = Vα Xα
then
Hn(X) = [direct sum series over α] Hn(Xα)
5. Exactness – Each pair (X, A) induces a long exact sequence in homology via the inclusions i : A → X and j : X → (X, A)
...→ Hn(A) →(i) Hn(X) →(j) Hn(X, A) →(∂) Hn – 1(A) → ...
If P is the one point space then H0(P) is called the coefficient group. For instance, singular homology has integer coefficients.
Some facts about homology groups can be derived directly from the axioms, such as the fact that homotopically equivalent spaces have isomorphic homology groups. The homology of some relatively simple spaces, such as n-spheres, can be calculated directly from the axioms. From this, it can be easily shown that the (n – 1)-sphere is not a retract from the n-disk.
The main cohomology theories that satisfy the Eilenberg-Steenrod axioms are
1. simplicial cohomology
2. singular cohomology
3. de Rham cohomology
4. Čech cohomology
5. Sheaf cohomology
6. Alexander-Spanier cohomology
Let’s say you take the Eilenberg-Steenrod axioms that I just listed, and you relax axiom #3, meaning that you allow cohomology theories that do not obey the third axiom but do obey the others. The third axiom is
3. Let P be the one-point space. Then Hn(P) = 0 for all n ≠ 0.
Relaxing this condition gives rise to a new type of cohomology theory called extraordinary cohomology theory. This allows theories based on K-theory and cobordism theory, as well as other theories coming from stable homotopy theory.
Since the axioms that extraordinary cohomology theories obey are less restrictive than those that traditional cohomology theories obey, there are therefore a lot more of them. The main extraordinary cohomology theories include
1. Group cohomology
2. Galois cohomology
3. Lie algebra cohomology
4. Harrison cohomology
5. Γ cohomology
6. Schur cohomology
7. Andre-Quillen cohomology
8. Hochschild cohomology
9. Cyclic cohomology
10. Topological Andre-Quillen cohomology
11. Topological Hochschild cohomology
12. Topological Cyclic cohomology
13. Coherent cohomology
14. Local cohomology
15. Etale cohomology
16. Crystalline cohomology
17. Flat cohomology
18. Motive cohomology
19. Deligne cohomology
20. Perverse cohomology
21. Intersection cohomology
22. Non-abelian cohomology
23. Gel’fand-Fuks cohomology
24. Spencer cohomology
25. Bonar-Claven cohomology
26. K-theory
The most famous cohomology theory is de Rham cohomology, invented by Georges de Rham, which is a cohomology theory based on the existence of differential forms with prescribed properties. It is, in different ways, dual to both singular homology and Alexander-Spanier cohomology.
Let’s say you have a set of smooth differentiable differential k-forms on any smooth manifold M which form an abelian group, which is a real vector space called
Ωk(M)
under addition. The exterior derivative d gives mappings
d : Ωk(M) → Ωk + 1(M)
There is a fundamental relationship
d2 = 0
which follows from the symmetry of second derivatives. Therefore vector spaces of k-forms along with the exterior derivative are a cochain complex called the de Rham complex
C∞(M) = Ω0(M) → Ω1(M) → Ω2(M) → Ω3(M) →...
Forms that are exterior derivatives are called exact. Forms whose exterior derivatives are zero are called closed. The relationship d2 = 0 then says that exact forms are closed.
Of course. Closed forms don’t have to be exact. The idea of de Rham cohomology is to classify the different types of closed forms on a manifold. Two closed forms α and β in Ωk(M) are called cohomologous if they differ by an exact form, in other words, if α - β is exact. This means that there is an equivalence relation on the space of closed forms in Ωk(M). You then define the k-th de Rham cohomology group
HdRk(M)
To be the set of equivalence classes, which is the set of closed forms in Ωk(M) modulo the exact forms.
For any manifold M with n connected components
HdR0 = Rn
Where the equal sign means that the two are homeomorphic. This follows from the fact that any C∞ function on M with zero derivative is locally constant on each of the connected components.
You can find the general de Rham cohomologies of a manifold by using the above fact about zero cohomology and the Mayer-Vietoris sequence, and also the fact that de Rham cohomology is homotopy invariant.
Here are the de Rham cohomology groups of some topological spaces.
1. sphere, Sn
HdRk(Sn) = R if k = 0, n, and = 0 if k ≠ 0, n
2. torus, Tn
HdRk(Tn) = R(n, k)
3. punctured plane, R2 - {(0, 0)}
HdRk( R2 - {(0, 0)}) = R if k = 0, n – 1, and = 0 if k ≠ 0, n – 1
4. Möbius strip, MS
HdRk = HdRk(S1)
De Rham’s theorem, proved by Georges de Rham in 1931, states that for a compact oriented smooth manifold M, the groups HdRk(M) are isomorphic as real vector spaces with singular cohomology groups.
HdRk = (M ; R)
The wedge product endows the direct sum with a ring structure. A further result of the theorem is that two cohomology rings are isomorphic, as graded rings, where the analogous product on singular cohomology is the cup product. The general Stokes theorem is an expression of duality between de Rham cohomology and the homology of chains.
The first extraordinary cohomology theory was K-theory. It includes both topological K-theory and algebraic K-theory. It leads to the construction of K-functors which contain useful information, which is, however, often difficult to compute. K-theory was originally discovered/invented by Alexander Grothendieck so he could formulate his Grothendieck-Riemann-Rock theorem. K-theory stands for Klassen-theory, where “Klassen” is the German word for “class”. Grothendieck needed to convert the commutative monoid of sheaves with an operation of direct sum into a group. Instead of attempting to work with sheaves directly, he took formal sums of certain classes of sheaves and formally added inverses. This is an explicit way of obtaining a left adjoint of a certain functor. This construction is called the Grothendieck group. It was taken up by Michael Atiyah and Friedrich Hirzebruch to define K(X) for a topological space X by means of the analogous sum construction for vector bundles. This was the basis of the first of the extraordinary cohomology theories of algebraic topology. It played a big role in the second proof of the Index Theorem in 1962. This approach also led to a noncommutative K-theory for C*-algebra.
In 1959, Jean-Pierre Serre used the analogy of vector bundles with projective modules to create algebraic K-theory. He formulated Serre’s conjecture, that projective modules over the ring of polynomials over a free field are modules. This resisted proof for 20 years. There then followed a period in which there were various partial definitions of higher K-functors until a comprehensive definition was given by Daniel Quillen using homotopy theory. The corresponding constructions involving an auxiliary quadratic form are called L-theory. It is a major tool in surgery theory.
Topological K-theory is the true K-theory in the sense that it came first. Topological K-theory has to do with vector bundles over topological spaces. Elements of K-theory are stable equivalence classes of vector bundles over a topological space. You can put a ring structure on the collection of stably equivalent bundles by defining addition through the Whitney sum, and multiplication through the tensor product of vector bundles. This defines the reduced real topological theory of a space. Of course, you could also use complex vector bundles instead of real vector bundles. Topological K-theory is significant because it forms a generalized cohomology theory, and it leads to a solution of the vector fields on a sphere problem, as well as to an understanding of the J-homeomorphism of homotopy theory.
In 1962, Swan noticed that there is a correspondence between the category of suitable topological spaces, such as Hausdorff spaces, and C*-algebras. The idea is to associate to every space the C*-algebra of continuous maps from that space to the reals. A vector bundle over a space has sections, and these sections can be multiplied by continuous functions to the reals. According the Swan’s correspondence, vector bundles correspond to modules over the C*-algebra of continuous functions, the modules being the modules of sections of the vector bundle. The study of modules over C*-algebras is the starting point of algebraic K-theory. The Quillen-Lichtenbaum conjecture connects algebraic K-theory to Etale cohomology.
Alexander Grothendieck suddenly disappeared in 1991. Nobody knew if he was alive or dead. His whereabouts or ultimate fate were the source of endless speculation in the mathematical community. However, it turned out that he became a hermit, and is living in seclusion because he’s embarrassed by the fact that he can no longer understand advanced math papers currently being published in the field he helped create, since the field has advanced so much since he made his contribution.
The Grothendieck group of a commutative monoid is the universal way of making the monoid into an abelian group. Let’s say M is a commutative monoid. Its Grothendieck group N should have the following universal property. There exists a map
i : M → N
such that for any map
f : M → A
from the commutative monoid M to an abelian group A, there is a unique map.
g : N → A
such that
f = gi
In other words, the forgetful functor from the category of abelian groups to the category of commutative monoids has a left adjoint.
To construct the Grothendieck group of a commutative monoid M, you take the cartesian product
M x M
The two coordinates (m, n) represent a positive part and a negative part, and are meant to correspond to m – n. Addition is defined coordinate-wise.
(m1, m2) + (n1, n2) = (m1 + n1, m2 + n2)
Next define an equivalence relation on M x M. (m1, m2) is equivalent to (n1, n2) if for some element k of M
m1 + n2 + k = m2 + n1 + k
It is easy to check that the addition is compatible with the equivalence relation. The identity is any element of the form (m, m), in other words, when m1 = m2. The inverse of (m1, m2) is (m2, m1).
In this form, the Grothendieck group is the fundamental construction of K-theory. The group K0(M) of a manifold M is defined to be the Grothendieck group of the commutative monoid of all vector bundles on M with the group operation given by the direct sum.
I’m now going to give a more technically rigorous discussion of K-theory. It’s not necessary to understand this level of technical detail to use K-theory to study string theory. Let’s say you have a topological space X. Consider all complex vector bundles over X. Including their homeomorphisms, these form a category Vect(X). There is a sort of semi-ring structure on that category in that vector bundles can be added, using the direct sum, and multiplied using the direct product. When we take equivalence classes in Vect(X), the result is an ordinary semi-ring. There is a standard way, called Grothendieck group completion, to throw in additive inverses so you can get an actual ring. This ring is denoted K0(X) and is called the K-theory of X. Instead of X, you can consider the suspension SX of X. This is sort of a sphere where the cross section at every latitude looks like X. More precisely, it is the space obtained by taking the product of X with the unit interval, and identifying all points of X attached to 0 ∈ [0, 1], and all those attached to 1 ∈ [0, 1]:
SX= (X x [0, 1])/((X x {0}) ∪ (X x {1}))
The n-th interation of taking the suspension is written SnX, and you define
Kn(X) = K0(SnX)
Bott periodicity says that Kn(X) is isomorphic to Kn + 2(X).
Kn(X) ~ Kn + 2(X)
If you had used real vector bundles instead of complex, you’d have periodicity of 8 instead of 2.
Often you would want to do away from topological spaces and replace them with their algebraic equivalent which is C*-algebras. Given the C*-algebra A = C(X) = {f : X → C} of continuous complex-valued functions on X, a vector bundle over X can equivalently be characterized as a projection in Mn(A), such as an n x n matrix P with values in A that satisfy P = P* = P2. So an alternative way to define K0, but now generalized to arbitrary C*-algebras A, is an abelian group K0(A) which has one generator [P] for every projector in Mn.(A) for all n subject to the identification of projections which can be continuously connected and to the relation
[p] + [q] = [p ⊕ q]
This formulation makes it easy to define a sort of dual K-theory. For every C*-algebra A, there is a dual C*-algebra Dρ(A) defined as the commutant up to compact operators of any representation of A. So given A, choose any representation ρ : A → B(H) of A in terms of bounded operators on some separable Hilbert space H, then
Dρ(A) = {T ∈ B(H) : [T, ρ(a)] ~0, for all a ∈ A}
Where T1 ~T2 means that T1 - T2 ∈ K(H) is a compact operator.
It may be that A didn’t have a unit. Let the result of making it unital by adjoining a unit be denoted [A tilde]. The K-theory of D([A tilde]) for any ρ is now called K-homology, and you write
The position of the indices indicates where the maps are covariant and contravariant, when these are regarded as functors from the category of C*-algebras to that of abelian groups. The reason for calling the map A → Kp(A) a homology is that in the case where A = C(X) is the C*-algebra of functors on X, this map can be shown to define what is called a generalized homology theory. This is any theory that associates a list of groups to any topological space such that a couple of crucial properties from simplicial homology are satisfied. What is the homology theory corresponding to Kp? It turns out this is just the K-theory that we started with
Kp(A) ~Hom(Kp(A), Z)
Kp(C(X)) is nothing but equivalence classes of generalized Dirac operators on rank-p vector bundles over X.
A Fredholm operator is a bounded operator F which admits the idea of an index.
Index(F) = Dim(Kernel(F)) – Dim(Cokernel(F))
A Fredholm module is like a spectral triple with a Fredholm operator instead of a Dirac operator. Specifically, it’s a triple (H, ρ(A), F) of a Hilbert space H on which F and the C*-algebra A are represented by ρ : A → B(H). If in addition, a Clifford algebra Cp is represented by H which commutes with F, this is called a p-multigraded Fredholm operator.
Kasparov defined the Kasparov K-homology group KK-p(A) to be the abelian group of equivalence classes of Fredholm modules. More precisely, KK-p(A) is the abelian group generated by p-multigraded Fredholm modules (H, ρ(A), F) up to unitary equivalence, and are modules which can be continuously connected by the relation
[a] + [b] = [a [direct sum] b]
where a and b are Fredholm operators, and [a] and [b] are their images in KK-1(X). It turns out that Kasparov groups are isomorphic to ordinary K-homology groups.
KK-p(A) ~ Kp(A)
This gives an interpretation for the pairing between K-homology and K-theory.