Cohomology
Cohomology can be viewed as a method of assigning richer algebraic invariants to a space than homology.
From its start in topology, this idea became a dominant method in the mathematics of the second half of the twentieth century.
The terminology tends to hide the fact that cohomology, a contravariant theory, is more natural than homology in many applications.
Unlike more subtle invariants such as homotopy groups, the cohomology ring tends to be computable in practice for spaces of interest.
is the homology of this chain complex (the kernel of one homomorphism modulo the image of the previous one).
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.
can move freely on X in the sense that N could be replaced by any continuous deformation of N inside M. In what follows, cohomology is taken with coefficients in the integers Z, unless stated otherwise.
Another interpretation of Poincaré duality is that the cohomology ring of a closed oriented manifold is self-dual in a strong sense.
More precisely, pulling back the class u gives a bijection for every space X with the homotopy type of a CW complex.
There is a related description of the first cohomology with coefficients in any abelian group A, say for a CW complex X. Namely,
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 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.
Although cohomology is fundamental to modern algebraic topology, its importance was not seen for some 40 years after the development of homology.
[15] In the mid-1920s, J. W. Alexander and Solomon Lefschetz founded intersection theory of cycles on manifolds.
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 Xi+1.
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 1948 Edwin Spanier, building on work of Alexander and Kolmogorov, developed Alexander–Spanier cohomology.
Grothendieck elegantly defined and characterized sheaf cohomology in the language of homological algebra.
Start with the functor taking a sheaf E on X to its abelian group of global sections over X, E(X).
From that point of view, sheaf cohomology becomes a sequence of functors from the derived category of sheaves on X to abelian groups.
For example, for a ring R, the Tor groups ToriR(M,N) form a "homology theory" in each variable, the left derived functors of the tensor product M⊗RN of R-modules.
The simplest case being the determination of cohomology for smooth projective varieties over a field of characteristic
This is defined as the projective limit If we have a scheme of finite type then there is an equality of dimensions for the Betti cohomology of
By definition, a generalized homology theory is a sequence of functors hi (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: hi(X, A) → hi−1(A) called the boundary homomorphism (here hi−1(A) is a shorthand for hi−1(A,∅)).
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.
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.
Cohomology theories in a broader sense (invariants of other algebraic or geometric structures, rather than of topological spaces) include:
