In this course, we will establish some key properties of these homology groups like the homotopy-invariance and the excision theorem. A convenient variant is provided by singular homology with coefficients - a framework which makes necessary a short discussion of basic homological algebra including tensor and torsion products. For CW complexes, there is also the more combinatorial cellular homology theory.
The course culminates in a proof that singular homology and cellular homology agree on CW complexes. This allows for more explicit calculations in examples of interest e. The basic philosophy of algebraic topology consists of assigning algebraic invariants to topological spaces.
prefomnalna.tkaic topology - What is the difference between homology and cohomology? - MathOverflow
These invariants are expected to be interesting enough to capture important geometric information and, at the same time, to be accessible to actual calculations. A prototype is given by singular homology theory in which case the invariants are abelian groups. In this course, we will study closely related invariants, namely singular homology with coefficients and, most of the time, singular cohomology.
It turns out that many of the key formal properties of singular homology homotopy invariance, excision, and so on are also enjoyed by these variants. A key difference occurs when one considers cohomology with coefficents in a commutative ring; in that case, the singular cohomology groups of a space can be turned into a graded-commutative ring, and this additional product structure is very useful in applications.
Topological K -theory, the first generalized cohomology theory to be studied thoroughly, was introduced in a paper by Atiyah and Hirzebruch, where they adapted the work of Grothendieck on algebraic varieties to a topological setting.
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This is an important intuition for understanding the statement of Poincare duality on a not-necessarily-compact manifold. Such multiplicative pairings on homology do often arise through cup product. However they usually not arise that naturally and also if so, it is harder to introduce them for example in terms of well-definedness. Maybe you want to look up the term "intersection form", which is dual to the cup product. However a functor also gives you maps, which is an additional tool which actually can give you additional information if you have both , you can look up an example here.
Lastly, it might be interesting for you to see an example about why the graded ring structure might be important. For example, you would expect the cup product to behave natural under wedge sum. So there is no way to distinguish them so. Now if we knew that cohomology is a functor to graded rings, i. There are a lot of different answers to this question in a lot of different levels of complexity, so I'm just going to pick one: Cohomology is a representable functor. I'm working in the category of, say, CW-complexes with basepoint throughout. There is a related description of the first cohomology with coefficients in any abelian group A , say for a CW complex X.
Namely, H 1 X , A 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 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.
Geometry & Topology
Although cohomology is fundamental to modern algebraic topology, its importance was not seen for some 40 years after the development of homology. There were various precursors to cohomology. Alexander and Solomon Lefschetz founded the intersection theory of cycles on manifolds. This leads to a multiplication of homology classes.
In , 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 , Lev Pontryagin proved the Pontryagin duality theorem; a result on topological groups. At a conference in Moscow , Andrey Kolmogorov and Alexander both introduced cohomology and tried to construct a cohomology product structure. Their theory was still limited to finite cell complexes.
In , Samuel Eilenberg overcame the technical limitations, and gave the modern definition of singular homology and cohomology. In , Eilenberg and Steenrod stated the axioms defining a homology or cohomology theory, discussed below. In their book, Foundations of Algebraic Topology , they proved that the existing homology and cohomology theories did indeed satisfy their axioms.
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 s, 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. This functor is left exact , but not necessarily right exact.
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.
Sheaf cohomology can be identified with a type of Ext group. There are numerous machines built for computing the cohomology of algebraic varieties. Tools from Hodge theory, called Hodge structures help give computations of cohomology of these types of varieties with the addition of more refined information.
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. This is defined as.
Related Algebraic Topology: Homology and Cohomology
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