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| General idea | |
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A torus with generators colored in pink and red.
To any topological space X and any natural number k, one can associate a set Hk(X), whose elements are called (k-dimensional) homology classes. There is a well-defined way to add and subtract homology classes, which makes Hk(X) into an abelian group, called the kth homology group of X. In heuristic terms, the size and structure of Hk(X) gives information about the number of k-dimensional holes in X. For example, if X is a figure eight, then it has two holes, which in this context count as being one-dimensional. The corresponding homology group H1(X) can be identified with the group of pairs of integers, with one copy of for each hole. While it seems very straightforward to say that X has two holes, it is surprisingly hard to formulate this in a mathematically rigorous way; this is a central purpose of homology theory.
For a more intricate example, if Y is a Klein bottle then H1(Y) can be identified with . This is not just a sum of copies of , so it gives more subtle information than just a count of holes.
The formal definition of H1(X) can be sketched as follows. The elements of H1(X) are one-dimensional cycles, except that two cycles are considered to represent the same element if they are homologous. The simplest kind of one-dimensional cycles are just closed curves in X, which could consist of one or more loops. If a closed curve C0 can be deformed continuously within X to another closed curve C1, then C0 and C1 are homologous and so determine the same element of H1(X). This captures the main geometric idea, but the full definition is somewhat more complex. For details, see singular homology. There is also a version (called simplicial homology) that works when X is presented as a simplicial complex; this is smaller and easier to understand, but technically less flexible.
For example, let T be a torus, as shown on the right. Let C be the pink curve, and let D be the red one. For integers n and m, we have another closed curve that goes n times around C and then m times around D; this is denoted by nC + mD. It can be shown that any closed curve in T is homologous to nC + mD for some n and m, and thus that H1(T) is again isomorphic to .
[edit] Tags:Homology,Topological Space,Singular Homology,Simplicial Homology,Simplicial Complex, | |
| Cohomology | |
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As well as the homology groups Hk(X), one can define cohomology groups Hk(X). In the common case where each group Hk(X) is isomorphic to for some , we just have , which is again isomorphic to , and , so Hk(X) and Hk(X) determine each other. In general, the relationship between Hk(X) and Hk(X) is only a little more complicated, and is controlled by the universal coefficient theorem. The main advantage of cohomology over homology is that it has a natural ring structure: there is a way to multiply an i-dimensional cohomology class by a j-dimensional cohomology class to get an i + j-dimensional cohomology class.
[edit] Tags:Universal Coefficient Theorem, | |
| Applications | |
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Notable theorems proved using homology include the following:
The Brouwer fixed point theorem: If f is any continuous map from the ball Bn to itself, then there is a fixed point with f(a) = a.
Invariance of domain: If U is an open subset of and is an injective continuous map, then V = f(U) is open and f is a homeomorphism between U and V.
The Hairy ball theorem: any vector field on the 2-sphere (or more generally, the 2k-sphere for any ) vanishes at some point.
The Borsuk–Ulam theorem: any continuous function from an n-sphere into Euclidean n-space maps some pair of antipodal points to the same point. (Two points on a sphere are called antipodal if they are in exactly opposite directions from the sphere's center.)
[edit] Tags:Invariance Of Domain,Injective,Continuous Map,Homeomorphism,Borsuk–ulam Theorem,Continuous Function,Euclidean ,Antipodal Points,Open Subset, | |
| Intersection theory and Poincaré duality | |
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Let M be a compact oriented manifold of dimension n. The Poincaré duality theorem gives a natural isomorphism , which we can use to transfer the ring structure from cohomology to homology. For any compact oriented submanifold of dimension d, one can define a so-called fundamental class . If L is another compact oriented submanifold which meets N transversely, it works out that . In many cases the group Hd(M) will have a basis consisting of fundamental classes of submanifolds, in which case the product rule gives a very clear geometric picture of the ring structure.
[edit] Tags:Compact,Oriented,Manifold, | |
| Connection with integration | |
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Suppose that X is an open subset of the complex plane, that f(z) is a holomorphic function on X, and that C is a closed curve in X. There is then a standard way to define the contour integral , which is a central idea in complex analysis. One formulation of Cauchy's integral theorem is as follows: if C0 and C1 are homologous, then . (Many authors consider only the case where X is simply connected, in which case every closed curve is homologous to the empty curve and so .) This means that we can make sense of when c is merely a homology class, or in other words an element of H1(X). It is also important that in the case where f(z) is the derivative of another function g(z), we always have (even when C is not homologous to zero).
This is the simplest case of a much more general relationship between homology and integration, which is most efficiently formulated in terms of differential forms and de Rham cohomology. To explain this briefly, suppose that X is an open subset of , or more generally, that X is a manifold. One can then define objects called n-forms on X. If X is open in , then the 0-forms are just the scalar fields, the 1-forms are the vector fields, the 2-forms are the same as the 1-forms, and the 3-forms are the same as the 0-forms. There is also a kind of differentiation operation called the exterior derivative: if α is an n-form, then the exterior derivative is an (n + 1)-form denoted by dα. The standard operators div, grad and curl from vector calculus can be seen as special cases of this. There is a procedure for integrating an n-form α over an n-cycle C to get a number . It can be shown that for any (n − 1)-form β, and that depends only on the homology class of C, provided that dα = 0. The classical Stokes's Theorem and Divergence Theorem can be seen as special cases of this.
We say that α is closed if dα = 0, and exact if α = dβ for some β. It can be shown that ddβ is always zero, so that exact forms are always closed. The de Rham cohomology group is the quotient of the group of closed forms by the subgroup of exact forms. It follows from the above that there is a well-defined pairing given by integration.
[edit] Tags:Complex Plane,Holomorphic,Complex Analysis,Differential Forms,Exterior Derivative,Div,Grad,Stokes's Theorem,Divergence Theorem, | |
| Axiomatics and generalised homology | |
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There are various ways to define cohomology groups (for example singular cohomology, Čech cohomology, Alexander–Spanier cohomology or Sheaf cohomology). These give different answers for some exotic 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 finite CW complexes, for example.
One of the axioms is the so-called dimension axiom: if P is a single point, then Hn(P) = 0 for all , and . We can generalise slightly by allowing an arbitrary abelian group A in dimension zero, but still insisting that the groups in nonzero dimension are trivial. It turns out that there is again an essentially unique system of groups satisfying these axioms, which are denoted by H * (X;A). In the common case where each group Hk(X) is isomorphic to for some , we just have . In general, the relationship between Hk(X) and Hk(X;A) is only a little more complicated, and is again controlled by the Universal coefficient theorem.
More significantly, we can drop the dimension axiom altogether. There are a number of different ways to define groups satisfying all the other axioms, including the following:
The stable homotopy groups
Various different flavours of cobordism groups: MO * (X), MSO * (X), MU * (X) and so on. The last of these (known as complex cobordism) is especially important, because of the link with formal group theory via a theorem of Daniel Quillen.
Various different flavours of K-theory: KO * (X) (real periodic K-theory), kO * (X) (real connective), KU * (X) (complex periodic), kU * (X) (complex connective) and so on.
Brown–Peterson homology, Morava K-theory, Morava E-theory, and other theories defined using the algebra of formal groups.
Various flavours of elliptic homology
These are called generalised homology theories; they carry much richer information than ordinary homology, but are often harder to compute. Their study is tightly linked (via the Brown representability theorem) to stable homotopy.
[edit] Tags:Axiomatic,Singular Cohomology,Čech Cohomology,Alexander–spanier Cohomology,Sheaf Cohomology,Eilenberg–steenrod Axioms,Cobordism,Complex Cobordism, | |
| Homological algebra and homology of other objects | |
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A chain complex consists of groups Ci (for all ) and homomorphisms satisfying dd = 0. This condition shows that the groups are contained in the groups , so one can form the quotient groups Hi = Zi / Bi, which are called the homology groups of the original complex. There is a similar theory of cochain complexes, consisting of groups Ci and homomorphisms . The simplicial, singular, Čech and Alexander–Spanier groups are all defined by first constructing a chain complex or cochain complex, and then taking its homology. Thus, a substantial part of the work in setting up these groups involves the general theory of chain and cochain complexes, which is known as homological algebra.
One can also associate (co)chain complexes to a wide variety of other mathematical objects, and then take their (co)homology. For example, there are cohomology modules for groups, Lie algebras and so on.
[edit] Tags:Homological Algebra, | |
| References | |
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Hilton, Peter (1988), "A Brief, Subjective History of Homology and Homotopy Theory in This Century", Mathematics Magazine 60 (5): 282–291, JSTOR 2689545
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Tags:Mathematics, | |
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