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Chris Hillman
Science Advisor
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Main Question or Discussion Point
This thread is similar to but more elementary than
https://www.physicsforums.com/showthread.php?t=379608
I plan to
Lets begin with a very natural question:
It helps to know from the outset that AT falls into two topics (which eventually turn out to be related):
And it helps to know from the outset that the origins of homology/cohomology lie with some natural approaches to some basic problems:
(I hope to eventually explain some of the terms I didn't define above, either in this BRS thread or the Cellular Homology BRS thread. In a later post in the BRS thread cited above, "Cellular Homology with Macaulay2", I hope to say more about Schubert calculus--- note that another Macaulay2 package allows us to compute with the cohomology ring of Grassmannian manifolds. I hope to eventually discuss Caratheodory's ideas in another BRS thread on the common ground shared by electrical networks and some formally identical problems involving vibrations, etc.)
It also helps to know from the outset that one of the most important developments in AT was the recognition that all varieties of homology/cohomology share a common ground in a more fundamental subject, commutative algebra and homological algebra, which deals with sequences of R-module homomorphisms between certain R-modules (some fixed ring R), forming a "chain complex". Here, the key idea is that as soon as you have a sequence of maps which obey the key property "boundary of a boundary vanishes" [itex]\partial^2 = 0[/itex], the image of each map in the sequence is a submodule of the kernel of the next map, both being submodules of a chain group, so one can define the quotient modules as the homology modules. Then one can define a graded R-module which is the direct sum of all the homology modules, so that we need only refer to "the homology", instead of a sequence of homology modules.
In algebraic topology, for a given topological space X, it turns out that the homology graded module H(X), but not the chain graded module C(X), is invariant under homeomorphisms, and thus a topological invariant of X. In fact, H(X) is invariant under a much more general equivalence relation which arises in homotopy theory, called homotopic equivalence. Because this relation is weaker than homeomorphism, homology cannot be a complete topological invariant; there are spaces which are not homeomorphic but which homology cannot distinguish. (I plan to give and sketch some examples in a later post in this thread.) It turns out that the abelianization of the fundamental group of X is just the homology in dimension one, [itex]H_1(X)[/itex].
And very roughly, the role of the various competing theories goes something like this:
https://www.physicsforums.com/showthread.php?t=379608
I plan to
- explain how to use the SimplicialComplexes package in Macaulay2 to construct simplicial complexes modeling some simple surfaces and then to compute their simplicial homology (starting with integer coefficients),
- try to offer some motivation for some of key definitions used in simplicial homology
- try to offer a very broad overview of some basic conceptual issues which few algebraic topology textbooks even try to address (no doubt the authors feel this is the job of the instructor)
Lets begin with a very natural question:
Well, in the interest of efficiency, few if any AT textbooks really bother to try to explain that AT began with all kinds of very concrete and sometimes not very rigorous arguments, and over decades leaders in the field gradually figured out how making various highly non-obvious definitions allows one to streamline and make rigorous the development.puzzled beginning student of algebraic topology said:I see that our textbook discusses a number of varieties of homology: simplicial, singular, cellular, and later discusses Czech and De Rham cohomology. I have heard that these are all mostly equivalent, at least for manifolds and spaces which are similar to manifolds, like orbifolds. So why so many varieties?
It helps to know from the outset that AT falls into two topics (which eventually turn out to be related):
- homotopy theory, where a basic equivalence notion is homotopic paths; it turns out that problems involving homotopic equivalence of paths and loops (think of "nice" maps from the circle S^1 into some space X) is a rather different subject from homotopic equivalence of "nice" maps from higher dimensional spheres S^q into X (a much harder question!),
- homology theory, where a basic equivalence notion if homologous one-chains (sorta like paths) and more generally, homologous q-chains (q is a dimension)
And it helps to know from the outset that the origins of homology/cohomology lie with some natural approaches to some basic problems:
- Kirchhoff's brilliant theory of electrical networks turns out to be closely related to homology, and is also closely connected to some lovely topics in the combinatorics of networks (I was delighted to see in a recent post in the n-category cafe that John Baez has finally gotten around to following up on my suggestion years ago that he look for a "categorization" of Kirchhoff's theory),
- Riemann's ideas on Riemann surfaces, Cauchy's integral theorem (which is related to winding numbers), and analytic continuation in the theory of functions of one complex variable led Betti to Betti numbers which eventually turned out to be the ranks of the R-modules appearing in the homology over a ring R (if R is the ring of integers, R-modules are just abelian groups); in turns out that in the integral theorems of complex analysis one is integrating over 1-chains, which gives some insight into the difference between "homologous chains" versus "homotopic paths"; the idea of analytic continuation leads naturally to the theory of topological manifolds and local coordinate charts on same,
- the idea of simplicial approximations to manifolds was suggested by Betti for studying all of these properties
- metric properties of Riemannian manifolds
- integration on smooth manifolds (eventually led to De Rham cohomology),
- topological properties of topological manifolds (eventually led to simplicial homology/cohomology)
- Schubert's ideas on counting things like the number of lines intersecting four given lines in [itex]\mathbb{R}^3[/itex], the number of lines lying on a generic cubic surface in [itex]\mathbb{R}^3[/itex] (questions which arise naturally in 19th century algebraic geometry) led him to his famous Schubert calculus, one of the most controversial topics in 19th century mathematics (yes, poor Schubert was called a "crank"!); attempts to put this on a rigorous footing led to the idea of intersection number and to the recognition that complex projective spaces are a more natural setting; one of the astounding early developments in cohomology was the recognition that intersection numbers can be understood in terms of computations in the cohomology ring of complex Grassmannian manifolds (in the case of the first type of problem, concerning configurations of k-flats in complex projective n-space) and certain sheaves over Grassmannians (in the second type of problem),
- around the turn of the century, Caratheodory noticed a connection between exterior calculus, homology theory, and classical thermodynamics, specifically the definition of entropy and the Second Law of Thermodynamics; here, a key lemma was the classification of exterior one-forms by Darboux, who was inspired in turn by Pfaff (cf. Pfaffians in the theory of PDEs).
(I hope to eventually explain some of the terms I didn't define above, either in this BRS thread or the Cellular Homology BRS thread. In a later post in the BRS thread cited above, "Cellular Homology with Macaulay2", I hope to say more about Schubert calculus--- note that another Macaulay2 package allows us to compute with the cohomology ring of Grassmannian manifolds. I hope to eventually discuss Caratheodory's ideas in another BRS thread on the common ground shared by electrical networks and some formally identical problems involving vibrations, etc.)
It also helps to know from the outset that one of the most important developments in AT was the recognition that all varieties of homology/cohomology share a common ground in a more fundamental subject, commutative algebra and homological algebra, which deals with sequences of R-module homomorphisms between certain R-modules (some fixed ring R), forming a "chain complex". Here, the key idea is that as soon as you have a sequence of maps which obey the key property "boundary of a boundary vanishes" [itex]\partial^2 = 0[/itex], the image of each map in the sequence is a submodule of the kernel of the next map, both being submodules of a chain group, so one can define the quotient modules as the homology modules. Then one can define a graded R-module which is the direct sum of all the homology modules, so that we need only refer to "the homology", instead of a sequence of homology modules.
In algebraic topology, for a given topological space X, it turns out that the homology graded module H(X), but not the chain graded module C(X), is invariant under homeomorphisms, and thus a topological invariant of X. In fact, H(X) is invariant under a much more general equivalence relation which arises in homotopy theory, called homotopic equivalence. Because this relation is weaker than homeomorphism, homology cannot be a complete topological invariant; there are spaces which are not homeomorphic but which homology cannot distinguish. (I plan to give and sketch some examples in a later post in this thread.) It turns out that the abelianization of the fundamental group of X is just the homology in dimension one, [itex]H_1(X)[/itex].
And very roughly, the role of the various competing theories goes something like this:
- cellular homology is by far most efficient for spaces which can be given a "cellular structure" (you can even do algebraic topology in your head, in simple cases), and then the "meaning" of the homology is most intuitive,
- the definition of the boundary maps and cup product are by far the simplest in simplicial homology; the main drawbacks are that rigorous development is hard (without following the textbook route of moving into singular homology and later proving an equivalence theorem) and that the simplicial chain complexes tend to be much larger than is needed in cellular homology,
- the most abstract but most powerful theory for developing what turn out to be key theorems (although their central role is not obvious!), especially Mayer-Vietoris theorem, is singular homology, which uses a vastly generalized notion of "singular simplex", resulting in huge chain spaces,
- the most useful theory for finite dimensional topological manifolds, including Lie groups is De Rham cohomology; the case of compact finite dimensional manifolds is especially nice; this subject provides a close connection with exterior calculus (which explains why the boundary map in simplicial homology looks so much like the definition of the exterior derivative in exterior calculus!),
- the most useful theory for sheaves (needed everywhere in algebraic geometry, and increasingly in other subjects) is Czech cohomology.
Well, it turns out to be much easier, in specific cases, to compute with coefficients in the finite field [itex]R = \mathbb{Z}/p[/itex] (where p is some prime). And in de Rham comology, we should work with real coefficients to see the connections with exterior calculus, which again make it possible to compute the de Rham cohomology of real Lie groups easily, using the Cartan-Maurer one-form. (In a later BRS thread, I hope to discuss "De Rham cohomology with Maple" and to compare results with cohomology computed using other approaches.) And by trying various [itex]R = \mathbb{Z}/p[/itex] we can sometimes distinguish between spaces which we could not otherwise distinguish.puzzled beginning student of algebraic topology said:OK, but why do we need to use so many rings? I hear that homology/cohomology over the integers gives more information, so why use any ring other than [itex]\mathbb{Z}[/itex]?
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