<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>\nIn article <cj9b85\\$gc7\\$1@lfa222122.richmond.edu>,\nUrs Schreiber <Urs.Schreiber@uni-essen.de> wrote:\n\n> John Baez worte:\n\n>> [the fact that]\n>> after the degree-4 Pontrjagin class an E8 bundle has no\n>> more characteristic classes until we get to degree 16\n>> makes it hard to tell if we\'re dealing with an E8 bundle\n>> or a 2-gerbe here. A 2-gerbe is just a degree-4 integral\n>> cohomology class; an E8 bundle is a degree-4 integral\n>> cohomology class plus some extra fuzz that you don\'t notice\n>> until you get to very high dimensions.\n\n>> Does the stuff I wrote above make sense?\n\n>I can follow it.\n\n>> One needs to see\n>> why n-gerbes on X are classified by the (n+2)nd integral\n>> cohomology of X.\n\nHere I meant "U(1) n-gerbes", by the way.\n\n>I have read that it does and it is plausible by extrapolating from the\n>0-gerbe case, but I am not able to prove this.\n\nThat\'s okay - nobody else is either! In fact, the concept of\n"U(1) n-gerbe" is still rather poorly understood except for\nn < 3. The one thing everybody agrees on is that this concept\nshould be defined to MAKE IT TRUE that U(1) n-gerbes on a space\nX are classified (up to equivalence) by elements of the integral\ncohomology group H^{n+2}(X,Z).\n\nThe reason is that people liked these facts:\n\nHomotopy classes of maps f: X -> U(1) are the same as\nelements of H^1(X,Z).\n\nIsomorphism classes of principal U(1) bundles over X\nare the same as elements of H^2(X,Z).\n\nand wanted the pattern to continue. So, they eventually\ndefined U(1) gerbes in such a way that:\n\nEquivalence classes of U(1) gerbes over X are the same\nas elements of H^3(X,Z).\n\nAnd then Larry Breen defined U(1) 2-gerbes in such a way\nthat:\n\nEquivalence classes of U(1) 2-gerbes over X are the same\nas elements of H^4(X,Z).\n\nBy this point, if not sooner, the pattern was clear.\nIt goes like this.\n\nFor any n and any abelian group A there is a space K(A,n)\ncalled an "Eilenberg-Mac Lane space" whose nth homotopy group\nis A, with all its other homotopy groups being zero.\nThis space is unique up to homotopy equivalence. For\nany space, elements of the cohomology group H^n(X,A)\nare the same as homotopy classes of maps\n\nf: X -> K(A,n)\n\n(All this was shown by Eilenberg and Mac Lane in the 1950s).\n\nThe great thing about this is that it gives a concrete\npicture of the cohomology group H^n(X,A) to the extent\nthat we understand K(A,n). Let\'s take A = Z and look at\nsome low n.\n\nK(Z,0) = Z\n\nThis says that the 0th integral cohomology group of X\nconsists of homotopy classes of maps from X to Z. But\nsince Z is discrete these are the same as maps from X\nto Z. By "maps" I always mean continuous maps when I\'m\ndoing topology! Since Z is discrete these maps must be\nconstant on each connected component of X. So, we\'re\nsaying that H^0(X,Z) is just Z^k where k is the number\nof connected components of X. Which is true!\n\nK(Z,1) = U(1)\n\nThis says that the 1st integral cohomology group of\nX consists of homotopy classes of maps from X to the\ncircle. And this is also true. You\'ll notice that\nthis is exactly backwards from how the fundamental\ngroup of X consists of homotopy classes of maps from\nthe circle to X! (Basepoint-preserving maps, anyway.)\n\nK(Z,2) = CP^infinity\n\nThis says that the 2nd integral cohomology group of\nX consists of homotopy classes of maps from X to\nthe infinite-dimensional complex projective space -\nor equivalently, the set of pure states in an\ninfinite-dimensional complex Hilbert space.\nThis is getting less easy to visualize, but luckily\nCP^infinity is the "classifying space for U(1) bundles".\n\nThis means that it has a principal U(1) bundle over it -\nwhich I\'ll gladly describe if asked - with the property\nthat *any* principal U(1) bundle over *any* space X can\nbe gotten (up to isomorphism) by pulling back this one\nvia some map f: X -> CP^infinity.\n\nMore precisely, homotopy classes of maps\n\nf: X -> CP^infinity\n\nare in 1-1 correspondence with isomorphism classes of\nprincipal U(1) bundles over X. So, elements of H^2(X,Z)\nhave a nice interpretation as isomorphism classes of\nprincipal U(1) bundles over X.\n\nIs it just a coincidence that CP^infinity is the\nclassifying space for U(1) bundles, where U(1) was\nthe previous space on our list? No! It\'s a theorem\nthat it keeps on going like this forever! K(A,n) is\nalways an abelian topological group, and K(A,n+1)\nis the classifying space for K(A,n) bundles!\nUnfortunately this only helps us simplify things\none step.\n\nTo see what I mean, consider the next one:\n\nK(Z,3) = U(Hilbert-Schmidt(infinity))/PU(infinity)\n\nThis says that the 3rd integral cohomology group\nof X consists of homotopy classes of maps from X to\nthe unitary group of the Hilbert space of Hilbert-Schmidt\noperators on an infinite-dimensional complex Hilbert space,\nmod the action of the projectivized unitary group of that\ninfinite-dimensional Hilbert space. I don\'t want to\nexplain this: I\'m only saying it to show how these\nEilenberg-Mac Lane spaces get more and more difficult\nto understand as n goes up - at least from *certain*\nviewpoints, like trying to get a model of them with a\nsimple relation to physics.\n\nHowever, we can use the trick I mentioned above to\nsimplify things one step: the 3rd integral cohomology\ngroup of X consists of isomorphism classes of principal\nCP^infinity bundles over X. This is a bit simpler than\nthe description given in the previous paragraph, but\nstill a bit scary. For one, most people aren\'t comfortable\nwith how CP^infinity becomes an abelian group! It\'s\nactually really cool - see below - but CP^infinity bundles\nwill never seem quite as simple as U(1) bundles.\n\nSo, at this point people resorted to another trick:\ngerbes!\n\nA U(1) gerbe is like a souped-up version of a U(1) bundle.\nTo build a U(1) bundle over X, first we pick a "good\ncover" of X - a cover by open sets U_i such that each open\nset, or any finite intersection of them, is contractible.\nThen we stick a trivial U(1) bundle over each patch U_i.\nThen we glue them together by transition functions\n\ng_{ij}: U_i intersect U_j -> U(1)\n\nFor consistency, these must satisfy a condition on\neach triple intersection\n\nU_i intersect U_j intersect U_k\n\nnamely:\n\ng_{ij} g_{jk} = g_{ik}\n\nExperts call this the "1-cocycle condition", and call\nthe whole collection of g_{ij}\'s a "U(1) Cech 1-cocycle"\nwhen it satisfies this condition. There\'s also a notion\nof "Cech coboundary" such that U(1) Cech 1-cocycles\ndiffering only by a coboundary define isomorphic U(1)\nbundles.\n\nIn fact, there\'s a whole big theory of the "nth Cech\ncohomology with coefficients in a sheaf of abelian groups" -\nwe\'re just doing the first cohomology where this sheaf\nconsists of smooth U(1)-valued functions.\n\nBut anyway: U(1) bundles can be classified by the 1st\nU(1) Cech cohomology group, and give a nice vivid picture\nof the 2nd integral cohomology group.\n\nSimilarly: U(1) gerbes can be classified by the 2nd\nU(1) Cech cohomology group, and give a nice (?) vivid (??)\npicture of the 3rd integral cohomology group.\n\nIn a bit more detail, this means that to build a\nU(1) gerbe over X, we first pick a good cover of X\nby open sets U_i. Then we stick a "trivial U(1) gerbe"\nover each open set U_i. But don\'t worry what that means!\nWhat really matters is that we glue these together by\ntransition functions\n\nh_{ijk} : U_i intersect U_j intersect U_k -> U(1)\n\nwhich must satisfy a 2-cocycle condition on quadruple\nintersections\n\nU_i intersect U_j intersect U_k intersect U_l\n\nI won\'t write this equation down, but you get it by staring\nat a picture of a tetrahedron with vertices labelled\nby i,j,k,l, so that each triangle gets labelled by the\nfunction h_{ijk} or h_{ikl} or h_{ijl} or h_{jkl}, and\nwe want these functions to multiply to the same thing\non the front of the tetrahedron as they do on that back!\nIn short, the obvious (?) generalization of the 1-cocycle\ncondition!\n\nPeople naturally want to carry this idea ever further, and\nnow in string theory we\'re seeing a certain desire for\n2-gerbes.\n\nSo let\'s see...\n\nK(Z,4) = ???\n\nNo description is known quite like the ones I\'ve given so far,\nwhich involve quantum mechanics in ever more complicated ways.\n\nBut, we can say this: "K(Z,4) is the classifying space for\nK(Z,3) bundles". So, elements of the fourth integral cohomology\ngroup of X are in 1-1 correspondence with isomorphism classes\nof principle K(Z,3) bundles over X.\n\nUnfortunately K(Z,3) is already pretty scary.\n\nLuckily, we also have: "K(Z,4) is the classifying space for\nK(Z,2) gerbes". We can generalize the concept of U(1) gerbe\nto any abelian topological group, and elements of the fourth\nintegral cohomology group of X are in 1-1 correspondence with\nequivalence classes of K(Z,2) gerbes over X.\n\nK(Z,2) = CP^infinity is less scary than K(Z,2), but gerbes are\nmore scary than bundles. So, it\'s a tradeoff.\n\nWe also have: "K(Z,4) is the classifying space for K(Z,1)\n2-gerbes". I\'m saying it this way to make the pattern clear,\nbut K(Z,1) is just U(1), so a nicer thing to say is just:\n"K(Z,4) is the classifying space for U(1) 2-gerbes".\n\nI haven\'t said what 2-gerbes are yet, and I don\'t want to\nright now, but the pattern should be clear: U(1) 2-gerbes\ncan be classified by the 3nd U(1) Cech cohomology group, and\nthey give a nice (??) vivid (???) picture of the 4rd integral\ncohomology group.\n\nThat\'s enough for now! Here\'s some more detail on K(Z,n)\'s\nif you want it, including a cute model of K(Z,2).\n\n....................................... ....................................\n\nAlso available at http://math.ucr.edu/home/baez/week151.html\n\nJune 26, 2000\nThis Week\'s Finds in Mathematical Physics (Week 151)\nJohn Baez\n\n[...]\n\nOkay, now on to K(Z,2)! I explained a bit about this space in "week149",\nbut I\'ve been pondering it a lot lately, so I\'d like to say a bit more.\n\nFirst let me review and elaborate on some basic stuff I said already.\nIf G is any topological group, there is a topological space BG with a\nbasepoint such that the space of loops in BG starting and ending at\nthis point is homotopy equivalent to G. This space BG is unique up\nto homotopy equivalence.\n\nBG is important because it\'s the "classifying space for G-bundles".\nWhat this means is that there\'s a principal G-bundle over BG called\nthe "universal G-bundle", with the marvelous property that *any*\nprincipal G-bundle over *any* space X is a pullback of this one by\nsome map\n\nf: X -> BG.\n\n(I explained in "week149" how to pull back complex line bundles, and\npulling back principal G-bundles works the same way.) Even better,\ntwo G-bundles that we get this way are isomorphic if and only if the maps\nthey come from are homotopic! So there is a one-to-one correspondence\nbetween:\n\nA) isomorphism classes of principal G-bundles over X\n\nand\n\nB) homotopy classes of maps from X to BG.\n\nNow, suppose G is an *abelian* topological group. Then BG is better\nthan a topological space with basepoint. It\'s an abelian topological\ngroup!\n\nThis means that we can *iterate* this trick. Starting with an abelian\ntopological group G we can form BG, and BBG, and BBBG, and so on. This\nis called "delooping", because the loop space of each of these spaces is\nthe previous one.\n\nIt\'s always fun to iterate any process whenever you can - Freud called\nthis "repetition compulsion" - but there\'s more going on here than just\nthat. In "week149" I said that when we have a list of spaces, each being\nthe loop space of the previous one, it\'s called a "spectrum". And I\nsaid that we can use a spectrum to get a generalized cohomology theory.\nSo we now have a trick for getting a generalized cohomology theory from\na topological abelian group!\n\nIn particular, suppose we start with a plain old abelian group A.\nWe can think of it as a topological group with the discrete topology -\nlet\'s call this K(A,0). Then we can define\n\nK(A,1) = B(K(A,0))\nK(A,2) = B(K(A,1))\nK(A,3) = B(K(A,2))\n\n.... and so on. We get a spectrum K(A,n) called an "Eilenberg-MacLane\nspectrum". The corresponding generalized cohomology theory is just\nordinary cohomology with coeffients in the abelian group A! This means\nthat\n\nH^n(X,A) = [X, K(A,n)]\n\nwhere the right-hand side is the set of homotopy classes of maps from X\nto K(A,n). In short, K(A,n) knows everything there is to know about the\nnth cohomology with coefficients in A.\n\nWe\'ve seen this trick a couple of times lately, and it\'s actually a big\ntheme in homotopy theory: whenever we have some interesting invariant of\nspaces, we try to cook up a space that "represents" this invariant. I\ncould say a LOT more about THIS idea, but that would propel us into\nfurther heights of abstraction, when what I really want is to come down\nto earth a bit. Just a little bit....\n\nSo: let\'s take A to be the integers, Z. As I said in "week149",\nwe then get\n\nK(Z,0) = Z,\n\nK(Z,1) = U(1),\n\nwhere U(1) is the group of "phases" or unit complex numbers, and\n\nK(Z,2) = CP^infinity\n\nwhere CP^infinity is infinite-dimensional complex projective space.\nThere are a couple of slightly different versions of this. Topologists\nlike to start with the direct limit of the spaces C^n, which they call\nC^infinity. Then they take the space of all 1-dimensional subspaces and\ncall that CP^infinity. Mathematical physicists prefer to start with a\nHilbert space of countable dimension. Then they take the space of unit\nvectors modulo phase. Both these versions are equally good models of\nK(Z,2). The first one is a lean, stripped-down version of the second.\n\nNow U(1) is very important in quantum theory, and so are unit vectors\nmodulo phase in a Hilbert space - physicists call these "pure states".\nSo something cool is going on here. For some mysterious reason, it\nlooks like K(Z,n)\'s are important quantum physics! This is especially\ninteresting because the abstract definition of the K(Z,n)\'s has nothing\nto do with the complex numbers - just the integers. The complex numbers\nshow up on their own accord. So maybe this hints at some explanation of\nwhy the complex numbers are important in quantum mechanics.\n\nWhy are K(Z,n)\'s connected to quantum theory? I don\'t really know.\nBut we can get some clues by asking some more specific questions.\n\nFirst of all, why is K(Z,2) the same as CP^infinity? In "week149" I\njust asserted this without proof. That\'s one of the fun things I\'m\nallowed to do in this column. But let me sketch why it\'s true.\n\nFirst I need to remind you of some more basic facts about topology.\nSuppose G is any topological group, and let P -> X be any principal\nG-bundle. This gives us a long exact sequence of homotopy groups:\n\n... -> pi_{n+1}(X) -> pi_n(G) -> pi_n(P) -> pi_n(X) -> pi_{n-1}(G) -> ...\n\nTwo-thirds of the arrows in this sequence come from the maps\n\nG -------> P -------> X\n\nwhile the less obvious remaining one-third come from the map\n\nLX -------> G\n\nsending each loop in the base space to the holonomy of some connection\non our bundle. Here LX means the space of based loops in X, and we\'re\nusing the fact that\n\npi_n(LX) = pi_{n+1}(X)\n\nwhich is obvious from the definition of the homotopy groups.\n\nBut now suppose P is contractible! Then all its homotopy groups vanish,\nso the above long exact sequence breaks up into lots of puny exact\nsequences like this:\n\n0 -> pi_{n+1}(X) -> pi_n(G) -> 0\n\nor in other words:\n\n0 -> pi_n(LX) -> pi_n(G) -> 0\n\nThis says that the map from LX to G induces isomorphisms on all homotopy\ngroups. By the Whitehead theorem, this implies that this map is a\nhomotopy equivalence! So LX is really just G!! So X is just BG!!!\n\nIn short: if we have a space X with a principal G-bundle P over it, and\nP is contractible, X must be BG. [1]\n\nNow let\'s use this fact to show that CP^infinity is K(Z,2). Remember\nthat by our recursive definition,\n\nK(Z,2) = B(K(Z,1)) = B(U(1))\n\nso to show that CP^infinity is K(Z,2), we just need to find a principal\nU(1)-bundle over it with a contractible total space.\n\nIn "week149" we discussed a complex line bundle over CP^infinity called\nthe "universal complex line bundle". If you take the space of unit\nvectors in a complex line bundle you get a principal U(1)-bundle. So\nlet\'s do this to the universal complex line bundle. What do we get?\nWe get a principal U(1)-bundle like this:\n\nS^infinity -> CP^infinity\n\nBeing a mathematical physicist, I\'m using S^infinity here to stand for\nthe unit sphere in some countable-dimensional Hilbert space, and the map\nsends each unit vector to the corresponding pure state, or unit vector\nmod phase. Since there\'s a circle of unit vectors for each pure state,\nthis is indeed a principal U(1)-bundle. But now for the cool part: the\nunit sphere in an infinite-dimensional Hilbert space is contractible!\nSo we\'ve got a principal U(1)-bundle with a contractible total space\nsitting over CP^infinity, proving that CP^infinity is K(Z,2). Even\nbetter, the bundle\n\nS^infinity -> CP^infinity\n\nis the universal principal U(1)-bundle.\n\nI can\'t resist explaining why the unit sphere in an infinite-dimensional\nHilbert space is contractible. It seems very odd that a sphere could be\ncontractible, but this is one of those funny things about infinite\ndimensions. Take our Hilbert space to be L^2[0,1] and consider any\nfunction f in the unit sphere of this Hilbert space:\n\nintegral |f(x)|^2 dx = 1\n\nFor t between 0 and 1, let f_t(x) be a function that equals 1 for x < t,\nand a sped-up version of f for x greater than or equal to t. If you do\nthis right f_t will still lie in the unit sphere, and you\'ll have a way\nof contracting the whole unit sphere down to a single point, namely the\nconstant function 1.\n\nCute, huh?\n\nNext question: how does CP^infinity become an abelian topological group?\nThere\'s a very pretty answer. Consider the space of rational functions\nof a single complex variable. This is a infinite-dimensional complex\nvector space, and there\'s a natural way to give it the topology of\nC^infinity. This gives us a nice way to think of CP^infinity: it\'s\njust the *nonzero* rational functions modulo multiplication by constants.\n\nBut nonzero rational functions form an abelian group under multiplication!\nAnd this is still true when we mod out by constant factors! So CP^infinity\nbecomes an abelian group - and in fact an abelian topological group.\n\nThis gives us a very concrete picture of CP^infinity. A rational\nfunction of a single complex variable has a bunch of zeros and poles -\na bunch of points on the Riemann sphere. We should really stick an\ninteger at each of these points: a positive integer for each zero, and\na negative integer for each pole, to tell us the order of the pole or\nzero. That gives us enough information to completely specify the\nrational function up to multiplication by a constant. So a point in\nCP^infinity is the same as a finite set of points on the sphere labelled\nby integers.\n\nOf course, we have to get the right topology on CP^infinity. As we\nmove our point in CP^infinity around in a continuous way, the\ncorresponding points on the sphere all move around continuously,\nlike a swarm of flies... but when points collide, their numbers add.\nFor example, when a point labelled by the number 7 collides with a\npoint labelled by the number -3, it turns into a point labelled by\nthe number 7 - 3 = 4.\n\nIn the lingo of physics, we\'ve got a picture of points in CP^infinity\nas "collections of particles and antiparticles on the sphere". The\ninteger at any point on the sphere tells us the number of particles\nsitting there - but if it\'s negative, it means we\'ve got *antiparticles*\nthere. Particle-antiparticle pairs can be created out of nothing, and\nthey annihilate when they collide... it\'s very nice!\n\nBy the way, there\'s something called the Thom-Dold theorem that lets\nus generalize the heck out of this. We just showed that if you take\nthe 2-sphere and consider the space of particle-antiparticle swarms in\nit, you get K(Z,2). But suppose instead we started with the n-sphere\nand considered the space of particle-antiparticle swarms in *that*.\nThen we\'d get K(Z,n)!\n\nMore generally, suppose we didn\'t use integers to say how many particles\nwere at each point in the n-sphere - suppose we used elements of some\nabelian group A. Then we\'d get K(A,n)!\n\nFor more tricks like this, try this paper:\n\n3) Dusa McDuff, Configuration spaces of positive and negative particles,\nTopology 14 (1975), 91-107.\n\nNow let me mention a different picture of K(Z,2), that\'s also nice,\nand also related to quantum theory. Take any countable-dimensional\nHilbert space H and let U(H) be the group of unitary operators on H.\nJust like the unit sphere in this Hilbert space is contractible, it\nturns out that U(H) is contractible if we give it the norm topology\nor the strong topology.\n\nAnyway, now let PU(H) be the "projective unitary group" of H, meaning\nthe group of unitary operators modulo phase. There\'s an obvious map\n\nU(H) -> PU(H)\n\nsending a circle\'s worth of points to each point in PU(H). It\'s\neasy to check that this is a principal U(1)-bundle. Since the total\nspace U(H) is contractible, it follows that PU(H) is K(Z,2)!\n\nThis give a *nonabelian* group structure on K(Z,2), which may seem\nkind of weird, given that we just made it into an *abelian* group a\nminute ago. But I guess this other product is "abelian up to homotopy"\nin a very strong sense, so it\'s just as good as abelian for the purposes\nof homotopy theory.\n\nAnyway, some people in Australia have figured out an extra trick you\ncan do with this PU(H) group:\n\n4) Alan L. Carey, Diarmuid Crowley and Michael K. Murray, Principal\nbundles and the Dixmier-Douady class, Comm. Math. Physics 193 (1998)\n171-196, preprint available as hep-th/9702147.\n\nHere\'s how it goes, at least in part. We say a linear operator\n\nA: H -> H\n\nis "Hilbert-Schmidt" if the trace of AA* is finite. The space\nof Hilbert-Schmidt operators is a Hilbert space in its own right,\nwith this inner product:\n\n<A,B> = tr(AB*)\n\nLet\'s call this Hilbert space X. U(H) acts on X by conjugation, and\nthis gives an action of PU(H) on X, because phases commute with everything.\nThis in turn gives an action of PU(H) on U(X)! Is your brain melting yet?\nAnyway, it turns out that this makes U(X) into the total space of a\nprincipal PU(H)-bundle:\n\nPU(H) -> U(X) -> U(X)/PU(H)\n\nBut X is a countable-dimensional Hilbert space, so U(X) is contractible,\nso this is the *universal* principal PU(H)-bundle. And as we\'ve seen,\nthis means that\n\nU(X)/PU(H) = B(PU(H))\n\nbut we just saw that\n\nPU(H) = K(Z,2)\n\nso\n\nU(X)/PU(H) = B(PU(H)) = B(K(Z,2)) = K(Z,3) !\n\nIn "week149", I said I\'d like K(Z,3) to be some sort of infinite-\ndimensional manifold closely related to quantum physics. I\'m\nhappier now, because here we are getting just that - technically,\nwe\'re getting it to be a "Banach manifold". Of course, I could\nstill complain that this description doesn\'t make the *abelian\ngroup structure* on K(Z,3) obvious. But it\'s definitely a big\nstep towards understanding what K(Z,n)\'s have to do with quantum\ntheory.\n\nWhile I\'m at it, I should report some other things people have told me\nvia email. If you ponder what I\'ve said, you can see that CP^infinity\nhas 2nd homology equal to Z, and that the generator of this homology\ngroup - the "universal cycle" - is given geometrically by the obvious\nway of sticking the sphere CP^1 inside CP^infinity. This is nice\nbecause CP^1 is actually a submanifold of the manifold CP^infinity.\nBut according to email from Mark Goresky, Rene Thom has shown that\nfor k > 6, we cannot make K(Z,n) into a manifold in such a way that\nthe universal cycle is represented by a submanifold!\n\nOn the other hand, Michael Murray reports that Pawel Gajer has managed\nto make K(Z,n) into something called a "differential space", which is\nnot quite a manifold, but good enough to do geometry on. I\'m not sure\nhow this relates to Thom\'s work... but anyway, I should read this stuff:\n\n5) Pawel Gajer, Geometry of Deligne cohomology, Invent. Math. 127\n(1997), 155-207, also available as alg-geom/9601025.\n\nPawel Gajer, Higher holonomies, geometric loop groups and smooth Deligne\ncohomology, Advances in Geometry, Birkhauser, Boston, 1999, pp. 195-235.\n\nNow, so far I\'ve been restraining myself from talking about "gerbes",\nbut if you\'ve gotten this far you must be pretty comfortable with\nabstract nonsense, so you\'ll probably like gerbes. Very roughly speaking,\na gerbe is a categorified version of a principal bundle! Actually it\'s\na categorified version of a sheaf, but sometimes we can think of it as\nanalogous to the sheaf of sections of a bundle. And just as K(Z,2) is\nthe classifying space for U(1) bundles, K(Z,3) is the classifying space\nfor a certain sort of gerbe!\n\nI sort of explained how this works in "week25", but you can read the\ndetails here:\n\n6) Jean-Luc Brylinski, Loop Spaces, Characteristic Classes and Geometric\nQuantization, Birkhauser, Boston, 1993. ISBN 0-176-3644-7\n\nWhat this means is that as we explore the meaning of these K(Z,n)\'s for\nquantum theory, we are really *categorifying* familiar ideas from quantum\ntheory. In particular, this story should keep going on forever: K(Z,4)\nshould be the classifying space for a certain sort of 2-gerbe, and so on.\nBut I don\'t think people have worked out the details beyond the case of\n2-gerbes. If you want to learn about 2-gerbes, you have to read this:\n\n7) Lawrence Breen, On the Classification of 2-Gerbes and 2-Stacks,\nAsterisque 225, 1994.\n\nFinally, for more applications to physics, try these papers:\n\n8) Alan L. Carey and Michael K. Murray, Faddeev\'s anomaly and bundle\ngerbes, Lett. Math. Phys. 37 (1996), 29-36.\n\nJouko Mickelsson, Gerbes and Hamiltonian quantization of chiral fermions,\nLie Theory and Its Applications in Physics, World Scientific, Singapore,\n1996, pp. 216-225.\n\nMichael K. Murray, Bundle gerbes, J. London Math. Soc. 54 (1996), 403-416.\n\nAlan L. Carey, Jouko Mickelsson and Michael K. Murray, Index theory,\ngerbes, and Hamiltonian quantization, Comm. Math. Phys. 183 (1997), 707-722,\npreprint available as hep-th/9511151.\n\nAlan L. Carey, Michael K. Murray and B. L. Wang, Higher bundle gerbes and\ncohomology classes in gauge theories, J. Geom. Phys. 21 (1997) 183-197,\npreprint available as hep-th/9511169.\n\nAlan L. Carey, Jouko Mickelsson and Michael K. Murray, Bundle gerbes\napplied to quantum field theory, Rev. Math. Phys. 12 (2000), 65-90,\npreprint available as hep-th/9711133.\n\nI thank N. Christopher Phillips of the University of Oregon,\nMichael K. Murray and Diarmuid Crowley of the University of\nAdelaide, and Mark Goresky of IHES for educating me about these\nmatters... all remaining errors are mine!\n\nFootnotes:\n\n[1] Moreover, P must be the universal principal G-bundle. Conversely,\nfor any topological group G the total space of the universal principal\nG-bundle is contractible. Everything fits together very neatly! But\nI don\'t need all this stuff now.\n\n-----------------------------------------------------------------------\nPrevious issues of "This Week\'s Finds" and other expository articles on\nmathematics and physics, as well as some of my research papers, can be\nobtained at\n\nhttp://math.ucr.edu/home/baez/\n\nFor a table of contents of all the issues of This Week\'s Finds, try\n\nhttp://math.ucr.edu/home/baez/twf.html\n\nA simple jumping-off point to the old issues is available at\n\nhttp://math.ucr.edu/home/baez/twfshort.html\n\nIf you just want the latest issue, go to\n\nhttp://math.ucr.edu/home/baez/this.week.html\n\n\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form"> View this Usenet post in original ASCII form </a></div><P></jabberwocky>In article <cj9b85$gc7$1@lfa222122.richmond.edu>,
Urs Schreiber <Urs.Schreiber@uni-essen.de> wrote:
> John Baez worte:
>> [the fact that]
>> after the degree-4 Pontrjagin class an E8 bundle has no
>> more characteristic classes until we get to degree 16
>> makes it hard to tell if we're dealing with an E8 bundle
>> or a 2-gerbe here. A 2-gerbe is just a degree-4 integral
>> cohomology class; an E8 bundle is a degree-4 integral
>> cohomology class plus some extra fuzz that you don't notice
>> until you get to very high dimensions.
>> Does the stuff I wrote above make sense?
>I can follow it.
>> One needs to see
>> why n-gerbes on X are classified by the (n+2)nd integral
>> cohomology of X.
Here I meant "U(1) n-gerbes", by the way.
>I have read that it does and it is plausible by extrapolating from the
>0-gerbe case, but I am not able to prove this.
That's okay - nobody else is either! In fact, the concept of
"U(1) n-gerbe" is still rather poorly understood except for
n < 3. The one thing everybody agrees on is that this concept
should be defined to MAKE IT TRUE that U(1) n-gerbes on a space
X are classified (up to equivalence) by elements of the integral
cohomology group H^{n+2}(X,Z).
The reason is that people liked these facts:
Homotopy classes of maps f: X -> U(1) are the same as
elements of H^1(X,Z).
Isomorphism classes of principal U(1) bundles over X
are the same as elements of H^2(X,Z).
and wanted the pattern to continue. So, they eventually
defined U(1) gerbes in such a way that:
Equivalence classes of U(1) gerbes over X are the same
as elements of H^3(X,Z).
And then Larry Breen defined U(1) 2-gerbes in such a way
that:
Equivalence classes of U(1) 2-gerbes over X are the same
as elements of H^4(X,Z).
By this point, if not sooner, the pattern was clear.
It goes like this.
For any n and any abelian group A there is a space K(A,n)
called an "Eilenberg-Mac Lane space" whose nth homotopy group
is A, with all its other homotopy groups being zero.
This space is unique up to homotopy equivalence. For
any space, elements of the cohomology group H^n(X,A)
are the same as homotopy classes of maps
f: X -> K(A,n)
(All this was shown by Eilenberg and Mac Lane in the 1950s).
The great thing about this is that it gives a concrete
picture of the cohomology group H^n(X,A) to the extent
that we understand K(A,n). Let's take A = Z and look at
some low n.
K(Z,0) = Z
This says that the 0th integral cohomology group of X
consists of homotopy classes of maps from X to Z. But
since Z is discrete these are the same as maps from X
to Z. By "maps" I always mean continuous maps when I'm
doing topology! Since Z is discrete these maps must be
constant on each connected component of X. So, we're
saying that H^0(X,Z) is just Z^k where k is the number
of connected components of X. Which is true!
K(Z,1) = U(1)
This says that the 1st integral cohomology group of
X consists of homotopy classes of maps from X to the
circle. And this is also true. You'll notice that
this is exactly backwards from how the fundamental
group of X consists of homotopy classes of maps from
the circle to X! (Basepoint-preserving maps, anyway.)
K(Z,2) = CP^infinity
This says that the 2nd integral cohomology group of
X consists of homotopy classes of maps from X to
the infinite-dimensional complex projective space -
or equivalently, the set of pure states in an
infinite-dimensional complex Hilbert space.
This is getting less easy to visualize, but luckily
CP^infinity is the "classifying space for U(1) bundles".
This means that it has a principal U(1) bundle over it -
which I'll gladly describe if asked - with the property
that *any* principal U(1) bundle over *any* space X can
be gotten (up to isomorphism) by pulling back this one
via some map f: X -> CP^infinity.
More precisely, homotopy classes of maps
f: X -> CP^infinity
are in 1-1 correspondence with isomorphism classes of
principal U(1) bundles over X. So, elements of H^2(X,Z)
have a nice interpretation as isomorphism classes of
principal U(1) bundles over X.
Is it just a coincidence that CP^infinity is the
classifying space for U(1) bundles, where U(1) was
the previous space on our list? No! It's a theorem
that it keeps on going like this forever! K(A,n) is
always an abelian topological group, and K(A,n+1)
is the classifying space for K(A,n) bundles!
Unfortunately this only helps us simplify things
one step.
To see what I mean, consider the next one:
K(Z,3) = U(Hilbert-Schmidt(infinity))/PU(infinity)
This says that the 3rd integral cohomology group
of X consists of homotopy classes of maps from X to
the unitary group of the Hilbert space of Hilbert-Schmidt
operators on an infinite-dimensional complex Hilbert space,
mod the action of the projectivized unitary group of that
infinite-dimensional Hilbert space. I don't want to
explain this: I'm only saying it to show how these
Eilenberg-Mac Lane spaces get more and more difficult
to understand as n goes up - at least from *certain*
viewpoints, like trying to get a model of them with a
simple relation to physics.
However, we can use the trick I mentioned above to
simplify things one step: the 3rd integral cohomology
group of X consists of isomorphism classes of principal
CP^infinity bundles over X. This is a bit simpler than
the description given in the previous paragraph, but
still a bit scary. For one, most people aren't comfortable
with how CP^infinity becomes an abelian group! It's
actually really cool - see below - but CP^infinity bundles
will never seem quite as simple as U(1) bundles.
So, at this point people resorted to another trick:
gerbes!
A U(1) gerbe is like a souped-up version of a U(1) bundle.
To build a U(1) bundle over X, first we pick a "good
cover" of X - a cover by open sets U_i such that each open
set, or any finite intersection of them, is contractible.
Then we stick a trivial U(1) bundle over each patch U_i.
Then we glue them together by transition functions
g_{ij}: U_i[/itex] intersect U_j -> U(1)
For consistency, these must satisfy a condition on
each triple intersection
U_i intersect U_j intersect U_k
namely:
g_{ij} g_{jk} = g_{ik}
Experts call this the "1-cocycle condition", and call
the whole collection of g_{ij}'s a "U(1) Cech 1-cocycle"
when it satisfies this condition. There's also a notion
of "Cech coboundary" such that U(1) Cech 1-cocycles
differing only by a coboundary define isomorphic U(1)
bundles.
In fact, there's a whole big theory of the "nth Cech
cohomology with coefficients in a sheaf of abelian groups" -
we're just doing the first cohomology where this sheaf
consists of smooth U(1)-valued functions.
But anyway: U(1) bundles can be classified by the 1st
U(1) Cech cohomology group, and give a nice vivid picture
of the 2nd integral cohomology group.
Similarly: U(1) gerbes can be classified by the 2nd
U(1) Cech cohomology group, and give a nice (?) vivid (??)
picture of the 3rd integral cohomology group.
In a bit more detail, this means that to build a
U(1) gerbe over X, we first pick a good cover of X
by open sets U_i. Then we stick a "trivial U(1) gerbe"
over each open set U_i. But don't worry what that means!
What really matters is that we glue these together by
transition functions
h_{ijk} : U_i intersect U_j intersect U_k -> U(1)
which must satisfy a 2-cocycle condition on quadruple
intersections
U_i intersect U_j intersect U_k intersect U_l
I won't write this equation down, but you get it by staring
at a picture of a tetrahedron with vertices labelled
by i,j,k,l, so that each triangle gets labelled by the
function h_{ijk} or h_{ikl} or h_{ijl} or h_{jkl}, and
we want these functions to multiply to the same thing
on the front of the tetrahedron as they do on that back!
In short, the obvious (?) generalization of the 1-cocycle
condition!
People naturally want to carry this idea ever further, and
now in string theory we're seeing a certain desire for
2-gerbes.
So let's see...
K(Z,4) = ???
No description is known quite like the ones I've given so far,
which involve quantum mechanics in ever more complicated ways.
But, we can say this: "K(Z,4) is the classifying space for
K(Z,3) bundles". So, elements of the fourth integral cohomology
group of X are in 1-1 correspondence with isomorphism classes
of principle K(Z,3) bundles over X.
Unfortunately K(Z,3) is already pretty scary.
Luckily, we also have: "K(Z,4) is the classifying space for
K(Z,2) gerbes". We can generalize the concept of U(1) gerbe
to any abelian topological group, and elements of the fourth
integral cohomology group of X are in 1-1 correspondence with
equivalence classes of K(Z,2) gerbes over X.
K(Z,2) = CP^infinity is less scary than K(Z,2), but gerbes are
more scary than bundles. So, it's a tradeoff.
We also have: "K(Z,4) is the classifying space for K(Z,1)
2-gerbes". I'm saying it this way to make the pattern clear,
but K(Z,1) is just U(1), so a nicer thing to say is just:
"K(Z,4) is the classifying space for U(1) 2-gerbes".
I haven't said what 2-gerbes are yet, and I don't want to
right now, but the pattern should be clear: U(1) 2-gerbes
can be classified by the 3nd U(1) Cech cohomology group, and
they give a nice (??) vivid (???) picture of the 4rd integral
cohomology group.
That's enough for now! Here's some more detail on K(Z,n)'s
if you want it, including a cute model of K(Z,2).
.................................................. .........................
Also available at http://math.ucr.edu/home/baez/week151.html
June 26, 2000
This Week's Finds in Mathematical Physics (Week 151)
John Baez
[...]
Okay, now on to K(Z,2)! I explained a bit about this space in "week149",
but I've been pondering it a lot lately, so I'd like to say a bit more.
First let me review and elaborate on some basic stuff I said already.
If G is any topological group, there is a topological space BG with a
basepoint such that the space of loops in BG starting and ending at
this point is homotopy equivalent to G. This space BG is unique up
to homotopy equivalence.
BG is important because it's the "classifying space for G-bundles".
What this means is that there's a principal G-bundle over BG called
the "universal G-bundle", with the marvelous property that *any*
principal G-bundle over *any* space X is a pullback of this one by
some map
f: X -> BG.
(I explained in "week149" how to pull back complex line bundles, and
pulling back principal G-bundles works the same way.) Even better,
two G-bundles that we get this way are isomorphic if and only if the maps
they come from are homotopic! So there is a one-to-one correspondence
between:
A) isomorphism classes of principal G-bundles over X
and
B) homotopy classes of maps from X to BG.
Now, suppose G is an *abelian* topological group. Then BG is better
than a topological space with basepoint. It's an abelian topological
group!
This means that we can *iterate* this trick. Starting with an abelian
topological group G we can form BG, and BBG, and BBBG, and so on. This
is called "delooping", because the loop space of each of these spaces is
the previous one.
It's always fun to iterate any process whenever you can - Freud called
this "repetition compulsion" - but there's more going on here than just
that. In "week149" I said that when we have a list of spaces, each being
the loop space of the previous one, it's called a "spectrum". And I
said that we can use a spectrum to get a generalized cohomology theory.
So we now have a trick for getting a generalized cohomology theory from
a topological abelian group!
In particular, suppose we start with a plain old abelian group A.
We can think of it as a topological group with the discrete topology -
let's call this K(A,0). Then we can define
K(A,1) = B(K(A,0))K(A,2) = B(K(A,1))K(A,3) = B(K(A,2))
.... and so on. We get a spectrum K(A,n) called an "Eilenberg-MacLane
spectrum". The corresponding generalized cohomology theory is just
ordinary cohomology with coeffients in the abelian group A! This means
that
H^n(X,A) = [X, K(A,n)]
where the right-hand side is the set of homotopy classes of maps from X
to K(A,n). In short, K(A,n) knows everything there is to know about the
nth cohomology with coefficients in A.
We've seen this trick a couple of times lately, and it's actually a big
theme in homotopy theory: whenever we have some interesting invariant of
spaces, we try to cook up a space that "represents" this invariant. I
could say a LOT more about THIS idea, but that would propel us into
further heights of abstraction, when what I really want is to come down
to earth a bit. Just a little bit....
So: let's take A to be the integers, Z. As I said in "week149",
we then get
K(Z,0) [itex]= Z,
K(Z,1) = U(1),
where U(1) is the group of "phases" or unit complex numbers, and
K(Z,2) = CP^infinity
where CP^infinity is infinite-dimensional complex projective space.
There are a couple of slightly different versions of this. Topologists
like to start with the direct limit of the spaces C^n, which they call
C^{infinity}. Then they take the space of all 1-dimensional subspaces and
call that CP^infinity. Mathematical physicists prefer to start with a
Hilbert space of countable dimension. Then they take the space of unit
vectors modulo phase. Both these versions are equally good models of
K(Z,2). The first one is a lean, stripped-down version of the second.
Now U(1) is very important in quantum theory, and so are unit vectors
modulo phase in a Hilbert space - physicists call these "pure states".
So something cool is going on here. For some mysterious reason, it
looks like K(Z,n)'s are important quantum physics! This is especially
interesting because the abstract definition of the K(Z,n)'s has nothing
to do with the complex numbers - just the integers. The complex numbers
show up on their own accord. So maybe this hints at some explanation of
why the complex numbers are important in quantum mechanics.
Why are K(Z,n)'s connected to quantum theory? I don't really know.
But we can get some clues by asking some more specific questions.
First of all, why is K(Z,2) the same as CP^infinity? In "week149" I
just asserted this without proof. That's one of the fun things I'm
allowed to do in this column. But let me sketch why it's true.
First I need to remind you of some more basic facts about topology.
Suppose G is any topological group, and let P -> X be any principal
G-bundle. This gives us a long exact sequence of homotopy groups:
... -> \pi_{n+1}(X) -> \pi_n(G) -> \pi_n(P) -> \pi_n(X) -> \pi_{n-1}(G) -> ...
Two-thirds of the arrows in this sequence come from the maps
G -------> P -------> X
while the less obvious remaining one-third come from the map
LX -------> G
sending each loop in the base space to the holonomy of some connection
on our bundle. Here LX means the space of based loops in X, and we're
using the fact that
\pi_n(LX) = \pi_{n+1}(X)
which is obvious from the definition of the homotopy groups.
But now suppose P is contractible! Then all its homotopy groups vanish,
so the above long exact sequence breaks up into lots of puny exact
sequences like this:
-> \pi_{n+1}(X) -> \pi_n(G) ->
or in other words:
-> \pi_n(LX) -> \pi_n(G) ->
This says that the map from LX to G induces isomorphisms on all homotopy
groups. By the Whitehead theorem, this implies that this map is a
homotopy equivalence! So LX is really just G!! So X is just BG!!!
In short: if we have a space X with a principal G-bundle P over it, and
P is contractible, X must be BG. [1]
Now let's use this fact to show that CP^infinity is K(Z,2). Remember
that by our recursive definition,
K(Z,2) = B(K(Z,1)) = B(U(1))
so to show that CP^infinity is K(Z,2), we just need to find a principal
U(1)-bundle over it with a contractible total space.
In "week149" we discussed a complex line bundle over CP^infinity called
the "universal complex line bundle". If you take the space of unit
vectors in a complex line bundle you get a principal U(1)-bundle. So
let's do this to the universal complex line bundle. What do we get?
We get a principal U(1)-bundle like this:
S^{infinity} -> CP^infinity
Being a mathematical physicist, I'm using S^{infinity} here to stand for
the unit sphere in some countable-dimensional Hilbert space, and the map
sends each unit vector to the corresponding pure state, or unit vector
mod phase. Since there's a circle of unit vectors for each pure state,
this is indeed a principal U(1)-bundle. But now for the cool part: the
unit sphere in an infinite-dimensional Hilbert space is contractible!
So we've got a principal U(1)-bundle with a contractible total space
sitting over CP^infinity, proving that CP^infinity is K(Z,2). Even
better, the bundle
S^{infinity} -> CP^infinity
is the universal principal U(1)-bundle.
I can't resist explaining why the unit sphere in an infinite-dimensional
Hilbert space is contractible. It seems very odd that a sphere could be
contractible, but this is one of those funny things about infinite
dimensions. Take our Hilbert space to be L^2[0,1] and consider any
function f in the unit sphere of this Hilbert space:
integral |f(x)|^2 dx = 1
For t between and 1, let f_t(x) be a function that equals 1 for x < t,
and a sped-up version of f for x greater than or equal to t. If you do
this right f_t will still lie in the unit sphere, and you'll have a way
of contracting the whole unit sphere down to a single point, namely the
constant function 1.
Cute, huh?
Next question: how does CP^infinity become an abelian topological group?
There's a very pretty answer. Consider the space of rational functions
of a single complex variable. This is a infinite-dimensional complex
vector space, and there's a natural way to give it the topology of
C^{infinity}. This gives us a nice way to think of CP^infinity: it's
just the *nonzero* rational functions modulo multiplication by constants.
But nonzero rational functions form an abelian group under multiplication!
And this is still true when we mod out by constant factors! So CP^infinity
becomes an abelian group - and in fact an abelian topological group.
This gives us a very concrete picture of CP^infinity. A rational
function of a single complex variable has a bunch of zeros and poles -
a bunch of points on the Riemann sphere. We should really stick an
integer at each of these points: a positive integer for each zero, and
a negative integer for each pole, to tell us the order of the pole or
zero. That gives us enough information to completely specify the
rational function up to multiplication by a constant. So a point in
CP^infinity is the same as a finite set of points on the sphere labelled
by integers.
Of course, we have to get the right topology on CP^infinity. As we
move our point in CP^infinity around in a continuous way, the
corresponding points on the sphere all move around continuously,
like a swarm of flies... but when points collide, their numbers add.
For example, when a point labelled by the number 7 collides with a
point labelled by the number -3, it turns into a point labelled by
the number 7 - 3 = 4.
In the lingo of physics, we've got a picture of points in CP^infinity
as "collections of particles and antiparticles on the sphere". The
integer at any point on the sphere tells us the number of particles
sitting there - but if it's negative, it means we've got *antiparticles*
there. Particle-antiparticle pairs can be created out of nothing, and
they annihilate when they collide... it's very nice!
By the way, there's something called the Thom-Dold theorem that lets
us generalize the heck out of this. We just showed that if you take
the 2-sphere and consider the space of particle-antiparticle swarms in
it, you get K(Z,2). But suppose instead we started with the n-sphere
and considered the space of particle-antiparticle swarms in *that*.
Then we'd get K(Z,n)!
More generally, suppose we didn't use integers to say how many particles
were at each point in the n-sphere - suppose we used elements of some
abelian group A. Then we'd get K(A,n)!
For more tricks like this, try this paper:
3) Dusa McDuff, Configuration spaces of positive and negative particles,
Topology 14 (1975), 91-107.
Now let me mention a different picture of K(Z,2), that's also nice,
and also related to quantum theory. Take any countable-dimensional
Hilbert space H and let U(H) be the group of unitary operators on H.
Just like the unit sphere in this Hilbert space is contractible, it
turns out that U(H) is contractible if we give it the norm topology
or the strong topology.
Anyway, now let PU(H) be the "projective unitary group" of H, meaning
the group of unitary operators modulo phase. There's an obvious map
U(H) -> PU(H)
sending a circle's worth of points to each point in PU(H). It's
easy to check that this is a principal U(1)-bundle. Since the total
space U(H) is contractible, it follows that PU(H) is K(Z,2)!
This give a *nonabelian* group structure on K(Z,2), which may seem
kind of weird, given that we just made it into an *abelian* group a
minute ago. But I guess this other product is "abelian up to homotopy"
in a very strong sense, so it's just as good as abelian for the purposes
of homotopy theory.
Anyway, some people in Australia have figured out an extra trick you
can do with this PU(H) group:
4) Alan L. Carey, Diarmuid Crowley and Michael K. Murray, Principal
bundles and the Dixmier-Douady class, Comm. Math. Physics 193 (1998)
171-196, preprint available as http://www.arxiv.org/abs/hep-th/9702147.
Here's how it goes, at least in part. We say a linear operator
A: H -> H
is "Hilbert-Schmidt" if the trace of AA* is finite. The space
of Hilbert-Schmidt operators is a Hilbert space in its own right,
with this inner product:
<A,B> = tr(AB*)
Let's call this Hilbert space X. U(H) acts on X by conjugation, and
this gives an action of PU(H) on X, because phases commute with everything.
This in turn gives an action of PU(H) on U(X)! Is your brain melting yet?
Anyway, it turns out that this makes U(X) into the total space of a
principal PU(H)-bundle:
PU(H) -> U(X) -> U(X)/PU(H)
But X is a countable-dimensional Hilbert space, so U(X) is contractible,
so this is the *universal* principal PU(H)-bundle. And as we've seen,
this means that
U(X)/PU(H) = B(PU(H))
but we just saw that
PU(H) = K(Z,2)
so
U(X)/PU(H) = B(PU(H)) = B(K(Z,2)) = K(Z,3) !
In "week149", I said I'd like K(Z,3) to be some sort of infinite-
dimensional manifold closely related to quantum physics. I'm
happier now, because here we are getting just that - technically,
we're getting it to be a "Banach manifold". Of course, I could
still complain that this description doesn't make the *abelian
group structure* on K(Z,3) obvious. But it's definitely a big
step towards understanding what K(Z,n)'s have to do with quantum
theory.
While I'm at it, I should report some other things people have told me
via email. If you ponder what I've said, you can see that CP^infinity
has 2nd homology equal to Z, and that the generator of this homology
group - the "universal cycle" - is given geometrically by the obvious
way of sticking the sphere CP^1 inside CP^infinity. This is nice
because CP^1 is actually a submanifold of the manifold CP^infinity.
But according to email from Mark Goresky, Rene Thom has shown that
for k > 6, we cannot make K(Z,n) into a manifold in such a way that
the universal cycle is represented by a submanifold!
On the other hand, Michael Murray reports that Pawel Gajer has managed
to make K(Z,n) into something called a "differential space", which is
not quite a manifold, but good enough to do geometry on. I'm not sure
how this relates to Thom's work... but anyway, I should read this stuff:
5) Pawel Gajer, Geometry of Deligne cohomology, Invent. Math. 127
(1997), 155-207, also available as alg-geom/9601025.
Pawel Gajer, Higher holonomies, geometric loop groups and smooth Deligne
cohomology, Advances in Geometry, Birkhauser, Boston, 1999, pp. 195-235.
Now, so far I've been restraining myself from talking about "gerbes",
but if you've gotten this far you must be pretty comfortable with
abstract nonsense, so you'll probably like gerbes. Very roughly speaking,
a gerbe is a categorified version of a principal bundle! Actually it's
a categorified version of a sheaf, but sometimes we can think of it as
analogous to the sheaf of sections of a bundle. And just as K(Z,2) is
the classifying space for U(1) bundles, K(Z,3) is the classifying space
for a certain sort of gerbe!
I sort of explained how this works in "week25", but you can read the
details here:
6) Jean-Luc Brylinski, Loop Spaces, Characteristic Classes and Geometric
Quantization, Birkhauser, Boston, 1993. ISBN 0-176-3644-7
What this means is that as we explore the meaning of these K(Z,n)'s for
quantum theory, we are really *categorifying* familiar ideas from quantum
theory. In particular, this story should keep going on forever: K(Z,4)
should be the classifying space for a certain sort of 2-gerbe, and so on.
But I don't think people have worked out the details beyond the case of
2-gerbes. If you want to learn about 2-gerbes, you have to read this:
7) Lawrence Breen, On the Classification of 2-Gerbes and 2-Stacks,
Asterisque 225, 1994.
Finally, for more applications to physics, try these papers:
8) Alan L. Carey and Michael K. Murray, Faddeev's anomaly and bundle
gerbes, Lett. Math. Phys. 37 (1996), 29-36.
Jouko Mickelsson, Gerbes and Hamiltonian quantization of chiral fermions,
Lie Theory and Its Applications in Physics, World Scientific, Singapore,
1996, pp. 216-225.
Michael K. Murray, Bundle gerbes, J. London Math. Soc. 54 (1996), 403-416.
Alan L. Carey, Jouko Mickelsson and Michael K. Murray, Index theory,
gerbes, and Hamiltonian quantization, Comm. Math. Phys. 183 (1997), 707-722,
preprint available as http://www.arxiv.org/abs/hep-th/9511151.
Alan L. Carey, Michael K. Murray and B. L. Wang, Higher bundle gerbes and
cohomology classes in gauge theories, J. Geom. Phys. 21 (1997) 183-197,
preprint available as http://www.arxiv.org/abs/hep-th/9511169.
Alan L. Carey, Jouko Mickelsson and Michael K. Murray, Bundle gerbes
applied to quantum field theory, Rev. Math. Phys. 12 (2000), 65-90,
preprint available as http://www.arxiv.org/abs/hep-th/9711133.
I thank N. Christopher Phillips of the University of Oregon,
Michael K. Murray and Diarmuid Crowley of the University of
Adelaide, and Mark Goresky of IHES for educating me about these
matters... all remaining errors are mine!
Footnotes:
[1] Moreover, P must be the universal principal G-bundle. Conversely,
for any topological group G the total space of the universal principal
G-bundle is contractible. Everything fits together very neatly! But
I don't need all this stuff now.
-----------------------------------------------------------------------
Previous issues of "This Week's Finds" and other expository articles on
mathematics and physics, as well as some of my research papers, can be
obtained at
http://math.ucr.edu/home/baez/
For a table of contents of all the issues of This Week's Finds, try
http://math.ucr.edu/home/baez/twf.html
A simple jumping-off point to the old issues is available at
http://math.ucr.edu/home/baez/twfshort.html
If you just want the latest issue, go to
http://math.ucr.edu/home/baez/this.week.html