In mathematics , with the help of the classifying space and the universal bundle of a topological group G, the principal bundles with G are classified as a structure group. The classifying space and the universal bundle are characterized by a universal property , an explicit construction goes back to John Milnor . Bundles and their classification play an important role in mathematics and theoretical physics.
Universal bundle
A - principal bundle is universal bundle when all (numerierbaren) -Prinzipalbündel by retracting can win the universal bundle; formal: if it has the following universal property for numerable G-principal bundles:
![{\ displaystyle \ xi \ colon EG \ to BG}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2db30eeea3d7f1761c5ed7700827335069a7cf4b)
![G](https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b)
- For every numerable -principal bundle there is a continuous mapping such that the bundles and are isomorphic .
![G](https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b)
![\ pi \ colon E \ to X](https://wikimedia.org/api/rest_v1/media/math/render/svg/5f9d6e10a97de4dd6d790d2ef4a13adf0e38052f)
![{\ displaystyle f \ colon X \ to BG}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d7fdb70806d00dc1f4230186b8122f3d921f960d)
![{\ displaystyle f ^ {*} \ xi}](https://wikimedia.org/api/rest_v1/media/math/render/svg/abf515682afb08e8d1f5e2a0a74f72e7af0086c5)
- For two mappings the bundles are isomorphic if and only if they are homotopic .
![{\ displaystyle f, g \ colon X \ to BG}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0dbe73c6da26d065adc0959f1271bec3937bdccc)
![{\ displaystyle f ^ {*} \ xi, g ^ {*} \ xi}](https://wikimedia.org/api/rest_v1/media/math/render/svg/220ff05d358c072458683ee795453c975e71f03b)
So you have a bijection
-
,
where denotes the homotopy classes of images .
![{\ displaystyle \ left [X, BG \ right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/391a1d273cb9d14c4760306944f6cf77651cdc57)
![{\ displaystyle X \ to BG}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5ab00a9aeca38e80f4567f05918c4418858784a2)
The basis of a universal bundle is called the classifying space of the topological group . Using general nonsense one can easily show that (if a universal bundle exists) it is uniquely determined except for homotopy equivalence . The following construction, which goes back to Milnor, also proves the existence of classifying space.
![BG](https://wikimedia.org/api/rest_v1/media/math/render/svg/773ca20b2080cb3766062a5451a01d2220e9b067)
![G](https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b)
![BG](https://wikimedia.org/api/rest_v1/media/math/render/svg/773ca20b2080cb3766062a5451a01d2220e9b067)
Milnor construction
The infinite union of
countably many copies of the topological group is called Milnor space. The elements are of the form with and only finitely many . (Note also for .)
![{\ displaystyle EG = G * G * \ ldots * G * \ ldots}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ade4ee19f5886e0f1b9cdc5cc20751313051488b)
![G](https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b)
![{\ displaystyle \ textstyle \ sum _ {i} t_ {i} g_ {i}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/97f0578fb4a3f7cbb8f14b5f75bf8b7b2d16b18c)
![{\ displaystyle \ textstyle g_ {i} \ in G, t_ {i} \ in \ left [0,1 \ right], \ sum _ {i} t_ {i} = 1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a6e5ac269a5c95aa0ddd2e6077c81a39ea08bbd9)
![{\ displaystyle t_ {i} \ not = 0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d2b8aa595719b05e5c3b26f572ee23809d27d602)
![{\ displaystyle 0g_ {i} = 0g_ {i} ^ {\ prime}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6aef3007687e5bd4a0418c7184c0969437ceb1c5)
![{\ displaystyle g_ {i} \ not = g_ {i} ^ {\ prime}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e0b544f88a2b3cc1958ec40861e04d920108eb8b)
The group works through the Milnor room . The quotient is the classifying space of the group , the principal bundle
![G](https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b)
![EG](https://wikimedia.org/api/rest_v1/media/math/render/svg/0a7ebfb4d29c3c955d9a1cabc6f2305b94c8bbce)
![{\ displaystyle \ textstyle (\ sum _ {i} t_ {i} g_ {i}) g = \ sum _ {i} t_ {i} (g_ {i} g)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cbe52076ba63470d4cec2d4dd3b895adcc543d14)
![{\ displaystyle BG: = EG / G}](https://wikimedia.org/api/rest_v1/media/math/render/svg/71947bc518ae5a9f776786ce7801b7287b94142f)
![G](https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b)
![{\ displaystyle \ xi \ colon EG \ to BG}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2db30eeea3d7f1761c5ed7700827335069a7cf4b)
is the universal bundle.
For various Lie groups, for example and there are simpler realizations of the classifying space using Graßmann manifolds, see below.
![O (n)](https://wikimedia.org/api/rest_v1/media/math/render/svg/34109fe397fdcff370079185bfdb65826cb5565a)
![U.N)](https://wikimedia.org/api/rest_v1/media/math/render/svg/e9dac86141aa23bec59b25ea2c986580753b6754)
In general, every free effect of gives a quotient on a contractible space , which is a classifying space (and thus in particular homotopy-equivalent to the above construction). The quotient mapping is then a universal -principal bundle.
![G](https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b)
![E.](https://wikimedia.org/api/rest_v1/media/math/render/svg/4232c9de2ee3eec0a9c0a19b15ab92daa6223f9b)
![{\ displaystyle B = E / G}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7529964fe229d09b2257509b0e5198382d408992)
![BG](https://wikimedia.org/api/rest_v1/media/math/render/svg/773ca20b2080cb3766062a5451a01d2220e9b067)
![E \ to B](https://wikimedia.org/api/rest_v1/media/math/render/svg/10db76bf8b1d965e8dceae2c308bb1db52f503c4)
![G](https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b)
Topology of the classifying space
is contractible . For the homotopy groups of true
![BG](https://wikimedia.org/api/rest_v1/media/math/render/svg/773ca20b2080cb3766062a5451a01d2220e9b067)
-
.
In particular, the following applies to groups provided with the discrete topology :
![\Gamma](https://wikimedia.org/api/rest_v1/media/math/render/svg/4cfde86a3f7ec967af9955d0988592f0693d2b19)
![{\ displaystyle \ pi _ {1} (B \ Gamma) = \ Gamma}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f3c524eecb73f35a4edfae3b98ee12f5de31a075)
-
for .![{\ displaystyle i \ not = 1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/77dfb852dd01c6a7a6c959efa503d2b6b9eb7378)
The classifying space of a discrete group is therefore an Eilenberg-MacLane space .
If there is a homotopy equivalence , then there is also a homotopy equivalence. In particular, is homotopy equivalent to .
![{\ displaystyle K \ to G}](https://wikimedia.org/api/rest_v1/media/math/render/svg/64ab2bf0fc3a336b79541c3b9969ffd5c5418a4e)
![{\ displaystyle BK \ to BG}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0043636283688aafbe8ef68a150d18bc5239bf15)
![{\ displaystyle BO (n)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5f65f1fc581a430570b8ca81a1aaa2f0d90820ee)
![{\ displaystyle BGL (n, \ mathbb {R})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3ca31b3b2e3dd3df684952edca16d529dec31a0d)
Examples of classifying spaces
The following list gives examples of classifying spaces with associated total space (of the universal bundle) . Note that for topological groups generally does not match (the classifying space for the same group with the discrete topology).
![BG](https://wikimedia.org/api/rest_v1/media/math/render/svg/773ca20b2080cb3766062a5451a01d2220e9b067)
![EG](https://wikimedia.org/api/rest_v1/media/math/render/svg/0a7ebfb4d29c3c955d9a1cabc6f2305b94c8bbce)
![BG](https://wikimedia.org/api/rest_v1/media/math/render/svg/773ca20b2080cb3766062a5451a01d2220e9b067)
![{\ displaystyle BG _ {\ delta}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/61437a54f2cd13899cb8ebdbad0a5bbf32c50dd8)
-
with total space (in particular )![S ^ {\ infty}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bc6ba232b7e000afed285fec1f303447da8bf617)
![{\ displaystyle B \ mathbb {Z} _ {2} = \ mathbb {R} P ^ {\ infty}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e9456c573ebf80d5a1232de8212a9b286971b274)
-
with total space
-
with total space
-
with total space (infinite tree of degree 4)![{\ mathcal {T}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8236d074e42310f5dc24d1d2b5b8f5981c3e87ba)
-
with total space
-
with total space
-
with total space (hyperbolic plane)![\ mathcal {H}](https://wikimedia.org/api/rest_v1/media/math/render/svg/19ef4c7b923a5125ac91aa491838a95ee15b804f)
![{\ displaystyle B (G_ {1} \ times G_ {2}) = BG_ {1} \ times BG_ {2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6c7d62d7dd7578f44985578e7d3b42ec06c291e3)
Vector bundle
For a real vector bundle of rank one has the frame bundle as a bundle over the same base. In particular , and because of the homotopy also a classifying space for real vector bundle of rank . Correspondingly, a classifying space for complex vector bundles is of rank .
![r](https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538)
![{\ displaystyle GL (r, \ mathbb {R})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7d4fafb17950e14e4a132b2a205e46d6c3a44987)
![{\ displaystyle BGL (r, \ mathbb {R})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/04a8e704086631f6085e08b0d87f1284fd4b8b5d)
![{\ displaystyle BO (r) \ simeq BGL (r, \ mathbb {R})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/114daa70b0bbdaa192e5a5b3ea1a317cf60d908f)
![{\ displaystyle BO (r)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4461a502442abf0d455d9d8bda4a84c3190e3280)
![r](https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538)
![{\ displaystyle BU (r) \ simeq BGL (r, \ mathbb {C})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4dcd163a51c1927061886fac38aa62f63a2e933d)
![r](https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538)
The Graßmann manifolds for or are explicit realizations of the classifying spaces or .
![{\ displaystyle Gr _ {\ mathbb {K}} (r, \ infty)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e08f212e21303e39a965c9d3fd8d9dfa8a7c16b1)
![{\ mathbb K} = \ R](https://wikimedia.org/api/rest_v1/media/math/render/svg/d0578e31db7003b87a028463eae1164c4ce717d0)
![{\ displaystyle {\ mathbb {K}} = \ mathbb {C}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d0410f2bd050ac2ff8ee7ab00ec3631aadb98bd3)
![{\ displaystyle BO (r)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4461a502442abf0d455d9d8bda4a84c3190e3280)
![{\ displaystyle BU (r)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d795fb18834249e63ffd986ff2aae7993d1a5ad6)
Analogously, oriented vector bundles can be classified from the rank by the universal bundle via the Graßmann manifold of the oriented sub-vector spaces.
![r](https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538)
![{\ displaystyle BSO (r) = Gr ^ {+} (r, \ infty)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/512320f08b42bbb82f8b055ec5837f049f46d985)
Characteristic classes
Cohomology classes of a classifying space are used to define characteristic classes.
For example, one obtains characteristic classes of oriented vector bundles of rank from the cohomology of . For a field F with applies
![n](https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b)
![{\ displaystyle BSO (n)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/38111d22c29c6da4ebd166587551265d88d26733)
![{\ displaystyle char (F) \ not = 2}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7d49c1d61e2660cfa20453b51acb657180a0cc5b)
![{\ displaystyle H ^ {*} (BSO (2n), F) = F \ left [e, p_ {1}, \ ldots, p_ {n} \ right] / (e ^ {2} -p_ {n} )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/83353da27118ced8c7b6444c5fdae5eb80b51be7)
-
,
where denotes the Euler class and the Pontryagin classes . For is
![e](https://wikimedia.org/api/rest_v1/media/math/render/svg/cd253103f0876afc68ebead27a5aa9867d927467)
![pi}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5bab39399bf5424f25d957cdc57c84a0622626d2)
![{\ displaystyle char (F) = 2}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6e305a383a80cf44c5bd096fd41ec0da320956b3)
-
,
being denoted the Stiefel-Whitney classes .
![w_ {i}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fe22f0329d3ecb2e1880d44d191aba0e5475db68)
literature
- John Milnor: Construction of universal bundles. Part I In: Ann. of Math. (2) 63 (1956), pp. 272-284. pdf ; Part II In: Ann. of Math. (2) 63 (1956), pp. 430-436. pdf
- Dale Husemoller: Fiber bundles. McGraw-Hill Book Co., New York / London / Sydney 1966, OCLC 909937420 .
- Tammo tom Dieck: Topology. (= de Gruyter textbook). Walter de Gruyter, Berlin 1991, ISBN 3-11-012463-7 .
Web links
Individual evidence
-
↑ An open covering of a topological space is called numerable if there is a locally finite decomposition of the one with . A principal bundle is called numerable if there is a numerable cover so that the bundle's restrictions on the can be trivialized . See Husemoller, op.cit., Section I.4.9.
![{\ displaystyle 1 = \ Sigma _ {i \ in I} u_ {i}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6ad9f34a474ddd65d46fb3a05e69588eafc8a76e)
![{\ displaystyle supp (u_ {i}) \ subset U_ {i}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3023287bed3ac04c3281dc6fcb1efc6298fe4d84)