727:: Given a connected p-compact group for each prime, all with the same rational type, there is an explicit double coset space of possible connected finite loop spaces with p-completion the give p-compact groups. As connected p-compact groups are classified combinatorially, this implies a classification of connected loop spaces as well.
170:, in the sense of homotopy theory, of (the classifying space of) a compact connected Lie group defines a connected p-compact group. (The Weyl group is just its ordinary Weyl group, now viewed as a p-adic reflection group by tensoring the coweight lattice by
710:
a similar statement holds but the new exotic 3-compact group is now group number 12 on the
Shepard-Todd list, of rank 2. For primes greater than 3, family 2 on the Shepard-Todd list will contain infinitely many exotic p-compact groups.
402:
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487:
343:
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74:-adic reflection group. They admit a classification in terms of root data, which mirrors the classification of compact Lie groups, but with the integers replaced by the
624:-reflection groups. Simple exotic p-compact groups are again in 1-1-correspondence with irreducible complex reflection groups whose character field can be embedded in
723:
is a pointed space BG such that the loop space ΩBG is homotopy equivalent to a finite CW-complex. The classification of connected p-compact groups implies a
528:
749:
The classification also implies a classification of which graded polynomial rings can occur as the cohomology ring of a space, the so-called
1017:
686:, where BH is the 2-completion of the classifying space of a connected compact Lie group, and BDI(4) denotes s copies of the "
127:, but then one needs to keep in mind that the loop space structure is part of the data (which then allows one to recover
994:
62:, which are defined purely homotopically in terms of the classifying space, but with the important difference that the
1012:
54:, making precise earlier notions of a mod p finite loop space. A p-compact group has many Lie-like properties like
767:
Andersen, K.K.S.; Grodal, J.; Møller, J.; Viruel, A. (2008), "The classification of p-compact groups for p odd",
571:, up to isomorphism. This is analogous to the classical classification of connected compact Lie groups, with the
360:
1022:
703:
355:
415:
695:
105:
915:
Dwyer, W.G; Wilkerson, C.W (1994), "Homotopy fixed-point methods for Lie groups and finite loop spaces",
734:: They are exactly those finite loop spaces that admit an integral maximal torus; this was the so-called
1037:
730:
Using the classification, one can identify the compact Lie groups inside finite loop spaces, giving a
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308:. These spheres turn out to have a unique loop space structure. They were first constructed by
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515:-compact group with this Weyl group is then relatively straightforward for large primes where
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902:
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969:
Grodal, J. (2010), "The classification of p-compact groups and homotopical group theory",
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598:-completion of a compact connected Lie group and BK is finite direct product of simple
408:-compact group for infinitely many primes, with the primes being determined by whether
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37:
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832:
890:
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17:
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Andersen, K.K.S.; Grodal, J. (2009), "The classification of 2-compact groups",
792:
602:
p-compact groups i.e., simple p-compact groups whose Weyl group group is not a
109:
63:
59:
945:
Dwyer, W.G; Wilkerson, C.W (1993), "A new finite loop space at the prime 2",
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874:
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146:(in general the group of components of G will be a finite p-group). The
938:
823:
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293:
930:
531:), but requires more sophisticated methods for the "modular primes"
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865:
312:
in his 1970 MIT notes. (The Weyl group is a cyclic group of order
682:
this implies that every connected 2-compact group can be written
971:
849:"The Steenrod problem of realizing polynomial cohomology rings"
690:-Wilkerson 2-compact group" BDI(4) of rank 3, constructed in
891:"The realization of polynomial algebras as cohomology rings"
732:
homotopical characterisation of compact connected Lie groups
556:
states that there is a 1-1 correspondence between connected
43:, but with all the local structure concentrated at a single
694:
with Weyl group corresponding to group number 24 in the
280:. A rank 1 connected p-compact group, for p odd, is a "
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defines a p-compact group. (Here the Weyl will be a
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560:-compact groups, up to homotopy equivalence, and
154:-compact group is the rank of its maximal torus.
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725:classification of connected finite loop spaces
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553:
51:
206:p-completion of a connected finite loop space
8:
996:Homotopy Lie Groups and Their Classification
582:It follows from the classification that any
237:-reflection group that may not come from a
524:
397:{\displaystyle W\leq GL_{r}(\mathbb {C} )}
119:homology. One sometimes also refer to the
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453:{\displaystyle GL_{r}(\mathbb {Z} _{p})}
715:Some consequences of the classification
404:can be realized as the Weyl group of a
743:
847:Andersen, K.K.S.; Grodal, J. (2008),
101:, which is local with respect to mod
7:
70:over the integers, is now a finite
14:
962:10.1090/S0894-0347-1993-1161306-9
586:-compact group can be written as
50:. This concept was introduced in
646:{\displaystyle \mathbb {Q} _{p}}
482:{\displaystyle \mathbb {Z} _{p}}
338:{\displaystyle \mathbb {Z} _{p}}
230:{\displaystyle \mathbb {Z} _{p}}
192:{\displaystyle \mathbb {Z} _{p}}
529:Chevalley–Shephard–Todd theorem
460:or not, with some embedding of
447:
432:
391:
383:
1:
988:Homotopy Lie Groups: A Survey
889:Clark, A.; Ewing, J. (1974),
833:10.1090/S0894-0347-08-00623-1
575:-adic integers replacing the
519:does not divide the order of
268:-compact group is either the
138:-compact group is said to be
66:, rather than being a finite
755:Andersen & Grodal (2008)
740:Andersen & Grodal (2009)
692:Dwyer & Wilkerson (1993)
668:{\displaystyle \mathbb {Q} }
617:{\displaystyle \mathbb {Z} }
554:Andersen & Grodal (2009)
504:{\displaystyle \mathbb {C} }
252:{\displaystyle \mathbb {Z} }
52:Dwyer & Wilkerson (1994)
1059:
793:10.4007/annals.2008.167.95
704:complex reflection groups
535:that divide the order of
736:maximal torus conjecture
525:Clark & Ewing (1974)
523:(carried out already in
511:. The construction of a
356:complex reflection group
908:10.2140/pjm.1974.50.425
412:and be conjugated into
108:, and such the pointed
1018:Topology of Lie groups
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548:The classification of
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875:10.1112/jtopol/jtn021
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552:-compact groups from
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948:J. Amer. Math. Soc.
810:J. Amer. Math. Soc.
678:For instance, when
284:sphere", i.e., the
264:A rank 1 connected
259:-reflection group.)
204:More generally the
97:is a pointed space
1013:Algebraic topology
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349:th root of unity.)
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123:-compact group by
22:algebraic topology
918:Ann. of Math. (2)
770:Ann. of Math. (2)
721:finite loop space
577:rational integers
288:-completion of a
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895:Pacific J. Math.
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751:Steenrod problem
684:BG = BH Ă— BDI(4)
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68:reflection group
20:, in particular
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1023:Homotopy theory
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702:enumeration of
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310:Dennis Sullivan
272:-completion of
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144:connected space
115:has finite mod
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901:(2): 425–434,
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859:(4): 747–760,
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817:(2): 387–436,
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569:-adic integers
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544:Classification
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79:-adic integers
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2:
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747:
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744:Grodal (2010)
741:
737:
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726:
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709:
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693:
689:
685:
681:
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653:, but is not
638:
601:
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95:compact group
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57:
53:
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42:
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36:version of a
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30:compact group
27:
23:
19:
1038:Group theory
995:
987:
970:
952:
946:
922:
916:
898:
894:
856:
852:
824:math/0611437
814:
808:
784:math/0302346
774:
768:
750:
748:
735:
731:
729:
724:
720:
718:
707:
683:
679:
677:
599:
595:
591:
588:BG = BH Ă— BK
587:
583:
581:
572:
566:
557:
549:
547:
536:
532:
520:
516:
512:
409:
405:
346:
316:, acting on
313:
305:
301:
297:
289:
285:
277:
273:
269:
265:
205:
165:
151:
147:
139:
135:
133:
128:
124:
120:
116:
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102:
98:
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90:
88:
76:
71:
56:maximal tori
47:
29:
25:
15:
925:: 395–442,
168:-completion
60:Weyl groups
34:homotopical
18:mathematics
1028:Lie groups
1007:Categories
777:: 95–210,
761:References
527:using the
142:if G is a
110:loop space
85:Definition
64:Weyl group
1033:Manifolds
980:1003.4010
955:: 37–64,
866:0704.4002
853:J. Topol.
801:119168267
564:over the
562:root data
368:≤
296:S, where
140:connected
41:Lie group
1043:Symmetry
841:17542829
304:−
300:divides
282:Sullivan
158:Examples
106:homology
939:2946585
883:1583621
753:. (See
738:. (See
696:Shepard
594:is the
345:via an
113:G = ΩBG
38:compact
937:
881:
839:
799:
706:. For
600:exotic
590:where
294:sphere
998:(PDF)
990:(PDF)
975:arXiv
935:JSTOR
879:S2CID
861:arXiv
837:S2CID
819:arXiv
797:S2CID
779:arXiv
688:Dwyer
278:SO(3)
274:SU(2)
150:of a
45:prime
32:is a
746:.)
742:and
700:Todd
290:2n-1
163:The
148:rank
131:).
58:and
24:, a
957:doi
927:doi
923:139
903:doi
871:doi
829:doi
789:doi
775:167
757:.)
708:p=3
680:p=2
489:in
276:or
16:In
1009::
973:,
951:,
933:,
921:,
899:50
897:,
893:,
877:,
869:,
855:,
851:,
835:,
827:,
815:22
813:,
795:,
787:,
773:,
719:A
675:.
592:BH
579:.
199:.)
134:A
129:BG
99:BG
89:A
81:.
977::
959::
953:6
929::
905::
873::
863::
857:1
831::
821::
791::
781::
698:-
662:Q
639:p
634:Q
611:Z
596:p
584:p
573:p
567:p
558:p
550:p
539:.
537:W
533:p
521:W
517:p
513:p
498:C
475:p
470:Z
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443:p
438:Z
433:(
428:r
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420:G
410:W
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388:C
384:(
379:r
375:L
371:G
365:W
347:n
331:p
326:Z
314:n
306:1
302:p
298:n
292:-
286:p
270:2
266:2
246:Z
223:p
218:Z
185:p
180:Z
166:p
152:p
136:p
125:G
121:p
117:p
103:p
93:-
91:p
77:p
72:p
48:p
28:-
26:p
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