Skew products of topological graphs and noncommutative duality
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1 Skew products of topological graphs and noncommutative duality V. Deaconu, S. Kaliszewski, J. Quigg*, and I. Raeburn work in progress AMS, Riverside, November 2009
2 Coverings can be done with small categories [Bridson-Haefliger]
3 Coverings can be done with small categories [Bridson-Haefliger] in particular: k-graphs [Pask-Q-Raeburn]
4 Coverings can be done with small categories [Bridson-Haefliger] in particular: k-graphs [Pask-Q-Raeburn] more particular: path categories of directed graphs [Deicke-Pask-Raeburn]
5 Definition D p C surjective functor
6 Definition D p C surjective functor v D 0 = p 1 (x) vd = r 1 (v) p bijective Dv x xc Cx p
7 Induced homomorphisms GD fundamental groupoid p GC
8 Induced homomorphisms GD fundamental groupoid p GC π(d, v) p π(c, x) injective fundamental group
9 Induced homomorphisms Theorem Image of fundamental group classifies covering.
10 Induced homomorphisms Theorem Image of fundamental group classifies covering. More precisely, coverings (D, p) and (E, q) of C are isomorphic p π(d, v) and q π(e, u) are conjugate subgroups of π(c, x).
11 Regular coverings Definition Simplifying assumption (for this talk): p : D C regular covering, i.e., p π(d, v) normal in π(c, x) Definition deck transformations: G := π(c, x)/p π(d, v) acts freely on (D, p), transitively on fibres
12 Skew product Definition cocycle: c : C G functor (G group)
13 Skew product Definition cocycle: c : C G functor (G group) Definition skew product covering: C c G (λ,g) (x, c(λ)g) (y, g) C x y λ
14 Skew product Definition cocycle: c : C G functor (G group) Definition skew product covering: C c G (λ,g) (x, c(λ)g) (y, g) C x y λ Theorem G acts freely on C c G via (λ, g) h = (λ, gh) (C c G)/G = C
15 Regular coverings = Skew products Theorem (Gross-Tucker well... ) If a group G acts freely on a small category D, then: D/G is a category the quotient map D D/G is a covering there is a cocycle c : D/G G such that D = quotient map D/G (D/G) c G skew product covering
16 Special case Specialize to: path categories of directed graphs (can also do for k-graphs) E = (E 0, E 1, r, s) E 1 r s E 0 edges edges go from sources to ranges: r(e) path category: e vertices s(e) e v 1 e 0 v 1 2 e v 2 v n n 1 v n
17 Cocycles for graphs ( labelings ) c : E 1 G any function into a group
18 Cocycles for graphs ( labelings ) c : E 1 G any function into a group can promote to a cocycle c : E G (E = path category): c(e 1 e n ) = c(e 1 ) c(e n )
19 Cocycles for graphs ( labelings ) c : E 1 G any function into a group can promote to a cocycle c : E G (E = path category): c(e 1 e n ) = c(e 1 ) c(e n ) Theorem (Kaliszewski-Q-Raeburn) There is a coaction δ : C (E) C (E) C (G) such that C (E c G) = C (E) δ G
20 Cocycles for graphs ( labelings ) c : E 1 G any function into a group can promote to a cocycle c : E G (E = path category): c(e 1 e n ) = c(e 1 ) c(e n ) Theorem (Kaliszewski-Q-Raeburn) There is a coaction δ : C (E) C (E) C (G) such that C (E c G) = C (E) δ G In fact, the coaction is easy to find: δ(s e ) = s(e) c(e) (e E 1 ) where s e is a typical generating partial isometry of C (E).
21 Topological graphs [Katsura] vertex and edge spaces E 0, E 1 locally compact Hausdorff
22 Topological graphs [Katsura] vertex and edge spaces E 0, E 1 locally compact Hausdorff range map r continuous
23 Topological graphs [Katsura] vertex and edge spaces E 0, E 1 locally compact Hausdorff range map r continuous source map s local homeomorphism
24 Topological graphs [Katsura] vertex and edge spaces E 0, E 1 locally compact Hausdorff range map r continuous source map s local homeomorphism C c (E 1 ) completes to C -correspondence X (E) over C 0 (E 0 )
25 Topological graphs [Katsura] vertex and edge spaces E 0, E 1 locally compact Hausdorff range map r continuous source map s local homeomorphism C c (E 1 ) completes to C -correspondence X (E) over C 0 (E 0 ) C (E) := O X (E) (Cuntz-Pimsner algebra)
26 Cocycle on topological graph c : E 1 G continuous (G locally compact group)
27 Cocycle on topological graph c : E 1 G continuous (G locally compact group) Theorem (Deaconu-Kaliszewski-Q-Raeburn) There is a coaction δ : C (E) M(C (E) C (G)) such that C (E c G) = C (E) δ G
28 Cocycle on topological graph c : E 1 G continuous (G locally compact group) Theorem (Deaconu-Kaliszewski-Q-Raeburn) There is a coaction δ : C (E) M(C (E) C (G)) such that C (E c G) = C (E) δ G But this time it s hard to get the coaction.
29 Cocycle on topological graph c : E 1 G continuous (G locally compact group) Theorem (Deaconu-Kaliszewski-Q-Raeburn) There is a coaction δ : C (E) M(C (E) C (G)) such that C (E c G) = C (E) δ G But this time it s hard to get the coaction. Two approaches: indirect: use Landstad duality direct: use multiplier bimodules
30 What about the covering theory? Question What is the appropriate notion of fundamental group for a topological graph E?
31 What about the covering theory? Question What is the appropriate notion of fundamental group for a topological graph E? Tentative Answer (Bridson-Haefliger) π(e, x) = free product of fundamental group defined as in discrete case and classical homotopy fundamental group of E 0
32 What about the covering theory? Question What is the appropriate notion of fundamental group for a topological graph E? Tentative Answer (Bridson-Haefliger) π(e, x) = free product of fundamental group defined as in discrete case and classical homotopy fundamental group of E 0 But: how to compute that?
33 What about the covering theory? Tentative Answer (Deaconu-Kumjian-Q) geometric realization: R(E) := ( (E 1 [0, 1] ) E 0) / Then define (e, 0) s(e) (e, 1) r(e) π(e, x) = π 1 (R(E), x) classical fundamental group
34 What about the covering theory? Tentative Answer (Deaconu-Kumjian-Q) geometric realization: R(E) := ( (E 1 [0, 1] ) E 0) / Then define (e, 0) s(e) (e, 1) r(e) π(e, x) = π 1 (R(E), x) classical fundamental group Hope to show these two approaches give isomorphic fundamental groups...
35 What about the covering theory? Question What about skew products and coverings?
36 What about the covering theory? Question What about skew products and coverings? Well, in the discrete case, the fibres of the covering were parameterized by the group G, so in the nondiscrete case, i.e., a cocycle from E into a locally compact group G, the projection map E c G E is certainly not a covering. Rather, it is a principal G-bundle.
37 What about the covering theory? Question What about skew products and coverings? Well, in the discrete case, the fibres of the covering were parameterized by the group G, so in the nondiscrete case, i.e., a cocycle from E into a locally compact group G, the projection map E c G E is certainly not a covering. Rather, it is a principal G-bundle. Fact This G bundle will be topologically trivial.
38 What about the covering theory? Question What about the Gross-Tucker theorem?
39 What about the covering theory? Question What about the Gross-Tucker theorem? The first part goes ok: Theorem (Deaconu-Kumjian-Q) If a locally compact group G acts freely and properly on a topological graph E, then the quotient E/G is a topological graph.
40 What about the covering theory? Question What about the Gross-Tucker theorem? The first part goes ok: Theorem (Deaconu-Kumjian-Q) If a locally compact group G acts freely and properly on a topological graph E, then the quotient E/G is a topological graph. But then we would want for some cocycle c : E 1 G. E = (E/G) c G
41 What about the covering theory? Question What about the Gross-Tucker theorem? The first part goes ok: Theorem (Deaconu-Kumjian-Q) If a locally compact group G acts freely and properly on a topological graph E, then the quotient E/G is a topological graph. But then we would want E = (E/G) c G for some cocycle c : E 1 G. And this will in general not be the case, because the principal G-bundle E E/G might not be trivial.
42 What about the covering theory? Question What about the Gross-Tucker theorem? The first part goes ok: Theorem (Deaconu-Kumjian-Q) If a locally compact group G acts freely and properly on a topological graph E, then the quotient E/G is a topological graph. But then we would want E = (E/G) c G for some cocycle c : E 1 G. And this will in general not be the case, because the principal G-bundle E E/G might not be trivial. Future work What to do with nontrivial principal G-bundles...
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