Motivation Algebraic ingredients Construction of the Gysin sequence Pimsner’s construction Conclusions The Gysin Sequence for Quantum Lens Spaces Some perspective Francesca Arici (SISSA) NGA2014 - Frascati (RM) 1/38 The Gysin Sequence for Quantum Lens Spaces Francesca Arici (SISSA)
Motivation Algebraic ingredients Construction of the Gysin sequence Pimsner’s construction Conclusions The Gysin Sequence for Quantum Lens Spaces F. Arici, S. Brain, G. Landi arXiv:1401.6788 [math.QA], to appear in JNCG . Pimsner Algebras and Gysin Sequences from Principal Circle Actions F. Arici, J. Kaad, G. Landi in preparation . 2/38 The Gysin Sequence for Quantum Lens Spaces Francesca Arici (SISSA)
Motivation Algebraic ingredients Construction of the Gysin sequence Pimsner’s construction Conclusions 1 Motivation 2 Algebraic ingredients 3 Construction of the Gysin sequence 4 Pimsner’s construction 5 Conclusions 3/38 The Gysin Sequence for Quantum Lens Spaces Francesca Arici (SISSA)
Motivation Algebraic ingredients Construction of the Gysin sequence Pimsner’s construction Conclusions 1 Topology: Quotient of odd dimensional spheres by an action of a finite cyclic group. L ( n , r ) := S 2 n + 1 / Z r (1) � L ( n , r ) � Torsion phenomena, e.g. π 1 = Z r . Total spaces of U ( 1 ) bundles over projective spaces. 4/38 The Gysin Sequence for Quantum Lens Spaces Francesca Arici (SISSA)
Motivation Algebraic ingredients Construction of the Gysin sequence Pimsner’s construction Conclusions 1 Topology: Quotient of odd dimensional spheres by an action of a finite cyclic group. L ( n , r ) := S 2 n + 1 / Z r (1) � L ( n , r ) � Torsion phenomena, e.g. π 1 = Z r . Total spaces of U ( 1 ) bundles over projective spaces. 2 Problems in high energy physics: T duality Chern Simons field theories 4/38 The Gysin Sequence for Quantum Lens Spaces Francesca Arici (SISSA)
Motivation Algebraic ingredients Construction of the Gysin sequence Pimsner’s construction Conclusions Topological formulation. Long exact sequence in cohomology, associated to any sphere bundle. → E → π X . In particular, for circle bundles: U ( 1 ) ֒ 5/38 The Gysin Sequence for Quantum Lens Spaces Francesca Arici (SISSA)
Motivation Algebraic ingredients Construction of the Gysin sequence Pimsner’s construction Conclusions Topological formulation. Long exact sequence in cohomology, associated to any sphere bundle. → E → π X . In particular, for circle bundles: U ( 1 ) ֒ ∪ c 1 ( E ) � H k + 1 ( X ) π ∗ � H k − 1 ( X ) π ∗ � H k ( E ) � H k + 1 ( E ) � · · · · · · 5/38 The Gysin Sequence for Quantum Lens Spaces Francesca Arici (SISSA)
Motivation Algebraic ingredients Construction of the Gysin sequence Pimsner’s construction Conclusions Topological formulation. Long exact sequence in cohomology, associated to any sphere bundle. → E → π X . In particular, for circle bundles: U ( 1 ) ֒ ∪ c 1 ( E ) � H k + 1 ( X ) π ∗ � H k − 1 ( X ) π ∗ � H k ( E ) � H k + 1 ( E ) � · · · · · · 5/38 The Gysin Sequence for Quantum Lens Spaces Francesca Arici (SISSA)
Motivation Algebraic ingredients Construction of the Gysin sequence Pimsner’s construction Conclusions The Gysin Sequence in K-Theory Main reference: Karoubi 1978. 6/38 The Gysin Sequence for Quantum Lens Spaces Francesca Arici (SISSA)
Motivation Algebraic ingredients Construction of the Gysin sequence Pimsner’s construction Conclusions The Gysin Sequence in K-Theory Main reference: Karoubi 1978. Cyclic Six Term exact sequence . 6/38 The Gysin Sequence for Quantum Lens Spaces Francesca Arici (SISSA)
Motivation Algebraic ingredients Construction of the Gysin sequence Pimsner’s construction Conclusions The Gysin Sequence in K-Theory Main reference: Karoubi 1978. Cyclic Six Term exact sequence . In our examples K 1 ( CP n ) = 0. π ∗ δ 10 α → K 1 ( L ( n , r )) → K 0 ( C P n ) → K 0 ( C P n ) → K 0 ( L ( n , r )) − 0 − − − − − − − → 0 , (2) 6/38 The Gysin Sequence for Quantum Lens Spaces Francesca Arici (SISSA)
Motivation Algebraic ingredients Construction of the Gysin sequence Pimsner’s construction Conclusions The Gysin Sequence in K-Theory Main reference: Karoubi 1978. Cyclic Six Term exact sequence . In our examples K 1 ( CP n ) = 0. π ∗ δ 10 α → K 1 ( L ( n , r )) → K 0 ( C P n ) → K 0 ( C P n ) → K 0 ( L ( n , r )) − 0 − − − − − − − → 0 , (2) where α is the mutiliplication by the Euler class χ ( L r ) = 1 − [ L r ] (3) of the bundle L r := ξ ⊗ r , where ξ is the tautological line bundle on C P n . 6/38 The Gysin Sequence for Quantum Lens Spaces Francesca Arici (SISSA)
Motivation Algebraic ingredients Construction of the Gysin sequence Pimsner’s construction Conclusions The Gysin Sequence in K-Theory Main reference: Karoubi 1978. Cyclic Six Term exact sequence . In our examples K 1 ( CP n ) = 0. π ∗ δ 10 α → K 1 ( L ( n , r )) → K 0 ( C P n ) → K 0 ( C P n ) → K 0 ( L ( n , r )) − 0 − − − − − − − → 0 , (2) where α is the mutiliplication by the Euler class χ ( L r ) = 1 − [ L r ] (3) of the bundle L r := ξ ⊗ r , where ξ is the tautological line bundle on C P n . ... Is there a quantum version? 6/38 The Gysin Sequence for Quantum Lens Spaces Francesca Arici (SISSA)
Motivation Algebraic ingredients Construction of the Gysin sequence Pimsner’s construction Conclusions Quantum spheres and quantum projective spaces Quantum spheres... L. Vaksman, Ya. Soibelman, 1991 M. Welk, 2000 The coordinate algebra A ( S 2 n + 1 ) quantum sphere S 2 n + 1 : q q ∗ -algebra generated by 2 n + 2 elements { z i , z ∗ i } i = 0 ,..., n s.t.: z i z j = q − 1 z j z i 0 ≤ i < j ≤ n , z ∗ i z j = qz j z ∗ i � = j , i n � [ z ∗ [ z ∗ i , z i ] = ( 1 − q 2 ) z j z ∗ n , z n ] = 0 , i = 0 , . . . , n − 1 , j j = i + 1 1 = z 0 z ∗ 0 + z 1 z ∗ 1 + . . . + z n z ∗ n . 7/38 The Gysin Sequence for Quantum Lens Spaces Francesca Arici (SISSA)
Motivation Algebraic ingredients Construction of the Gysin sequence Pimsner’s construction Conclusions Quantum spheres and quantum projective spaces ...and quantum projective spaces The ∗ -subalgebra of A ( S 2 n + 1 ) generated by p ij := z ∗ i z j is the coordinate q algebra A ( C P n q ) of the quantum projective space C P n q invariant elements for the U ( 1 ) -action on the algebra A ( S 2 n + 1 ) : q ( z 0 , z 1 , . . . , z n ) �→ ( λ z 0 , λ z 1 , . . . , λ z n ) , λ ∈ U ( 1 ) . 8/38 The Gysin Sequence for Quantum Lens Spaces Francesca Arici (SISSA)
Motivation Algebraic ingredients Construction of the Gysin sequence Pimsner’s construction Conclusions Quantum spheres and quantum projective spaces ...and quantum projective spaces The ∗ -subalgebra of A ( S 2 n + 1 ) generated by p ij := z ∗ i z j is the coordinate q algebra A ( C P n q ) of the quantum projective space C P n q invariant elements for the U ( 1 ) -action on the algebra A ( S 2 n + 1 ) : q ( z 0 , z 1 , . . . , z n ) �→ ( λ z 0 , λ z 1 , . . . , λ z n ) , λ ∈ U ( 1 ) . The C ∗ -algebras C ( S 2 n + 1 ) and C ( C P n q ) of continuous functions: q completions of A ( S 2 n + 1 ) and A ( C P n q ) in the universal C ∗ -norms q 8/38 The Gysin Sequence for Quantum Lens Spaces Francesca Arici (SISSA)
Motivation Algebraic ingredients Construction of the Gysin sequence Pimsner’s construction Conclusions Quantum spheres and quantum projective spaces ...and quantum projective spaces The ∗ -subalgebra of A ( S 2 n + 1 ) generated by p ij := z ∗ i z j is the coordinate q algebra A ( C P n q ) of the quantum projective space C P n q invariant elements for the U ( 1 ) -action on the algebra A ( S 2 n + 1 ) : q ( z 0 , z 1 , . . . , z n ) �→ ( λ z 0 , λ z 1 , . . . , λ z n ) , λ ∈ U ( 1 ) . The C ∗ -algebras C ( S 2 n + 1 ) and C ( C P n q ) of continuous functions: q completions of A ( S 2 n + 1 ) and A ( C P n q ) in the universal C ∗ -norms q These are graph algebras J.H. Hong, W. Szymański 2002. 8/38 The Gysin Sequence for Quantum Lens Spaces Francesca Arici (SISSA)
Motivation Algebraic ingredients Construction of the Gysin sequence Pimsner’s construction Conclusions Quantum spheres and quantum projective spaces ...and quantum projective spaces The ∗ -subalgebra of A ( S 2 n + 1 ) generated by p ij := z ∗ i z j is the coordinate q algebra A ( C P n q ) of the quantum projective space C P n q invariant elements for the U ( 1 ) -action on the algebra A ( S 2 n + 1 ) : q ( z 0 , z 1 , . . . , z n ) �→ ( λ z 0 , λ z 1 , . . . , λ z n ) , λ ∈ U ( 1 ) . The C ∗ -algebras C ( S 2 n + 1 ) and C ( C P n q ) of continuous functions: q completions of A ( S 2 n + 1 ) and A ( C P n q ) in the universal C ∗ -norms q These are graph algebras J.H. Hong, W. Szymański 2002. Their K-theory can be computed out of the incidence matrix . 8/38 The Gysin Sequence for Quantum Lens Spaces Francesca Arici (SISSA)
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