coefficients of equivariant complex cobordism

play

Coefficients of equivariant complex cobordism Yunze Lu University - PowerPoint PPT Presentation

Jul 01, 2023 •107 likes •814 views

Coefficients of equivariant complex cobordism Yunze Lu University of Michigan August, 2019 1 / 24 Complex cobordism Complex manifolds: Compact smooth manifolds, with a tangential stable almost complex structure. Two closed manifolds are

Coefficients of equivariant complex cobordism Yunze Lu University of Michigan August, 2019 1 / 24
Complex cobordism Complex manifolds: Compact smooth manifolds, with a tangential stable almost complex structure. Two closed manifolds are cobordant, if their disjoint union is the boundary of a third manifold. This is an equivalent relation. Complex cobordism ring Ω U ∗ (graded), under disjoint union and Cartesian product. 2 / 24
Complex cobordism Complex manifolds: Compact smooth manifolds, with a tangential stable almost complex structure. Two closed manifolds are cobordant, if their disjoint union is the boundary of a third manifold. This is an equivalent relation. Complex cobordism ring Ω U ∗ (graded), under disjoint union and Cartesian product. 2 / 24
Complex cobordism Complex manifolds: Compact smooth manifolds, with a tangential stable almost complex structure. Two closed manifolds are cobordant, if their disjoint union is the boundary of a third manifold. This is an equivalent relation. Complex cobordism ring Ω U ∗ (graded), under disjoint union and Cartesian product. 2 / 24
Complex cobordism Complex manifolds: Compact smooth manifolds, with a tangential stable almost complex structure. Two closed manifolds are cobordant, if their disjoint union is the boundary of a third manifold. This is an equivalent relation. Complex cobordism ring Ω U ∗ (graded), under disjoint union and Cartesian product. 2 / 24
Thom’s Theorem Thom space Th ( ξ ) . Universal n -complex bundle γ n . Thom’s homomorphism: τ : π k + 2 n Th ( γ n ) → Ω U k . Theorem (Thom, 54) τ is an isomorphism for large n . Those Thom spaces could be assembled to form a spectrum called ∗ ∼ MU , and Ω U = π ∗ MU . Theorem (Milnor, Novikov, 60) MU ∗ = π ∗ MU = Z [ x 1 , x 2 , ... ] where x i ∈ π 2 i MU . 3 / 24
Thom’s Theorem Thom space Th ( ξ ) . Universal n -complex bundle γ n . Thom’s homomorphism: τ : π k + 2 n Th ( γ n ) → Ω U k . Theorem (Thom, 54) τ is an isomorphism for large n . Those Thom spaces could be assembled to form a spectrum called ∗ ∼ MU , and Ω U = π ∗ MU . Theorem (Milnor, Novikov, 60) MU ∗ = π ∗ MU = Z [ x 1 , x 2 , ... ] where x i ∈ π 2 i MU . 3 / 24
Thom’s Theorem Thom space Th ( ξ ) . Universal n -complex bundle γ n . Thom’s homomorphism: τ : π k + 2 n Th ( γ n ) → Ω U k . Theorem (Thom, 54) τ is an isomorphism for large n . Those Thom spaces could be assembled to form a spectrum called ∗ ∼ MU , and Ω U = π ∗ MU . Theorem (Milnor, Novikov, 60) MU ∗ = π ∗ MU = Z [ x 1 , x 2 , ... ] where x i ∈ π 2 i MU . 3 / 24
Thom’s Theorem Thom space Th ( ξ ) . Universal n -complex bundle γ n . Thom’s homomorphism: τ : π k + 2 n Th ( γ n ) → Ω U k . Theorem (Thom, 54) τ is an isomorphism for large n . Those Thom spaces could be assembled to form a spectrum called ∗ ∼ MU , and Ω U = π ∗ MU . Theorem (Milnor, Novikov, 60) MU ∗ = π ∗ MU = Z [ x 1 , x 2 , ... ] where x i ∈ π 2 i MU . 3 / 24
Thom’s Theorem Thom space Th ( ξ ) . Universal n -complex bundle γ n . Thom’s homomorphism: τ : π k + 2 n Th ( γ n ) → Ω U k . Theorem (Thom, 54) τ is an isomorphism for large n . Those Thom spaces could be assembled to form a spectrum called ∗ ∼ MU , and Ω U = π ∗ MU . Theorem (Milnor, Novikov, 60) MU ∗ = π ∗ MU = Z [ x 1 , x 2 , ... ] where x i ∈ π 2 i MU . 3 / 24
Thom’s Theorem Thom space Th ( ξ ) . Universal n -complex bundle γ n . Thom’s homomorphism: τ : π k + 2 n Th ( γ n ) → Ω U k . Theorem (Thom, 54) τ is an isomorphism for large n . Those Thom spaces could be assembled to form a spectrum called ∗ ∼ MU , and Ω U = π ∗ MU . Theorem (Milnor, Novikov, 60) MU ∗ = π ∗ MU = Z [ x 1 , x 2 , ... ] where x i ∈ π 2 i MU . 3 / 24
Homotopical equivariant complex cobordism MU G Compact Lie group G . Complete universe U . B U ( n ) : G -space of n -dimensional complex subspaces of U . Universal n -complex G -vector bundle γ n G . Complex finite dimensional representation V : G -vector bundle over a point. 4 / 24
Homotopical equivariant complex cobordism MU G Compact Lie group G . Complete universe U . B U ( n ) : G -space of n -dimensional complex subspaces of U . Universal n -complex G -vector bundle γ n G . Complex finite dimensional representation V : G -vector bundle over a point. 4 / 24
Homotopical equivariant complex cobordism MU G Compact Lie group G . Complete universe U . B U ( n ) : G -space of n -dimensional complex subspaces of U . Universal n -complex G -vector bundle γ n G . Complex finite dimensional representation V : G -vector bundle over a point. 4 / 24
Homotopical equivariant complex cobordism MU G Compact Lie group G . Complete universe U . B U ( n ) : G -space of n -dimensional complex subspaces of U . Universal n -complex G -vector bundle γ n G . Complex finite dimensional representation V : G -vector bundle over a point. 4 / 24
Homotopical equivariant complex cobordism MU G Compact Lie group G . Complete universe U . B U ( n ) : G -space of n -dimensional complex subspaces of U . Universal n -complex G -vector bundle γ n G . Complex finite dimensional representation V : G -vector bundle over a point. 4 / 24
Homotopical equivariant complex cobordism MU G Construction (tom Dieck, 70) For V ⊂ W , there are classifying map ( W − V ) × γ | V | → γ | W | G . G We have Th (( W − V ) × γ | V | = Σ W − V Th ( γ | V | G ) → Th ( γ | W | G ) ∼ G ) . Let D V = Th ( γ | V | G ) with the structured maps described above, then spectrify to obtain MU G . MU G is a genuine multiplicative G -specturm. It is complex stable: ( S V ∧ X ) . = MU ∗ + 2 | V | G ( X ) ∼ MU ∗ G 5 / 24
Homotopical equivariant complex cobordism MU G Construction (tom Dieck, 70) For V ⊂ W , there are classifying map ( W − V ) × γ | V | → γ | W | G . G We have Th (( W − V ) × γ | V | = Σ W − V Th ( γ | V | G ) → Th ( γ | W | G ) ∼ G ) . Let D V = Th ( γ | V | G ) with the structured maps described above, then spectrify to obtain MU G . MU G is a genuine multiplicative G -specturm. It is complex stable: ( S V ∧ X ) . = MU ∗ + 2 | V | G ( X ) ∼ MU ∗ G 5 / 24
Homotopical equivariant complex cobordism MU G Construction (tom Dieck, 70) For V ⊂ W , there are classifying map ( W − V ) × γ | V | → γ | W | G . G We have Th (( W − V ) × γ | V | = Σ W − V Th ( γ | V | G ) → Th ( γ | W | G ) ∼ G ) . Let D V = Th ( γ | V | G ) with the structured maps described above, then spectrify to obtain MU G . MU G is a genuine multiplicative G -specturm. It is complex stable: ( S V ∧ X ) . = MU ∗ + 2 | V | G ( X ) ∼ MU ∗ G 5 / 24
Homotopical equivariant complex cobordism MU G Construction (tom Dieck, 70) For V ⊂ W , there are classifying map ( W − V ) × γ | V | → γ | W | G . G We have Th (( W − V ) × γ | V | = Σ W − V Th ( γ | V | G ) → Th ( γ | W | G ) ∼ G ) . Let D V = Th ( γ | V | G ) with the structured maps described above, then spectrify to obtain MU G . MU G is a genuine multiplicative G -specturm. It is complex stable: ( S V ∧ X ) . = MU ∗ + 2 | V | G ( X ) ∼ MU ∗ G 5 / 24
Geometric equivariant complex cobordism Tangential stable almost complex structure for a smooth G -manifold M : equivariant isomorphism to a G -complex vector bundle ξ over M : TM × R k ∼ = ξ. Geometric equivariant complex cobordism ring Ω G ∗ . However, Ω G ∗ ≇ π ∗ MU G . The Euler class e V ∈ π − 2 | V | MU G of V is S 0 → S V → Th ( γ | V | G ) . Fact: e V � = 0 if V G = 0. 6 / 24
Geometric equivariant complex cobordism Tangential stable almost complex structure for a smooth G -manifold M : equivariant isomorphism to a G -complex vector bundle ξ over M : TM × R k ∼ = ξ. Geometric equivariant complex cobordism ring Ω G ∗ . However, Ω G ∗ ≇ π ∗ MU G . The Euler class e V ∈ π − 2 | V | MU G of V is S 0 → S V → Th ( γ | V | G ) . Fact: e V � = 0 if V G = 0. 6 / 24
Geometric equivariant complex cobordism Tangential stable almost complex structure for a smooth G -manifold M : equivariant isomorphism to a G -complex vector bundle ξ over M : TM × R k ∼ = ξ. Geometric equivariant complex cobordism ring Ω G ∗ . However, Ω G ∗ ≇ π ∗ MU G . The Euler class e V ∈ π − 2 | V | MU G of V is S 0 → S V → Th ( γ | V | G ) . Fact: e V � = 0 if V G = 0. 6 / 24
Pontryagin-Thom construction Take a cobordant class [ M ] . Equivariant Whiteny’s embedding: M ֒ → V . The normal bundle ν embeds as a tubular neighborhood. Pontryagin-Thom constuction gives a composite map S V → Th ( ν ) → Th ( γ | ν | G ) , which induces a homomorphism Ω G ∗ → π ∗ MU G . The opposite of Thom’s homomorphism does not exist, due to transversality issues. 7 / 24
Pontryagin-Thom construction Take a cobordant class [ M ] . Equivariant Whiteny’s embedding: M ֒ → V . The normal bundle ν embeds as a tubular neighborhood. Pontryagin-Thom constuction gives a composite map S V → Th ( ν ) → Th ( γ | ν | G ) , which induces a homomorphism Ω G ∗ → π ∗ MU G . The opposite of Thom’s homomorphism does not exist, due to transversality issues. 7 / 24

Download Presentation

Download Policy: The content available on the website is offered to you 'AS IS' for your personal information and use only. It cannot be commercialized, licensed, or distributed on other websites without prior consent from the author. To download a presentation, simply click this link. If you encounter any difficulties during the download process, it's possible that the publisher has removed the file from their server.

Recommend

More recommend