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. U Complex cobordism ring Ω∗ (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. U Complex cobordism ring Ω∗ (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. U Complex cobordism ring Ω∗ (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. U Complex cobordism ring Ω∗ (graded), under disjoint union and Cartesian product. 2 / 24 Thom’s Theorem Thom space Th(ξ). Universal n-complex bundle γn. n U Thom’s homomorphism: τ : πk+2nTh(γ ) ! Ωk . Theorem (Thom, 54) τ is an isomorphism for large n. Those Thom spaces could be assembled to form a spectrum called U ∼ MU, and Ω∗ = π∗MU: Theorem (Milnor, Novikov, 60) MU∗ = π∗MU = Z[x1; x2; :::] where xi 2 π2i MU: 3 / 24 Thom’s Theorem Thom space Th(ξ). Universal n-complex bundle γn. n U Thom’s homomorphism: τ : πk+2nTh(γ ) ! Ωk . Theorem (Thom, 54) τ is an isomorphism for large n. Those Thom spaces could be assembled to form a spectrum called U ∼ MU, and Ω∗ = π∗MU: Theorem (Milnor, Novikov, 60) MU∗ = π∗MU = Z[x1; x2; :::] where xi 2 π2i MU: 3 / 24 Thom’s Theorem Thom space Th(ξ). Universal n-complex bundle γn. n U Thom’s homomorphism: τ : πk+2nTh(γ ) ! Ωk . Theorem (Thom, 54) τ is an isomorphism for large n. Those Thom spaces could be assembled to form a spectrum called U ∼ MU, and Ω∗ = π∗MU: Theorem (Milnor, Novikov, 60) MU∗ = π∗MU = Z[x1; x2; :::] where xi 2 π2i MU: 3 / 24 Thom’s Theorem Thom space Th(ξ). Universal n-complex bundle γn. n U Thom’s homomorphism: τ : πk+2nTh(γ ) ! Ωk . Theorem (Thom, 54) τ is an isomorphism for large n. Those Thom spaces could be assembled to form a spectrum called U ∼ MU, and Ω∗ = π∗MU: Theorem (Milnor, Novikov, 60) MU∗ = π∗MU = Z[x1; x2; :::] where xi 2 π2i MU: 3 / 24 Thom’s Theorem Thom space Th(ξ). Universal n-complex bundle γn. n U Thom’s homomorphism: τ : πk+2nTh(γ ) ! Ωk . Theorem (Thom, 54) τ is an isomorphism for large n. Those Thom spaces could be assembled to form a spectrum called U ∼ MU, and Ω∗ = π∗MU: Theorem (Milnor, Novikov, 60) MU∗ = π∗MU = Z[x1; x2; :::] where xi 2 π2i MU: 3 / 24 Thom’s Theorem Thom space Th(ξ). Universal n-complex bundle γn. n U Thom’s homomorphism: τ : πk+2nTh(γ ) ! Ωk . Theorem (Thom, 54) τ is an isomorphism for large n. Those Thom spaces could be assembled to form a spectrum called U ∼ MU, and Ω∗ = π∗MU: Theorem (Milnor, Novikov, 60) MU∗ = π∗MU = Z[x1; x2; :::] where xi 2 π2i MU: 3 / 24 Homotopical equivariant complex cobordism MUG Compact Lie group G. Complete universe U. BU(n): G-space of n-dimensional complex subspaces of U. n Universal n-complex G-vector bundle γG. Complex finite dimensional representation V : G-vector bundle over a point. 4 / 24 Homotopical equivariant complex cobordism MUG Compact Lie group G. Complete universe U. BU(n): G-space of n-dimensional complex subspaces of U. n Universal n-complex G-vector bundle γG. Complex finite dimensional representation V : G-vector bundle over a point. 4 / 24 Homotopical equivariant complex cobordism MUG Compact Lie group G. Complete universe U. BU(n): G-space of n-dimensional complex subspaces of U. n Universal n-complex G-vector bundle γG. Complex finite dimensional representation V : G-vector bundle over a point. 4 / 24 Homotopical equivariant complex cobordism MUG Compact Lie group G. Complete universe U. BU(n): G-space of n-dimensional complex subspaces of U. n Universal n-complex G-vector bundle γG. Complex finite dimensional representation V : G-vector bundle over a point. 4 / 24 Homotopical equivariant complex cobordism MUG Compact Lie group G. Complete universe U. BU(n): G-space of n-dimensional complex subspaces of U. n Universal n-complex G-vector bundle γG. Complex finite dimensional representation V : G-vector bundle over a point. 4 / 24 Homotopical equivariant complex cobordism MUG Construction (tom Dieck, 70) jV j jW j For V ⊂ W , there are classifying map (W − V ) × γG ! γG : We have jV j ∼ W −V jV j jW j Th((W − V ) × γG ) = Σ Th(γG ) ! Th(γG ): jV j Let DV = Th(γG ) with the structured maps described above, then spectrify to obtain MUG. MUG is a genuine multiplicative G-specturm. It is complex stable: ∗ ∼ ∗+2jV j V MUG(X) = MUG (S ^ X): 5 / 24 Homotopical equivariant complex cobordism MUG Construction (tom Dieck, 70) jV j jW j For V ⊂ W , there are classifying map (W − V ) × γG ! γG : We have jV j ∼ W −V jV j jW j Th((W − V ) × γG ) = Σ Th(γG ) ! Th(γG ): jV j Let DV = Th(γG ) with the structured maps described above, then spectrify to obtain MUG. MUG is a genuine multiplicative G-specturm. It is complex stable: ∗ ∼ ∗+2jV j V MUG(X) = MUG (S ^ X): 5 / 24 Homotopical equivariant complex cobordism MUG Construction (tom Dieck, 70) jV j jW j For V ⊂ W , there are classifying map (W − V ) × γG ! γG : We have jV j ∼ W −V jV j jW j Th((W − V ) × γG ) = Σ Th(γG ) ! Th(γG ): jV j Let DV = Th(γG ) with the structured maps described above, then spectrify to obtain MUG. MUG is a genuine multiplicative G-specturm. It is complex stable: ∗ ∼ ∗+2jV j V MUG(X) = MUG (S ^ X): 5 / 24 Homotopical equivariant complex cobordism MUG Construction (tom Dieck, 70) jV j jW j For V ⊂ W , there are classifying map (W − V ) × γG ! γG : We have jV j ∼ W −V jV j jW j Th((W − V ) × γG ) = Σ Th(γG ) ! Th(γG ): jV j Let DV = Th(γG ) with the structured maps described above, then spectrify to obtain MUG. MUG is a genuine multiplicative G-specturm. It is complex stable: ∗ ∼ ∗+2jV j V MUG(X) = MUG (S ^ X): 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: k ∼ TM × R = ξ: G Geometric equivariant complex cobordism ring Ω∗ : However, G Ω∗ π∗MUG: The Euler class eV 2 π−2jV jMUG of V is 0 V jV j S ! S ! Th(γG ): G Fact: eV 6= 0 if V = 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: k ∼ TM × R = ξ: G Geometric equivariant complex cobordism ring Ω∗ : However, G Ω∗ π∗MUG: The Euler class eV 2 π−2jV jMUG of V is 0 V jV j S ! S ! Th(γG ): G Fact: eV 6= 0 if V = 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: k ∼ TM × R = ξ: G Geometric equivariant complex cobordism ring Ω∗ : However, G Ω∗ π∗MUG: The Euler class eV 2 π−2jV jMUG of V is 0 V jV j S ! S ! Th(γG ): G Fact: eV 6= 0 if V = 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 V jνj S ! Th(ν) ! Th(γG ); G which induces a homomorphism Ω∗ ! π∗MUG. 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 V jνj S ! Th(ν) ! Th(γG ); G which induces a homomorphism Ω∗ ! π∗MUG. 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 V jνj S ! Th(ν) ! Th(γG ); G which induces a homomorphism Ω∗ ! π∗MUG. 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 V jνj S ! Th(ν) ! Th(γG ); G which induces a homomorphism Ω∗ ! π∗MUG. The opposite of Thom’s homomorphism does not exist, due to transversality issues.