WHAT IS K-HOMOLOGY? Paul Baum Penn State. Texas A&M University College Station, Texas, USA. April 2, 2014

Transcription

1 WHAT IS K-HOMOLOGY? Paul Baum Penn State Texas A&M University College Station, Texas, USA April 2, 2014 Paul Baum (Penn State) WHAT IS K-HOMOLOGY? April 2, / 56

2 Let X be a compact C manifold without boundary. X is not required to be oriented. X is not required to be even dimensional. On X let δ : C (X, E 0 ) C (X, E 1 ) be an elliptic differential (or pseudo-differential) operator. (S(T X 1 R ), E σ ) K 0 ( ), and Index(D Eσ ) = Index(δ). Paul Baum (Penn State) WHAT IS K-HOMOLOGY? April 2, / 56

3 (S(T X 1 R ), E σ ) Index(δ) = (ch(e σ ) Td((S(T X 1 R )))[(S(T X 1 R )] and this is the general Atiyah-Singer formula. S(T X 1 R ) is the unit sphere bundle of T X 1 R. S(T X 1 R ) is even dimensional and is in a natural way a Spin c manifold. E σ is the C vector bundle on S(T X 1 R ) obtained by doing a clutching construction using the symbol σ of δ. Paul Baum (Penn State) WHAT IS K-HOMOLOGY? April 2, / 56

4 Minicourse of five lectures: 1. Dirac operator 2. Atiyah-Singer revisited 3. What is K-homology? 4. Beyond ellipticity 5. The Riemann-Roch theorem The minicourse is based on joint work with Ron Douglas and is dedicated to Ron Douglas. Paul Baum (Penn State) WHAT IS K-HOMOLOGY? April 2, / 56

5 K-homology in algebraic geometry Let X be a (possibly singular) projective algebraic variety / C. Grothendieck defined two abelian groups: Kalg 0 (X) = Grothendieck group of algebraic vector bundles on X. K alg 0 (X) = Grothendieck group of coherent algebraic sheaves on X. Kalg 0 (X) = the algebraic geometry K-theory of X (contravariant). K alg 0 (X) = the algebraic geometry K-homology of X (covariant). Paul Baum (Penn State) WHAT IS K-HOMOLOGY? April 2, / 56

6 K-homology in topology Problem How can K-homology be taken from algebraic geometry to topology? Paul Baum (Penn State) WHAT IS K-HOMOLOGY? April 2, / 56

7 K-homology is the dual theory to K-theory. There are three ways in which K-homology in topology has been defined: Homotopy Theory K-theory is the cohomology theory and K-homology is the homology theory determined by the Bott (i.e. K-theory) spectrum. This is the spectrum..., Z BU, U, Z BU, U,... K-Cycles K-homology is the group of K-cycles. C -algebras K-homology is the Atiyah-BDF-Kasparov group KK (A, C). Paul Baum (Penn State) WHAT IS K-HOMOLOGY? April 2, / 56

8 Let X be a finite CW complex. The three versions of K-homology are isomorphic. K homotopy j (X) K j(x) KK j (C(X), C) homotopy theory K-cycles Atiyah-BDF-Kasparov j = 0, 1 Paul Baum (Penn State) WHAT IS K-HOMOLOGY? April 2, / 56

9 K-homology in topology Problem How can K-homology be taken from algebraic geometry to topology? There are three ways in which this has been done: Homotopy Theory K-homology is the homology theory determined by the Bott spectrum. K-Cycles K-homology is the group of K-cycles. C -algebras K-homology is the Kasparov group KK (A, C). Paul Baum (Penn State) WHAT IS K-HOMOLOGY? April 2, / 56

10 Atiyah-BDF-Kasparov K-homology M.F. Atiyah Brown-Douglas-Fillmore G.Kasparov Let X be a finite CW complex. C(X) = {α : X C α is continuous} L(H) = {bounded operators T : H H} Any element in the Atiyah-BDF-Kasparov K-homology group KK 0 (C(X), C) is given by a 5-tuple (H 0, ψ 0, H 1, ψ 1, T ) such that : Paul Baum (Penn State) WHAT IS K-HOMOLOGY? April 2, / 56

11 H 0 and H 1 are separable Hilbert spaces. ψ 0 : C(X) L(H 0 ) and ψ 1 : C(X) L(H 1 ) are unital -homomorphisms. T : H 0 H 1 is a (bounded) Fredholm operator. For every α C(X) the commutator T ψ 0 (α) ψ 1 (α) T L(H 0, H 1 ) is compact. KK 0 (C(X), C) := {(H 0, ψ 0, H 1, ψ 1, T )}/ Paul Baum (Penn State) WHAT IS K-HOMOLOGY? April 2, / 56

12 KK 0 (C(X), C) := {(H 0, ψ 0, H 1, ψ 1, T )}/ (H 0, ψ 0, H 1, ψ 1, T ) + (H 0, ψ 0, H 1, ψ 1, T ) = (H 0 H 0, ψ 0 ψ 0, H 1 H 1, ψ 1 ψ 1, T T ) (H 0, ψ 0, H 1, ψ 1, T ) = (H 1, ψ 1, H 0, ψ 0, T ) Paul Baum (Penn State) WHAT IS K-HOMOLOGY? April 2, / 56

13 Let X be a finite CW complex. Any element in the Atiyah-BDF-Kasparov K-homology group KK 1 (C(X), C) is given by a 3-tuple (H, ψ, T ) such that : H is a separable Hilbert space. ψ : C(X) L(H) is a unital -homomorphism. T : H H is a (bounded) self-adjoint Fredholm operator. For every α C(X) the commutator T ψ(α) ψ(α) T L(H) is compact. Paul Baum (Penn State) WHAT IS K-HOMOLOGY? April 2, / 56

14 KK 1 (C(X), C) := {(H, ψ, T )}/ (H, ψ, T ) + (H, ψ, T ) = (H H, ψ ψ, T T ) (H, ψ, T ) = (H, ψ, T ) Paul Baum (Penn State) WHAT IS K-HOMOLOGY? April 2, / 56

15 Let X, Y be CW complexes and let f : X Y be a continuous map. Denote by f : C(X) C(Y ) the -homomorphism f (α) := α f α C(Y ) Then f : KK j (C(X), C) KK j (C(Y ), C) is f (H, ψ, T ) := (H, ψ f, T ) j = 1 f (H 0, ψ 0, H 1, ψ 1, T ) := (H 0, ψ 0 f, H 1, ψ 1 f, T ) j = 0 Paul Baum (Penn State) WHAT IS K-HOMOLOGY? April 2, / 56

16 Cycles for K-homology Let X be a CW complex. Definition A K-cycle on X is a triple (M, E, ϕ) such that : 1 M is a compact Spin c manifold without boundary. 2 E is a C vector bundle on M. 3 ϕ: M X is a continuous map from M to X. Paul Baum (Penn State) WHAT IS K-HOMOLOGY? April 2, / 56

17 Set K (X) = {(M, E, ϕ)}/ where the equivalence relation is generated by the three elementary steps Bordism Direct sum - disjoint union Vector bundle modification Paul Baum (Penn State) WHAT IS K-HOMOLOGY? April 2, / 56

18 Isomorphism (M, E, ϕ) is isomorphic to (M, E, ϕ ) iff a diffeomorphism ψ : M M preserving the Spin c -structures on M, M and with and commutativity in the diagram ψ (E ) = E ψ M M ϕ ϕ X Paul Baum (Penn State) WHAT IS K-HOMOLOGY? April 2, / 56

19 Bordism (M 0, E 0, ϕ 0 ) is bordant to (M 1, E 1, ϕ 1 ) iff (Ω, E, ϕ) such that: 1 Ω is a compact Spin c manifold with boundary. 2 E is C vector bundle on Ω. 3 ( Ω, E Ω, ϕ Ω ) = (M 0, E 0, ϕ 0 ) ( M 1, E 1, ϕ 1 ) M 1 is M 1 with the Spin c structure reversed. Paul Baum (Penn State) WHAT IS K-HOMOLOGY? April 2, / 56

20 (M 0, E 0, ϕ 0 ) ( M 1, E 1, ϕ 1 ) X Paul Baum (Penn State) WHAT IS K-HOMOLOGY? April 2, / 56

21 Direct sum - disjoint union Let E, E be two C vector bundles on M (M, E, ϕ) (M, E, ϕ) (M, E E, ϕ) Paul Baum (Penn State) WHAT IS K-HOMOLOGY? April 2, / 56

22 Vector bundle modification (M, E, ϕ) Let F be a Spin c vector bundle on M Assume that dim R (F p ) 0 mod 2 p M for every fiber F p of F 1 R = M R S(F 1 R ) := unit sphere bundle of F 1 R (M, E, ϕ) (S(F 1 R ), β π E, ϕ π) Paul Baum (Penn State) WHAT IS K-HOMOLOGY? April 2, / 56

23 S(F 1 R ) π M This is a fibration with even-dimensional spheres as fibers. F 1 R is a Spin c vector bundle on M with odd-dimensional fibers. Let Σ be the spinor bundle for F 1 Cliff C (F p R) Σ p Σ p π Σ = β β (M, E, ϕ) (S(F 1 R ), β π E, ϕ π) Paul Baum (Penn State) WHAT IS K-HOMOLOGY? April 2, / 56

24 {(M, E, ϕ)}/ = K 0 (X) K 1 (X) K j (X) = subgroup of {(M, E, ϕ)}/ consisting of all (M, E, ϕ) such that every connected component of M has dimension j mod 2, j = 0, 1 Paul Baum (Penn State) WHAT IS K-HOMOLOGY? April 2, / 56

25 Addition in K j (X) is disjoint union. (M, E, ϕ) + (M, E, ϕ ) = (M M, E E, ϕ ϕ ) Additive inverse of (M, E, ϕ) is obtained by reversing the Spin c structure of M. (M, E, ϕ) = ( M, E, ϕ) Paul Baum (Penn State) WHAT IS K-HOMOLOGY? April 2, / 56

26 DEFINITION. (M, E, ϕ) bounds (Ω, Ẽ, ϕ) with : 1 Ω is a compact Spin c manifold with boundary. 2 Ẽ is a C vector bundle on Ω. 3 ϕ: Ω X is a continuous map. 4 ( Ω, Ẽ Ω, ϕ Ω ) = (M, E, ϕ) REMARK. (M, E, ϕ) = 0 in K (X) (M, E, ϕ) (M, E, ϕ ) where (M, E, ϕ ) bounds. Paul Baum (Penn State) WHAT IS K-HOMOLOGY? April 2, / 56

27 Let X, Y be CW complexes and let f : X Y be a continuous map. Then f : K j (X) K j (Y ) is f (M, E, ϕ) := (M, E, f ϕ) Paul Baum (Penn State) WHAT IS K-HOMOLOGY? April 2, / 56

28 Chern character in K-theory X finite CW complex ch: K j (X) l j = 0, 1 Q Z K j (X) l H j+2l (X; Q) H j+2l (X; Q) is an isomorphism of Q vector spaces. Paul Baum (Penn State) WHAT IS K-HOMOLOGY? April 2, / 56

29 Chern character in K-homology X finite CW complex (M, E, ϕ) ϕ (ch(e) T d(m) [M]) ch: K j (X) l j = 0, 1 Q Z K j (X) l H j+2l (X; Q) H j+2l (X; Q) is an isomorphism of Q vector spaces. Paul Baum (Penn State) WHAT IS K-HOMOLOGY? April 2, / 56

30 Chern character and module structure X a finite CW complex. K (X) is a ring and K (X) is a module over this ring. Chern character respects the ring and module structure. Paul Baum (Penn State) WHAT IS K-HOMOLOGY? April 2, / 56

31 Theorem (PB and R.Douglas and M.Taylor, PB and N. Higson and T. Schick) Let X be a finite CW complex. Then for j = 0, 1 the natural map of abelian groups is an isomorphism. K j (X) KK j (C(X), C) Paul Baum (Penn State) WHAT IS K-HOMOLOGY? April 2, / 56

32 For j = 0, 1 the natural map of abelian groups is (M, E, ϕ) ϕ [D E ] where K j (X) KK j (C(X), C) 1 D E is the Dirac operator of M tensored with E. 2 [D E ] KK j (C(M), C) is the element in the Atiyah-BDF-Kasparov K-homology of M determined by D E. 3 ϕ : KK j (C(M), C) KK j (C(X), C) is the homomorphism of abelian groups determined by ϕ: M X. Paul Baum (Penn State) WHAT IS K-HOMOLOGY? April 2, / 56