From Baum-Connes to the Novikov conjecture
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What is the Baum-Connes conjecture?
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What is the Novikov conjecture?
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How does Baum-Connes ⇒ Novikov ?
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Table of contents
Brief description of these conjectures
Details on the Baum-Connes conjecture
Details on the Novikov conjecture
Baum-Connes ⇒ Novikov
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The Baum-Connes conjecture
The Baum-Connes conjecture expresses a relationship
K-theory of the C*-algebra of a group
−→ K-homology of the classifying space of proper actions of the group.
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The Baum-Connes conjecture
The Baum-Connes conjecture expresses a relationship
K-theory of the C*-algebra of a group
−→ K-homology of the classifying space of proper actions of the group.
G locally compact group
Cr∗ (G ) reduced group C*-algebra of G
E G classifying space for proper actions of G
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The Baum-Connes conjecture
The Baum-Connes conjecture expresses a relationship
K-theory of the C*-algebra of a group
−→ K-homology of the classifying space of proper actions of the group.
G locally compact group
Cr∗ (G ) reduced group C*-algebra of G
E G classifying space for proper actions of G
Analytic assembly map
µ : KjG (E G ) −→ Kj (Cr∗ (G ))
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The Baum-Connes conjecture
The Baum-Connes conjecture expresses a relationship
K-theory of the C*-algebra of a group
−→ K-homology of the classifying space of proper actions of the group.
G locally compact group
Cr∗ (G ) reduced group C*-algebra of G
E G classifying space for proper actions of G
Analytic assembly map
µ : KjG (E G ) −→ Kj (Cr∗ (G ))
Conjecture (P. Baum, A. Connes; 1982)
The assembly map µ is an isomorphism of abelian groups.
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The Baum-Connes conjecture: µ : KjG (E G ) −→ Kj (Cr∗ (G )) is an isomorphism.
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a way of calculating the operator K-theory of Cr∗ (G ) by means of group homology
and representation theory
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The Baum-Connes conjecture: µ : KjG (E G ) −→ Kj (Cr∗ (G )) is an isomorphism.
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a way of calculating the operator K-theory of Cr∗ (G ) by means of group homology
and representation theory
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relates an analytical object Kj (Cr∗ (G )) with a geometrical/topological object
KjG (E G )
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The Baum-Connes conjecture: µ : KjG (E G ) −→ Kj (Cr∗ (G )) is an isomorphism.
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a way of calculating the operator K-theory of Cr∗ (G ) by means of group homology
and representation theory
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relates an analytical object Kj (Cr∗ (G )) with a geometrical/topological object
KjG (E G )
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implies several other conjectures in diverse fields (topology, geometry, functional
analysis, ..)
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The Baum-Connes conjecture: µ : KjG (E G ) −→ Kj (Cr∗ (G )) is an isomorphism.
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a way of calculating the operator K-theory of Cr∗ (G ) by means of group homology
and representation theory
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relates an analytical object Kj (Cr∗ (G )) with a geometrical/topological object
KjG (E G )
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implies several other conjectures in diverse fields (topology, geometry, functional
analysis, ..)
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injectivity of µ has geometrical/topological implications: e.g. Novikov conj,
Gromov-Lawson-Rosenberg conj
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The Baum-Connes conjecture: µ : KjG (E G ) −→ Kj (Cr∗ (G )) is an isomorphism.
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a way of calculating the operator K-theory of Cr∗ (G ) by means of group homology
and representation theory
I
relates an analytical object Kj (Cr∗ (G )) with a geometrical/topological object
KjG (E G )
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implies several other conjectures in diverse fields (topology, geometry, functional
analysis, ..)
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injectivity of µ has geometrical/topological implications: e.g. Novikov conj,
Gromov-Lawson-Rosenberg conj
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surjectivity of µ has analytical implications: e.g. Kaplansky-Kadison conj
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Groups for which Baum-Connes conjecture has been verified
Some examples of classes of groups
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Compact groups
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Abelian groups
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Lie groups with finitely many conn. comps.
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hyperbolic groups
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a-T-menable groups (includes amenable groups, groups acting properly on trees,
...)
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p-adic groups
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adelic groups
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Groups for which Baum-Connes conjecture has been verified
Some examples of classes of groups
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Compact groups
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Abelian groups
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Lie groups with finitely many conn. comps.
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hyperbolic groups
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a-T-menable groups (includes amenable groups, groups acting properly on trees,
...)
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p-adic groups
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adelic groups
Counterexamples? Have been established for generalized forms of the conjecture:
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Baum-Connes conjecture with coefficients. µ : KjG (E G , A) −→ Kj (A or G ).
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coarse Baum-Connes conjecture: µ : K∗ (X ) −→ K∗ (C ∗ (X )).
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The Novikov conjecture
The Novikov conjecture is a statement about homotopy invariance of higher signatures
associated to a discrete group and closed, connected, oriented, smooth manifolds.
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The Novikov conjecture
The Novikov conjecture is a statement about homotopy invariance of higher signatures
associated to a discrete group and closed, connected, oriented, smooth manifolds.
Let Γ be a discrete group and BΓ its classifying space.
M a smooth, closed, connected, oriented manifold.
LM ∈ H 4∗ (M; Q) the total L-class of M.
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The Novikov conjecture
The Novikov conjecture is a statement about homotopy invariance of higher signatures
associated to a discrete group and closed, connected, oriented, smooth manifolds.
Let Γ be a discrete group and BΓ its classifying space.
M a smooth, closed, connected, oriented manifold.
LM ∈ H 4∗ (M; Q) the total L-class of M.
For x ∈ H ∗ (BΓ; Q) and f : M −→ BΓ continuous map, define
σx (M, f ) := hLM ∪ f ∗ (x), [M]i ∈ Q
”higher signature” .
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The Novikov conjecture
The Novikov conjecture is a statement about homotopy invariance of higher signatures
associated to a discrete group and closed, connected, oriented, smooth manifolds.
Let Γ be a discrete group and BΓ its classifying space.
M a smooth, closed, connected, oriented manifold.
LM ∈ H 4∗ (M; Q) the total L-class of M.
For x ∈ H ∗ (BΓ; Q) and f : M −→ BΓ continuous map, define
σx (M, f ) := hLM ∪ f ∗ (x), [M]i ∈ Q
”higher signature” .
Conjecture (S. Novikov; 1965)
All higher signatures determined by Γ are oriented homotopy invariants.
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The Novikov conjecture
The Novikov conjecture is a statement about homotopy invariance of higher signatures
associated to a discrete group and closed, connected, oriented, smooth manifolds.
Let Γ be a discrete group and BΓ its classifying space.
M a smooth, closed, connected, oriented manifold.
LM ∈ H 4∗ (M; Q) the total L-class of M.
For x ∈ H ∗ (BΓ; Q) and f : M −→ BΓ continuous map, define
σx (M, f ) := hLM ∪ f ∗ (x), [M]i ∈ Q
”higher signature” .
Conjecture (S. Novikov; 1965)
All higher signatures determined by Γ are oriented homotopy invariants. Meaning:
for any x ∈ H ∗ (BΓ; Q), any orientation preserving homotopy equivalence h : N −→ M
of closed oriented manifolds N and M, and f : M −→ BΓ,
σx (M, f ) = σx (N, f ◦ h).
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The Novikov conjecture: for x ∈ H ∗ (BΓ; Q) and reference map f : M −→ BΓ, the
characteristic number hLM ∪ f ∗ (x), [M]i ∈ Q is an oriented homotopy invariant.
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The Novikov conjecture: for x ∈ H ∗ (BΓ; Q) and reference map f : M −→ BΓ, the
characteristic number hLM ∪ f ∗ (x), [M]i ∈ Q is an oriented homotopy invariant.
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characteristic numbers are interesting and important for classification of manifolds
(e.g. Euler characteristic)
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The Novikov conjecture: for x ∈ H ∗ (BΓ; Q) and reference map f : M −→ BΓ, the
characteristic number hLM ∪ f ∗ (x), [M]i ∈ Q is an oriented homotopy invariant.
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characteristic numbers are interesting and important for classification of manifolds
(e.g. Euler characteristic)
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the conjecture expresses a relationship between a characteristic class of the
manifold to the underlying homotopy theory
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The Novikov conjecture: for x ∈ H ∗ (BΓ; Q) and reference map f : M −→ BΓ, the
characteristic number hLM ∪ f ∗ (x), [M]i ∈ Q is an oriented homotopy invariant.
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characteristic numbers are interesting and important for classification of manifolds
(e.g. Euler characteristic)
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the conjecture expresses a relationship between a characteristic class of the
manifold to the underlying homotopy theory
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Hirzebruch signature theorem: σ(M) = hLM, [M]i. The signature is an oriented
homotopy invariant of M
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The Novikov conjecture: for x ∈ H ∗ (BΓ; Q) and reference map f : M −→ BΓ, the
characteristic number hLM ∪ f ∗ (x), [M]i ∈ Q is an oriented homotopy invariant.
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characteristic numbers are interesting and important for classification of manifolds
(e.g. Euler characteristic)
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the conjecture expresses a relationship between a characteristic class of the
manifold to the underlying homotopy theory
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Hirzebruch signature theorem: σ(M) = hLM, [M]i. The signature is an oriented
homotopy invariant of M
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LM is characteristic class, a certain polynomial in rational Pontrjagin classes.
These are themselves not homotopy invariants
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The Novikov conjecture: for x ∈ H ∗ (BΓ; Q) and reference map f : M −→ BΓ, the
characteristic number hLM ∪ f ∗ (x), [M]i ∈ Q is an oriented homotopy invariant.
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characteristic numbers are interesting and important for classification of manifolds
(e.g. Euler characteristic)
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the conjecture expresses a relationship between a characteristic class of the
manifold to the underlying homotopy theory
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Hirzebruch signature theorem: σ(M) = hLM, [M]i. The signature is an oriented
homotopy invariant of M
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LM is characteristic class, a certain polynomial in rational Pontrjagin classes.
These are themselves not homotopy invariants
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There are no known counterexamples to the Novikov conjecture
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Details on the Baum-Connes conjecture
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Details on the Baum-Connes conjecture
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The classifying space E G for proper actions
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Details on the Baum-Connes conjecture
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The classifying space E G for proper actions
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KK and KK G
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Details on the Baum-Connes conjecture
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The classifying space E G for proper actions
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KK and KK G
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K-theory and K-homology
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Details on the Baum-Connes conjecture
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The classifying space E G for proper actions
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KK and KK G
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K-theory and K-homology
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Reduced group C*-algebra Cr∗ (G )
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Details on the Baum-Connes conjecture
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The classifying space E G for proper actions
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KK and KK G
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K-theory and K-homology
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Reduced group C*-algebra Cr∗ (G )
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G -equivariant K-homology KjG
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Details on the Baum-Connes conjecture
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The classifying space E G for proper actions
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KK and KK G
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K-theory and K-homology
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Reduced group C*-algebra Cr∗ (G )
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G -equivariant K-homology KjG
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The analytic assembly map µ as the G -index IndexG
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The classifying space E G
G locally compact, Hausdorff, second countable group.
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The classifying space E G
G locally compact, Hausdorff, second countable group.
X is called a G -space if equipped with a continuous action G × X −→ X
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The classifying space E G
G locally compact, Hausdorff, second countable group.
X is called a G -space if equipped with a continuous action G × X −→ X
Interested in proper actions (the G -action is locally induced from action of a compact
subgroup).
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The classifying space E G
G locally compact, Hausdorff, second countable group.
X is called a G -space if equipped with a continuous action G × X −→ X
Interested in proper actions (the G -action is locally induced from action of a compact
subgroup).
E G := a universal example for proper actions of G , meaning
E G is a proper G -space such that:
if X is any proper G -space, there exists a unique (up to G -homotopy) G -map
X −→ E G .
E G can be constructed (a G-CW-complex) as an infinite join W ∗ W ∗ W ∗ · · · where
W = ∪G /H is the disjoint union over compact subgroups H ⊂ G .
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Kasparov’s operator K-theory KK and KK G
Homotopy invariant bifunctor KKj (−, −) : C*-algebras −→ abelian groups. j = 0, 1.
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Kasparov’s operator K-theory KK and KK G
Homotopy invariant bifunctor KKj (−, −) : C*-algebras −→ abelian groups. j = 0, 1.
Contravariant in first argument, covariant in second.
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Kasparov’s operator K-theory KK and KK G
Homotopy invariant bifunctor KKj (−, −) : C*-algebras −→ abelian groups. j = 0, 1.
Contravariant in first argument, covariant in second.
KKj (A, B) = homotopy classes of triples (E , π, F ) where
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Kasparov’s operator K-theory KK and KK G
Homotopy invariant bifunctor KKj (−, −) : C*-algebras −→ abelian groups. j = 0, 1.
Contravariant in first argument, covariant in second.
KKj (A, B) = homotopy classes of triples (E , π, F ) where
I E is count.gen. Hilbert B-module
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Kasparov’s operator K-theory KK and KK G
Homotopy invariant bifunctor KKj (−, −) : C*-algebras −→ abelian groups. j = 0, 1.
Contravariant in first argument, covariant in second.
KKj (A, B) = homotopy classes of triples (E , π, F ) where
I E is count.gen. Hilbert B-module
I π : A −→ LA (E ) *-homomorphism
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Kasparov’s operator K-theory KK and KK G
Homotopy invariant bifunctor KKj (−, −) : C*-algebras −→ abelian groups. j = 0, 1.
Contravariant in first argument, covariant in second.
KKj (A, B) = homotopy classes of triples (E , π, F ) where
I E is count.gen. Hilbert B-module
I π : A −→ LA (E ) *-homomorphism
I F ∈ LA (E ) satisfies π(a)(F 2 − 1), [π(a), F ] ∈ KA (E )
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Kasparov’s operator K-theory KK and KK G
Homotopy invariant bifunctor KKj (−, −) : C*-algebras −→ abelian groups. j = 0, 1.
Contravariant in first argument, covariant in second.
KKj (A, B) = homotopy classes of triples (E , π, F ) where
I E is count.gen. Hilbert B-module
I π : A −→ LA (E ) *-homomorphism
I F ∈ LA (E ) satisfies π(a)(F 2 − 1), [π(a), F ] ∈ KA (E )
I grading: for j = 0, E has Z2 -grading, for j = 1 no grading
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Kasparov’s operator K-theory KK and KK G
Homotopy invariant bifunctor KKj (−, −) : C*-algebras −→ abelian groups. j = 0, 1.
Contravariant in first argument, covariant in second.
KKj (A, B) = homotopy classes of triples (E , π, F ) where
I E is count.gen. Hilbert B-module
I π : A −→ LA (E ) *-homomorphism
I F ∈ LA (E ) satisfies π(a)(F 2 − 1), [π(a), F ] ∈ KA (E )
I grading: for j = 0, E has Z2 -grading, for j = 1 no grading
I addition: by direct sum ⊕
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Kasparov’s operator K-theory KK and KK G
Homotopy invariant bifunctor KKj (−, −) : C*-algebras −→ abelian groups. j = 0, 1.
Contravariant in first argument, covariant in second.
KKj (A, B) = homotopy classes of triples (E , π, F ) where
I E is count.gen. Hilbert B-module
I π : A −→ LA (E ) *-homomorphism
I F ∈ LA (E ) satisfies π(a)(F 2 − 1), [π(a), F ] ∈ KA (E )
I grading: for j = 0, E has Z2 -grading, for j = 1 no grading
I addition: by direct sum ⊕
I homotopy: a continuous path of triples
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Kasparov’s operator K-theory KK and KK G
Homotopy invariant bifunctor KKj (−, −) : C*-algebras −→ abelian groups. j = 0, 1.
Contravariant in first argument, covariant in second.
KKj (A, B) = homotopy classes of triples (E , π, F ) where
I E is count.gen. Hilbert B-module
I π : A −→ LA (E ) *-homomorphism
I F ∈ LA (E ) satisfies π(a)(F 2 − 1), [π(a), F ] ∈ KA (E )
I grading: for j = 0, E has Z2 -grading, for j = 1 no grading
I addition: by direct sum ⊕
I homotopy: a continuous path of triples
KK G (−, −) : G-C*-algebras −→ abelian groups.
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Kasparov’s operator K-theory KK and KK G
Homotopy invariant bifunctor KKj (−, −) : C*-algebras −→ abelian groups. j = 0, 1.
Contravariant in first argument, covariant in second.
KKj (A, B) = homotopy classes of triples (E , π, F ) where
I E is count.gen. Hilbert B-module
I π : A −→ LA (E ) *-homomorphism
I F ∈ LA (E ) satisfies π(a)(F 2 − 1), [π(a), F ] ∈ KA (E )
I grading: for j = 0, E has Z2 -grading, for j = 1 no grading
I addition: by direct sum ⊕
I homotopy: a continuous path of triples
KK G (−, −) : G-C*-algebras −→ abelian groups.
KKjG (A, B) = homotopy classes of triples (E , π, F ) as above, in addition
I E carries a unitary representation G −→ LA (E )
I π is G -equivariant
I F is G -continuous (g 7→ g · F norm cont.) and G -equivariant
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Kasparov’s operator K-theory KK and KK G
Homotopy invariant bifunctor KKj (−, −) : C*-algebras −→ abelian groups. j = 0, 1.
Contravariant in first argument, covariant in second.
KKj (A, B) = homotopy classes of triples (E , π, F ) where
I E is count.gen. Hilbert B-module
I π : A −→ LA (E ) *-homomorphism
I F ∈ LA (E ) satisfies π(a)(F 2 − 1), [π(a), F ] ∈ KA (E )
I grading: for j = 0, E has Z2 -grading, for j = 1 no grading
I addition: by direct sum ⊕
I homotopy: a continuous path of triples
KK G (−, −) : G-C*-algebras −→ abelian groups.
KKjG (A, B) = homotopy classes of triples (E , π, F ) as above, in addition
I E carries a unitary representation G −→ LA (E )
I π is G -equivariant
I F is G -continuous (g 7→ g · F norm cont.) and G -equivariant
{e}
KKj (A, B) = KKj
(A, B) with {e} trivial group.
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K-theory and K-homology
C*-algebra A.
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K-theory and K-homology
C*-algebra A.
K-theory. Kj (A) = KKj (C, A). j = 0, 1.
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K-theory and K-homology
C*-algebra A.
K-theory. Kj (A) = KKj (C, A). j = 0, 1.
More conventional definition:
K0 (A) built from equivalence classes of idempotents in matrix algebras ∪Mn (A)
i.e. built from finitely generated projective A-modules.
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K-theory and K-homology
C*-algebra A.
K-theory. Kj (A) = KKj (C, A). j = 0, 1.
More conventional definition:
K0 (A) built from equivalence classes of idempotents in matrix algebras ∪Mn (A)
i.e. built from finitely generated projective A-modules.
Proper G -space X .
G -equivariant K-homology. KjG (X ) := KKjG (C0 (X ), C).
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K-theory and K-homology
C*-algebra A.
K-theory. Kj (A) = KKj (C, A). j = 0, 1.
More conventional definition:
K0 (A) built from equivalence classes of idempotents in matrix algebras ∪Mn (A)
i.e. built from finitely generated projective A-modules.
Proper G -space X .
G -equivariant K-homology. KjG (X ) := KKjG (C0 (X ), C).
A generalized index map. The G -index construction IndexG : KjG (X ) −→ Kj (Cr∗ (G )),
in other words
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K-theory and K-homology
C*-algebra A.
K-theory. Kj (A) = KKj (C, A). j = 0, 1.
More conventional definition:
K0 (A) built from equivalence classes of idempotents in matrix algebras ∪Mn (A)
i.e. built from finitely generated projective A-modules.
Proper G -space X .
G -equivariant K-homology. KjG (X ) := KKjG (C0 (X ), C).
A generalized index map. The G -index construction IndexG : KjG (X ) −→ Kj (Cr∗ (G )),
in other words
IndexG : KKjG (C0 (X ), C)
−−−−→
KKj (C, Cr∗ (G ))
(E , π, F )
G-module
−−−−→
(E, F)
−−−−−−−−−−→ Cr∗ (G )-module
completed to
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The reduced group C*-algebra Cr∗ (G )
Locally compact G with a fixed Haar measure.
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The reduced group C*-algebra Cr∗ (G )
Locally compact G with a fixed Haar measure.
Hilbert space L2 (G ). Left regular representation
λ : G −→ B(L2 (G )),
λs (ξ)(t) = ξ(s −1 t).
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The reduced group C*-algebra Cr∗ (G )
Locally compact G with a fixed Haar measure.
Hilbert space L2 (G ). Left regular representation
λs (ξ)(t) = ξ(s −1 t).
R
Consider L1 (G ) with convolution product, (f × g )(t) = G f (s)g (s −1 t) ds, and
involution f ∗ (t) = ∆(t)−1 f (t −1 ). This makes L1 (G ) a Banach algebra.
λ : G −→ B(L2 (G )),
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The reduced group C*-algebra Cr∗ (G )
Locally compact G with a fixed Haar measure.
Hilbert space L2 (G ). Left regular representation
λs (ξ)(t) = ξ(s −1 t).
R
Consider L1 (G ) with convolution product, (f × g )(t) = G f (s)g (s −1 t) ds, and
involution f ∗ (t) = ∆(t)−1 f (t −1 ). This makes L1 (G ) a Banach algebra.
λ : G −→ B(L2 (G )),
The left regular representation induces a *-homomorphism
λ : L1 (G ) −→ B(L2 (G )), f 7→ λf ,
Z
λf (ξ)(t) =
f (s)(λs ξ)(t) ds.
G
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The reduced group C*-algebra Cr∗ (G )
Locally compact G with a fixed Haar measure.
Hilbert space L2 (G ). Left regular representation
λs (ξ)(t) = ξ(s −1 t).
R
Consider L1 (G ) with convolution product, (f × g )(t) = G f (s)g (s −1 t) ds, and
involution f ∗ (t) = ∆(t)−1 f (t −1 ). This makes L1 (G ) a Banach algebra.
λ : G −→ B(L2 (G )),
The left regular representation induces a *-homomorphism
λ : L1 (G ) −→ B(L2 (G )), f 7→ λf ,
Z
λf (ξ)(t) =
f (s)(λs ξ)(t) ds.
G
The reduced group C*-algebra of G is defined
Cr∗ (G ) := λ(L1 (G )) ⊆ B(L2 (G )).
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The G -equivariant K-homology KjG (X )
Let X be a G -compact, proper G -space.
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The G -equivariant K-homology KjG (X )
Let X be a G -compact, proper G -space.
G -equivariant K-homology of X
KjG (X ) := KKjG (C0 (X ), C) = ”G-equivariant abstract elliptic operators”
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The G -equivariant K-homology KjG (X )
Let X be a G -compact, proper G -space.
G -equivariant K-homology of X
KjG (X ) := KKjG (C0 (X ), C) = ”G-equivariant abstract elliptic operators”
KjG (X ) consists of homotopy classes of triples (H, π, F ) where
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The G -equivariant K-homology KjG (X )
Let X be a G -compact, proper G -space.
G -equivariant K-homology of X
KjG (X ) := KKjG (C0 (X ), C) = ”G-equivariant abstract elliptic operators”
KjG (X ) consists of homotopy classes of triples (H, π, F ) where
I H is Hilbert space with unitary representation G −→ B(H)
I G -covariant representation π : C0 (X ) −→ B(H)
I F is G -equivariant operator, F ∈ B(H)
I the operators π(f )(F 2 − 1) and [π(f ), F ] are compact
I j = 0 : H has Z2 -grading and representations are grading preserving, F is grading
reversing. j = 1, no grading.
Homotopy: norm continuous path (Ft )t∈[0,1] connecting the triples (H, π, F0 ) and
(H, π, F1 )
Addition operation ⊕: form the direct sum of the triples
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The G -equivariant K-homology KjG (X )
X is assumed G -compact proper G -space.
homotopy classes of G -equivariant
G
Kj (X ) =
abstract elliptic operators on X
These are abelian groups with addition ⊕.
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The G -equivariant K-homology KjG (X )
X is assumed G -compact proper G -space.
homotopy classes of G -equivariant
G
Kj (X ) =
abstract elliptic operators on X
These are abelian groups with addition ⊕.
In general, when Z is a proper G -space not G -compact,
KjG (Z ) =
lim
−→
G -compact X ⊂ Z
KjG (X )
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The assembly map µ
µ : KjG (X ) −→ Kj (Cr∗ (G )) = KKj (C, Cr∗ (G ))
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The assembly map µ
µ : KjG (X ) −→ Kj (Cr∗ (G )) = KKj (C, Cr∗ (G ))
(H, π, F )
7→
(H, F)
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The assembly map µ
µ : KjG (X ) −→ Kj (Cr∗ (G )) = KKj (C, Cr∗ (G ))
(H, π, F )
Given an element (H, π, F ) ∈
KjG (X )
7→
(H, F)
we construct an element (H, F) ∈ Kj (Cr∗ (G )).
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The assembly map µ
µ : KjG (X ) −→ Kj (Cr∗ (G )) = KKj (C, Cr∗ (G ))
(H, π, F )
Given an element (H, π, F ) ∈
KjG (X )
7→
(H, F)
we construct an element (H, F) ∈ Kj (Cr∗ (G )).
Let H0 = π(Cc (X ))H. This is a right module for the convolution algebra
Cc (G ) ⊂ Cr∗ (G ),
Z
ξ · f :=
f (s) s −1 · ξ ds, ξ ∈ H0 , f ∈ Cc (G ),
G
15 / 30
The assembly map µ
µ : KjG (X ) −→ Kj (Cr∗ (G )) = KKj (C, Cr∗ (G ))
(H, π, F )
Given an element (H, π, F ) ∈
KjG (X )
7→
(H, F)
we construct an element (H, F) ∈ Kj (Cr∗ (G )).
Let H0 = π(Cc (X ))H. This is a right module for the convolution algebra
Cc (G ) ⊂ Cr∗ (G ),
Z
ξ · f :=
f (s) s −1 · ξ ds, ξ ∈ H0 , f ∈ Cc (G ),
G
and carries the Cc (G )-valued inner product hv1 , v2 i(g ) = hv1 , g · v2 i.
15 / 30
The assembly map µ
µ : KjG (X ) −→ Kj (Cr∗ (G )) = KKj (C, Cr∗ (G ))
(H, π, F )
Given an element (H, π, F ) ∈
KjG (X )
7→
(H, F)
we construct an element (H, F) ∈ Kj (Cr∗ (G )).
Let H0 = π(Cc (X ))H. This is a right module for the convolution algebra
Cc (G ) ⊂ Cr∗ (G ),
Z
ξ · f :=
f (s) s −1 · ξ ds, ξ ∈ H0 , f ∈ Cc (G ),
G
and carries the Cc (G )-valued inner product hv1 , v2 i(g ) = hv1 , g · v2 i.
Define H = completion of H0 in the norm associated to this inner product, pass from
the dense subspace Cc (G ) to its completion Cr∗ (G ).
The operator F : H −→ H passes to an operator F : H −→ H.
15 / 30
The assembly map µ
This gives us a map
µ : KjG (X ) −→ Kj (Cr∗ (G ))
µ([(H, π, F )]) = [(H, F)]
16 / 30
The assembly map µ
This gives us a map
µ : KjG (X ) −→ Kj (Cr∗ (G ))
µ([(H, π, F )]) = [(H, F)]
which is the G -index construction
µ((H, π, F )) = IndexG (F ).
16 / 30
The assembly map µ
This gives us a map
µ : KjG (X ) −→ Kj (Cr∗ (G ))
µ([(H, π, F )]) = [(H, F)]
which is the G -index construction
µ((H, π, F )) = IndexG (F ).
Construction passes to direct limits, and we get the analytic assembly map
µ : KjG (E G ) −→ Kj (Cr∗ (G )).
16 / 30
Details on the Novikov conjecture
17 / 30
Details on the Novikov conjecture
I
Signature of a manifold
17 / 30
Details on the Novikov conjecture
I
Signature of a manifold
I
Signature theorem and the signature operator
17 / 30
Details on the Novikov conjecture
I
Signature of a manifold
I
Signature theorem and the signature operator
I
Higher signatures and homological formulation of Novikov conjecture
17 / 30
Details on the Novikov conjecture
I
Signature of a manifold
I
Signature theorem and the signature operator
I
Higher signatures and homological formulation of Novikov conjecture
I
Dual chern character
17 / 30
Details on the Novikov conjecture
I
Signature of a manifold
I
Signature theorem and the signature operator
I
Higher signatures and homological formulation of Novikov conjecture
I
Dual chern character
I
Homotopy invariance of analytic signature
17 / 30
Details on the Novikov conjecture
I
Signature of a manifold
I
Signature theorem and the signature operator
I
Higher signatures and homological formulation of Novikov conjecture
I
Dual chern character
I
Homotopy invariance of analytic signature
I
Strong Novikov conjecture
17 / 30
Signature of manifold
Let M be a closed, connected, oriented smooth manifold of dimension n = 4k.
18 / 30
Signature of manifold
Let M be a closed, connected, oriented smooth manifold of dimension n = 4k.
The intersection form (cup product)
Z
2k
2k
α∧β
H (M) × H (M) −→ R, ([α], [β]) 7→
M
is a symmetric bilinear form.
18 / 30
Signature of manifold
Let M be a closed, connected, oriented smooth manifold of dimension n = 4k.
The intersection form (cup product)
Z
2k
2k
α∧β
H (M) × H (M) −→ R, ([α], [β]) 7→
M
is a symmetric bilinear form.
Define the signature of the manifold
σ(M) = signature of the intersection form
18 / 30
Signature of manifold
Let M be a closed, connected, oriented smooth manifold of dimension n = 4k.
The intersection form (cup product)
Z
2k
2k
α∧β
H (M) × H (M) −→ R, ([α], [β]) 7→
M
is a symmetric bilinear form.
Define the signature of the manifold
σ(M) = signature of the intersection form
Hirzebruch Signature Theorem gives a cohomological formula for σ(M), in terms of
pairing between L-class and fundamental class of the manifold.
18 / 30
Signature of manifold
Let M be a closed, connected, oriented smooth manifold of dimension n = 4k.
The intersection form (cup product)
Z
2k
2k
α∧β
H (M) × H (M) −→ R, ([α], [β]) 7→
M
is a symmetric bilinear form.
Define the signature of the manifold
σ(M) = signature of the intersection form
Hirzebruch Signature Theorem gives a cohomological formula for σ(M), in terms of
pairing between L-class and fundamental class of the manifold.
The fundamental class [M] ∈ Hn (M; Z) = Z is the generator element. It pairs with
top-degree differential forms
Z
hω, [M]i =
ω.
M
18 / 30
Signature of manifold
The total L-class LM ∈
L
k≥0 H
4k (M; Q)
LM = 1 + L1 (p1 ) + L2 (p1 , p2 ) + L3 (p1 , p2 , p3 ) + . . .
consists of homogeneous polynomials Lk in the rational Pontrjagin classes
pi = pi (M; Q) ∈ H 4i (M; Q)
19 / 30
Signature of manifold
The total L-class LM ∈
L
k≥0 H
4k (M; Q)
LM = 1 + L1 (p1 ) + L2 (p1 , p2 ) + L3 (p1 , p2 , p3 ) + . . .
consists of homogeneous polynomials Lk in the rational Pontrjagin classes
pi = pi (M; Q) ∈ H 4i (M; Q)
Theorem (Hirzebruch)
σ(M) = hLM, [M]i.
19 / 30
Signature of manifold
The total L-class LM ∈
L
k≥0 H
4k (M; Q)
LM = 1 + L1 (p1 ) + L2 (p1 , p2 ) + L3 (p1 , p2 , p3 ) + . . .
consists of homogeneous polynomials Lk in the rational Pontrjagin classes
pi = pi (M; Q) ∈ H 4i (M; Q)
Theorem (Hirzebruch)
σ(M) = hLM, [M]i.
One way of approaching this formula is
I
Define the signature operator ∂, a classical elliptic differential operator
I
Identify the signature of M with the index of ∂ via Hodge theory and harmonic
forms
I
Compute index(∂) by Atiyah-Singer index thm
19 / 30
Signature operator of M
Let Ω = ⊕Ωk be sections of exterior algebra of differential forms, Ωk = C ∞ (Λk T ∗ M).
d : Ωi −→ Ωi+1 exterior differentiation
with formal adjoint d ∗ : Ωi+1 −→ Ωi with
R
respect to pairing (α, β) = M α ∧ ∗β, Hodge star ∗ : Ωi −→ Ωn−i
d + d ∗ : Ω −→ Ω is an elliptic first-order operator and (d + d ∗ )2 = ∆ Laplacian.
20 / 30
Signature operator of M
Let Ω = ⊕Ωk be sections of exterior algebra of differential forms, Ωk = C ∞ (Λk T ∗ M).
d : Ωi −→ Ωi+1 exterior differentiation
with formal adjoint d ∗ : Ωi+1 −→ Ωi with
R
respect to pairing (α, β) = M α ∧ ∗β, Hodge star ∗ : Ωi −→ Ωn−i
d + d ∗ : Ω −→ Ω is an elliptic first-order operator and (d + d ∗ )2 = ∆ Laplacian.
Denote by Ω± the ±1-eigenspaces for ∗, then d + d ∗ interchanges Ω+ and Ω− .
20 / 30
Signature operator of M
Let Ω = ⊕Ωk be sections of exterior algebra of differential forms, Ωk = C ∞ (Λk T ∗ M).
d : Ωi −→ Ωi+1 exterior differentiation
with formal adjoint d ∗ : Ωi+1 −→ Ωi with
R
respect to pairing (α, β) = M α ∧ ∗β, Hodge star ∗ : Ωi −→ Ωn−i
d + d ∗ : Ω −→ Ω is an elliptic first-order operator and (d + d ∗ )2 = ∆ Laplacian.
Denote by Ω± the ±1-eigenspaces for ∗, then d + d ∗ interchanges Ω+ and Ω− . Define
the signature operator ∂ as the restriction of d + d ∗ as an operator from Ω+ to Ω− , i.e.
∂ : Ω+ −→ Ω−
20 / 30
Signature operator of M
Let Ω = ⊕Ωk be sections of exterior algebra of differential forms, Ωk = C ∞ (Λk T ∗ M).
d : Ωi −→ Ωi+1 exterior differentiation
with formal adjoint d ∗ : Ωi+1 −→ Ωi with
R
respect to pairing (α, β) = M α ∧ ∗β, Hodge star ∗ : Ωi −→ Ωn−i
d + d ∗ : Ω −→ Ω is an elliptic first-order operator and (d + d ∗ )2 = ∆ Laplacian.
Denote by Ω± the ±1-eigenspaces for ∗, then d + d ∗ interchanges Ω+ and Ω− . Define
the signature operator ∂ as the restriction of d + d ∗ as an operator from Ω+ to Ω− , i.e.
∂ : Ω+ −→ Ω−
Recall index(∂) = dim(ker ∂) − dim(ker ∂ ∗ ).
20 / 30
Signature operator of M
Let Ω = ⊕Ωk be sections of exterior algebra of differential forms, Ωk = C ∞ (Λk T ∗ M).
d : Ωi −→ Ωi+1 exterior differentiation
with formal adjoint d ∗ : Ωi+1 −→ Ωi with
R
respect to pairing (α, β) = M α ∧ ∗β, Hodge star ∗ : Ωi −→ Ωn−i
d + d ∗ : Ω −→ Ω is an elliptic first-order operator and (d + d ∗ )2 = ∆ Laplacian.
Denote by Ω± the ±1-eigenspaces for ∗, then d + d ∗ interchanges Ω+ and Ω− . Define
the signature operator ∂ as the restriction of d + d ∗ as an operator from Ω+ to Ω− , i.e.
∂ : Ω+ −→ Ω−
Recall index(∂) = dim(ker ∂) − dim(ker ∂ ∗ ).
ker ∂ = ker ∂ ∗ ∂ = (ker ∆) ∩ Ω+ . Put H+ := ker ∆ ∩ Ω+ . Likewise
ker ∂ ∗ = H− := ker ∆ ∩ Ω− . Then re-phrase
index(∂) = dim(H+ ) − dim(H− ).
20 / 30
Signature operator of M
Let Ω = ⊕Ωk be sections of exterior algebra of differential forms, Ωk = C ∞ (Λk T ∗ M).
d : Ωi −→ Ωi+1 exterior differentiation
with formal adjoint d ∗ : Ωi+1 −→ Ωi with
R
respect to pairing (α, β) = M α ∧ ∗β, Hodge star ∗ : Ωi −→ Ωn−i
d + d ∗ : Ω −→ Ω is an elliptic first-order operator and (d + d ∗ )2 = ∆ Laplacian.
Denote by Ω± the ±1-eigenspaces for ∗, then d + d ∗ interchanges Ω+ and Ω− . Define
the signature operator ∂ as the restriction of d + d ∗ as an operator from Ω+ to Ω− , i.e.
∂ : Ω+ −→ Ω−
Recall index(∂) = dim(ker ∂) − dim(ker ∂ ∗ ).
ker ∂ = ker ∂ ∗ ∂ = (ker ∆) ∩ Ω+ . Put H+ := ker ∆ ∩ Ω+ . Likewise
ker ∂ ∗ = H− := ker ∆ ∩ Ω− . Then re-phrase
index(∂) = dim(H+ ) − dim(H− ).
j
H+
There are isomorphisms
degree forms, degree 2k,
4k−j
∼
= H− , which give cancellations except for the middle
20 / 30
Signature operator of M
Let Ω = ⊕Ωk be sections of exterior algebra of differential forms, Ωk = C ∞ (Λk T ∗ M).
d : Ωi −→ Ωi+1 exterior differentiation
with formal adjoint d ∗ : Ωi+1 −→ Ωi with
R
respect to pairing (α, β) = M α ∧ ∗β, Hodge star ∗ : Ωi −→ Ωn−i
d + d ∗ : Ω −→ Ω is an elliptic first-order operator and (d + d ∗ )2 = ∆ Laplacian.
Denote by Ω± the ±1-eigenspaces for ∗, then d + d ∗ interchanges Ω+ and Ω− . Define
the signature operator ∂ as the restriction of d + d ∗ as an operator from Ω+ to Ω− , i.e.
∂ : Ω+ −→ Ω−
Recall index(∂) = dim(ker ∂) − dim(ker ∂ ∗ ).
ker ∂ = ker ∂ ∗ ∂ = (ker ∆) ∩ Ω+ . Put H+ := ker ∆ ∩ Ω+ . Likewise
ker ∂ ∗ = H− := ker ∆ ∩ Ω− . Then re-phrase
index(∂) = dim(H+ ) − dim(H− ).
4k−j
∼
There are isomorphisms
= H− , which give cancellations except for the middle
degree forms, degree 2k, thus
j
H+
2k
2k
index(∂) = dim(H+
) − dim(H−
).
20 / 30
Signature operator of M
Consider the quadratic form of the intersection form Q : α 7→
R
M
α ∧ ∗α.
21 / 30
Signature operator of M
R
Consider the quadratic form of the intersection form Q : α 7→ M α ∧ ∗α. This form is
2k and negative definite on H 2k , and so the signature of M can
positive definite on H+
−
be computed as
21 / 30
Signature operator of M
R
Consider the quadratic form of the intersection form Q : α 7→ M α ∧ ∗α. This form is
2k and negative definite on H 2k , and so the signature of M can
positive definite on H+
−
be computed as
σ(M) = signature of Q
= #{+1 eigenvalues of Q} − #{−1 eigenvalues of Q}
2k
2k
= dim(H+
) − dim(H−
)
= index(∂).
21 / 30
Signature operator of M
Atiyah-Singer index formula
22 / 30
Signature operator of M
Atiyah-Singer index formula gives
Z
LM.
index(∂) =
M
22 / 30
Signature operator of M
Atiyah-Singer index formula gives
Z
LM.
index(∂) =
M
Hence
Z
LM = hLM, [M]i.
σ(M) = index(∂) =
M
22 / 30
Homological formulation of the Novikov conjecture
Let Γ = π1 (M) be a discrete group, and BΓ its classifying space.
23 / 30
Homological formulation of the Novikov conjecture
Let Γ = π1 (M) be a discrete group, and BΓ its classifying space.
e −→ M.
Let τ : M −→ BΓ denote the classifying map for the universal cover M
23 / 30
Homological formulation of the Novikov conjecture
Let Γ = π1 (M) be a discrete group, and BΓ its classifying space.
e −→ M.
Let τ : M −→ BΓ denote the classifying map for the universal cover M
∗
Let x ∈ H (BΓ; Q).
23 / 30
Homological formulation of the Novikov conjecture
Let Γ = π1 (M) be a discrete group, and BΓ its classifying space.
e −→ M.
Let τ : M −→ BΓ denote the classifying map for the universal cover M
∗
Let x ∈ H (BΓ; Q).
Novikov conjecture:
σx (M) = hLM ∪ τ ∗ x, [M]i ∈ Q
is an oriented homotopy invariant.
23 / 30
Homological formulation of the Novikov conjecture
Let Γ = π1 (M) be a discrete group, and BΓ its classifying space.
e −→ M.
Let τ : M −→ BΓ denote the classifying map for the universal cover M
∗
Let x ∈ H (BΓ; Q).
Novikov conjecture:
σx (M) = hLM ∪ τ ∗ x, [M]i ∈ Q
is an oriented homotopy invariant.
Then observe: hLM ∪ τ ∗ x, [M]i = hτ ∗ x, LM ∩ [M]i = hx, τ∗ (LM ∩ [M])i.
23 / 30
Homological formulation of the Novikov conjecture
Let Γ = π1 (M) be a discrete group, and BΓ its classifying space.
e −→ M.
Let τ : M −→ BΓ denote the classifying map for the universal cover M
∗
Let x ∈ H (BΓ; Q).
Novikov conjecture:
σx (M) = hLM ∪ τ ∗ x, [M]i ∈ Q
is an oriented homotopy invariant.
Then observe: hLM ∪ τ ∗ x, [M]i = hτ ∗ x, LM ∩ [M]i = hx, τ∗ (LM ∩ [M])i.
So an equivalent formulation is:
τ∗ (LM ∩ [M]) ∈ H∗ (BΓ; Q) is an oriented homotopy invariant.
23 / 30
e
e in K Γ (M)
The element [∂] in K∗ (M) and [∂]
∗
Consider signature operator ∂ of M.
24 / 30
e
e in K Γ (M)
The element [∂] in K∗ (M) and [∂]
∗
Consider signature operator ∂ of M. It gives canonically an element
[∂] := (L2 (Λ∗ M), ∂(1 + ∂ 2 )−1/2 ) ∈ K0 (M).
24 / 30
e
e in K Γ (M)
The element [∂] in K∗ (M) and [∂]
∗
Consider signature operator ∂ of M. It gives canonically an element
[∂] := (L2 (Λ∗ M), ∂(1 + ∂ 2 )−1/2 ) ∈ K0 (M).
e −→ M. The signature operator ∂ lifts
Γ = π1 (M) acts on the universal cover M
e
e This gives the Γ-equivariant (abstract)
canonically to a Γ-equivariant operator ∂ on M.
elliptic operator
e ∈ K Γ (M).
e
[∂]
0
24 / 30
e
e in K Γ (M)
The element [∂] in K∗ (M) and [∂]
∗
Consider signature operator ∂ of M. It gives canonically an element
[∂] := (L2 (Λ∗ M), ∂(1 + ∂ 2 )−1/2 ) ∈ K0 (M).
e −→ M. The signature operator ∂ lifts
Γ = π1 (M) acts on the universal cover M
e
e This gives the Γ-equivariant (abstract)
canonically to a Γ-equivariant operator ∂ on M.
elliptic operator
e ∈ K Γ (M).
e
[∂]
0
The Γ-index
e −→ Kj (C ∗ Γ)
IndexΓ : KjΓ (M)
r
is constructed as before; a triple (H, π, F ) carrying a representation of Γ is completed
to a triple built on a Hilbert C*-module for Cr∗ Γ.
24 / 30
e −→ M with classifying map τ : M −→ BΓ. Universal principal
Universal cover M
Γ-bundle E Γ −→ BΓ.
e −−−−→
M


y
EΓ


y
M −−−−→ BΓ
τ
25 / 30
e −→ M with classifying map τ : M −→ BΓ. Universal principal
Universal cover M
Γ-bundle E Γ −→ BΓ.
e −−−−→
M


y
EΓ


y
M −−−−→ BΓ
τ
By universal property of E Γ, there is a map σ : E Γ −→ E Γ.
e
[∂]
e
IndexΓ (∂)
µ
σ∗
e −−−−→ K Γ (E Γ) −−−
K∗Γ (M)
−→ K∗Γ (E Γ) −−−−→ K∗ (Cr∗ Γ)
∗

∼
y=
K∗ (M) −−−−→ K∗ (BΓ)
τ∗
[∂]
25 / 30
Dual Chern character homomorphism
Chern character isomorphism ch∗ : K ∗ (X ) ⊗ Q −→ H ∗ (X ; Q).
26 / 30
Dual Chern character homomorphism
Chern character isomorphism ch∗ : K ∗ (X ) ⊗ Q −→ H ∗ (X ; Q).
Dually there is
ch∗ : K∗ (X ) ⊗ Q −→ H∗ (X ; Q)
hch∗ (x), ch∗ (y )i = hx, y i,
x ∈ K ∗ (X ) ⊗ Q, y ∈ K∗ (X ) ⊗ Q.
26 / 30
Dual Chern character homomorphism
Chern character isomorphism ch∗ : K ∗ (X ) ⊗ Q −→ H ∗ (X ; Q).
Dually there is
ch∗ : K∗ (X ) ⊗ Q −→ H∗ (X ; Q)
hch∗ (x), ch∗ (y )i = hx, y i,
x ∈ K ∗ (X ) ⊗ Q, y ∈ K∗ (X ) ⊗ Q.
One can deduce from the signature/index theorem:
ch∗ ([∂]) = LM ∩ [M].
26 / 30
Homotopy invariance of analytic signature
e −→ M denote the universal cover and Γ = π1 (M). Then the signature operator
Let M
e
∂ of M lifts to a Γ-equivariant elliptic operator ∂e on M.
27 / 30
Homotopy invariance of analytic signature
e −→ M denote the universal cover and Γ = π1 (M). Then the signature operator
Let M
e
∂ of M lifts to a Γ-equivariant elliptic operator ∂e on M.
Theorem (Kasparov. Mischchenko. Lusztig. Kaminker-Miller)
Let M and N be closed connected oriented manifolds and f : M −→ N an orientation
preserving homotopy equivalence. Let Γ = π1 (M) = π1 (N). Denote the signature
operators by ∂M and ∂N .
27 / 30
Homotopy invariance of analytic signature
e −→ M denote the universal cover and Γ = π1 (M). Then the signature operator
Let M
e
∂ of M lifts to a Γ-equivariant elliptic operator ∂e on M.
Theorem (Kasparov. Mischchenko. Lusztig. Kaminker-Miller)
Let M and N be closed connected oriented manifolds and f : M −→ N an orientation
preserving homotopy equivalence. Let Γ = π1 (M) = π1 (N). Denote the signature
operators by ∂M and ∂N . Then
IndexΓ (∂eM ) = IndexΓ (∂eN ) ∈ Kj (Cr∗ Γ).
27 / 30
Homotopy invariance of analytic signature
e −→ M denote the universal cover and Γ = π1 (M). Then the signature operator
Let M
e
∂ of M lifts to a Γ-equivariant elliptic operator ∂e on M.
Theorem (Kasparov. Mischchenko. Lusztig. Kaminker-Miller)
Let M and N be closed connected oriented manifolds and f : M −→ N an orientation
preserving homotopy equivalence. Let Γ = π1 (M) = π1 (N). Denote the signature
operators by ∂M and ∂N . Then
IndexΓ (∂eM ) = IndexΓ (∂eN ) ∈ Kj (Cr∗ Γ).
I
Using general theory of Fredholm complexes; co-chain complexes of Hilbert
C*-modules
27 / 30
Homotopy invariance of analytic signature
e −→ M denote the universal cover and Γ = π1 (M). Then the signature operator
Let M
e
∂ of M lifts to a Γ-equivariant elliptic operator ∂e on M.
Theorem (Kasparov. Mischchenko. Lusztig. Kaminker-Miller)
Let M and N be closed connected oriented manifolds and f : M −→ N an orientation
preserving homotopy equivalence. Let Γ = π1 (M) = π1 (N). Denote the signature
operators by ∂M and ∂N . Then
IndexΓ (∂eM ) = IndexΓ (∂eN ) ∈ Kj (Cr∗ Γ).
I
I
Using general theory of Fredholm complexes; co-chain complexes of Hilbert
C*-modules
Associate a generalized signature operator to a complex; and its analytic index in
K-theory
27 / 30
Homotopy invariance of analytic signature
e −→ M denote the universal cover and Γ = π1 (M). Then the signature operator
Let M
e
∂ of M lifts to a Γ-equivariant elliptic operator ∂e on M.
Theorem (Kasparov. Mischchenko. Lusztig. Kaminker-Miller)
Let M and N be closed connected oriented manifolds and f : M −→ N an orientation
preserving homotopy equivalence. Let Γ = π1 (M) = π1 (N). Denote the signature
operators by ∂M and ∂N . Then
IndexΓ (∂eM ) = IndexΓ (∂eN ) ∈ Kj (Cr∗ Γ).
I
I
I
Using general theory of Fredholm complexes; co-chain complexes of Hilbert
C*-modules
Associate a generalized signature operator to a complex; and its analytic index in
K-theory
Homotopy equivalent Fredholm complexes are shown to have same signature, i.e.
same analytic index of their signature operators
27 / 30
Homotopy invariance of analytic signature
e −→ M denote the universal cover and Γ = π1 (M). Then the signature operator
Let M
e
∂ of M lifts to a Γ-equivariant elliptic operator ∂e on M.
Theorem (Kasparov. Mischchenko. Lusztig. Kaminker-Miller)
Let M and N be closed connected oriented manifolds and f : M −→ N an orientation
preserving homotopy equivalence. Let Γ = π1 (M) = π1 (N). Denote the signature
operators by ∂M and ∂N . Then
IndexΓ (∂eM ) = IndexΓ (∂eN ) ∈ Kj (Cr∗ Γ).
I
I
I
I
Using general theory of Fredholm complexes; co-chain complexes of Hilbert
C*-modules
Associate a generalized signature operator to a complex; and its analytic index in
K-theory
Homotopy equivalent Fredholm complexes are shown to have same signature, i.e.
same analytic index of their signature operators
e −→ M and lifted signature operator ∂e is the signature operator
Our scenario: M
of a Fredholm complex Ω∗ (M, ψ Γ ) where ψ Γ is a bundle with fibers Cr∗ Γ. The
27 / 30
Strong Novikov conjecture
By universal property of E Γ there is a Γ-map σ : E Γ −→ E Γ.
28 / 30
Strong Novikov conjecture
By universal property of E Γ there is a Γ-map σ : E Γ −→ E Γ.
Conjecture (Strong Novikov conjecture)
The composition
∼
=
σ
µ
Kj (BΓ) −−−−→ KjΓ (E Γ) −−−∗−→ KjΓ (E Γ) −−−−→ Kj (Cr∗ Γ)
is rationally injective.
28 / 30
Strong Novikov conjecture
By universal property of E Γ there is a Γ-map σ : E Γ −→ E Γ.
Conjecture (Strong Novikov conjecture)
The composition
∼
=
σ
µ
Kj (BΓ) −−−−→ KjΓ (E Γ) −−−∗−→ KjΓ (E Γ) −−−−→ Kj (Cr∗ Γ)
is rationally injective.
Comments
I
Kj (BΓ) ∼
= KjΓ (E Γ) follows since Γ acts freely on E Γ (use induction hom, or
descent hom)
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Strong Novikov conjecture
By universal property of E Γ there is a Γ-map σ : E Γ −→ E Γ.
Conjecture (Strong Novikov conjecture)
The composition
∼
=
σ
µ
Kj (BΓ) −−−−→ KjΓ (E Γ) −−−∗−→ KjΓ (E Γ) −−−−→ Kj (Cr∗ Γ)
is rationally injective.
Comments
I
I
Kj (BΓ) ∼
= KjΓ (E Γ) follows since Γ acts freely on E Γ (use induction hom, or
descent hom)
σ∗ is rational injective, follows from equivariant Chern character of Baum and
Connes to group homology
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Strong Novikov conjecture
By universal property of E Γ there is a Γ-map σ : E Γ −→ E Γ.
Conjecture (Strong Novikov conjecture)
The composition
∼
=
σ
µ
Kj (BΓ) −−−−→ KjΓ (E Γ) −−−∗−→ KjΓ (E Γ) −−−−→ Kj (Cr∗ Γ)
is rationally injective.
Comments
I
I
Kj (BΓ) ∼
= KjΓ (E Γ) follows since Γ acts freely on E Γ (use induction hom, or
descent hom)
σ∗ is rational injective, follows from equivariant Chern character of Baum and
Connes to group homology
Thus: if µ is rationally injective (e.g. if Baum-Connes conj holds for Γ), then Strong
Novikov conj holds for Γ
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Baum-Connes ⇒ Novikov
Let Γ be a discrete group satisfying the Baum-Connes conjecture, i.e.
µ : KjΓ (E Γ) −→ Kj (Cr∗ Γ)
is an isomorphism.
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Baum-Connes ⇒ Novikov
Let Γ be a discrete group satisfying the Baum-Connes conjecture, i.e.
µ : KjΓ (E Γ) −→ Kj (Cr∗ Γ)
is an isomorphism.
Then Γ satisfies the Novikov conjecture, i.e. for any closed, connected, oriented,
smooth manifold M with π1 (M) = Γ, then
τ∗ (LM ∩ [M]) ∈ H∗ (BΓ; Q)
is an oriented homotopy invariant.
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Baum-Connes ⇒ Novikov
Let Γ be a discrete group satisfying the Baum-Connes conjecture, i.e.
µ : KjΓ (E Γ) −→ Kj (Cr∗ Γ)
is an isomorphism.
Then Γ satisfies the Novikov conjecture, i.e. for any closed, connected, oriented,
smooth manifold M with π1 (M) = Γ, then
τ∗ (LM ∩ [M]) ∈ H∗ (BΓ; Q)
is an oriented homotopy invariant.
Approach:
Baum-Connes:
⇒ Strong Novikov:
⇒ Novikov:
µ : KjΓ (E Γ) −→ Kj (Cr∗ Γ) is isomorphism
µσ∗ : Kj (BΓ) −→ Kj (Cr∗ Γ) is rationally injective
τ∗ (LM ∩ [M]) ∈ H∗ (BΓ; Q) is an oriented homotopy invariant
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Sketch of Baum-Connes ⇒ Novikov
LM ∩ [M]
[∂]
e
IndexΓ (∂)
ch
H∗ (M; Q) ←−−∗−− K∗ (M) −−−−→ K∗ (Cr∗ Γ)




τ
τ
y
y∗
y∗
H∗ (BΓ; Q) ←−−−− K∗ (BΓ) −−−−→ K∗ (Cr∗ Γ)
µσ∗
ch∗
τ∗ (LM ∩ [M])
I
τ∗ [∂]
e
IndexΓ (∂)
Start by considering [∂] ∈ K∗ (M)
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Sketch of Baum-Connes ⇒ Novikov
LM ∩ [M]
[∂]
e
IndexΓ (∂)
ch
H∗ (M; Q) ←−−∗−− K∗ (M) −−−−→ K∗ (Cr∗ Γ)




τ
τ
y
y∗
y∗
H∗ (BΓ; Q) ←−−−− K∗ (BΓ) −−−−→ K∗ (Cr∗ Γ)
µσ∗
ch∗
τ∗ (LM ∩ [M])
I
I
τ∗ [∂]
e
IndexΓ (∂)
Start by considering [∂] ∈ K∗ (M)
e ∈ K∗ (C ∗ Γ) in bottom right is oriented homotopy invariant by previous
IndexΓ (∂)
r
Thm
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Sketch of Baum-Connes ⇒ Novikov
LM ∩ [M]
[∂]
e
IndexΓ (∂)
ch
H∗ (M; Q) ←−−∗−− K∗ (M) −−−−→ K∗ (Cr∗ Γ)




τ
τ
y
y∗
y∗
H∗ (BΓ; Q) ←−−−− K∗ (BΓ) −−−−→ K∗ (Cr∗ Γ)
µσ∗
ch∗
τ∗ (LM ∩ [M])
I
I
I
τ∗ [∂]
e
IndexΓ (∂)
Start by considering [∂] ∈ K∗ (M)
e ∈ K∗ (C ∗ Γ) in bottom right is oriented homotopy invariant by previous
IndexΓ (∂)
r
Thm
As µσ∗ is rationally injective (SNC), the element τ∗ [∂] ∈ K∗ (BΓ) is an oriented
homotopy invariant modulo torsion
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Sketch of Baum-Connes ⇒ Novikov
LM ∩ [M]
[∂]
e
IndexΓ (∂)
ch
H∗ (M; Q) ←−−∗−− K∗ (M) −−−−→ K∗ (Cr∗ Γ)




τ
τ
y
y∗
y∗
H∗ (BΓ; Q) ←−−−− K∗ (BΓ) −−−−→ K∗ (Cr∗ Γ)
µσ∗
ch∗
τ∗ (LM ∩ [M])
I
I
τ∗ [∂]
e
IndexΓ (∂)
Start by considering [∂] ∈ K∗ (M)
e ∈ K∗ (C ∗ Γ) in bottom right is oriented homotopy invariant by previous
IndexΓ (∂)
r
Thm
I
As µσ∗ is rationally injective (SNC), the element τ∗ [∂] ∈ K∗ (BΓ) is an oriented
homotopy invariant modulo torsion
I
Then ch∗ (τ∗ [∂]) = τ∗ (LM ∩ [M]) ∈ H∗ (BΓ; Q) in bottom left must be an oriented
homotopy invariant
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