Smooth extensions of cohomology theories (Oxford,

May 2005)

U. Bunke and Th. Schick ^{∗}

May 8, 2005

Contents

1 Smooth integral cohomology 2 1.1 Cheeger-Simons characters . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2 Charactristic classes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.3 Generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2 The smooth extension of complex K-theory 4 2.1 The construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.2 The smooth Grothendieck-Riemann-Roch . . . . . . . . . . . . . . . . . . . 6 2.3 Secondary classes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.4 Chern classes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.5 Additional material . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

3 Other theories 9 3.1 Bordism theories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1 SMOOTH INTEGRAL COHOMOLOGY 2

∗Mathematisches Institut, Universität Göttingen, Bunsenstr. 3-5, 37073 Göttingen, GERMANY, bunke@uni-math.gwdg.de, schick@uni-math.gwdg.de

1 Smooth integral cohomology

1.1 Cheeger-Simons characters

1.1.1 Let H_{ℤ} denote integral cohomology. A model of its smooth extension Ĥ_{ℤ} is given by Cheeger-Simons characters. It is defined on the category of smooth manifolds. Definition 1.1 The group of Cheeger-Simons characters is defined by Ĥ_{ℤ}^{k}(M) := {ϕ : Hom(Z_{k−1}(M), ℝ/ℤ)|{∃R(ϕ) ∈ A_{d=0}^{k}(M)∀C ∈ C_{k}(M)|ϕ(∂C) = [∫_{C} R(ϕ)]}} . 1.1.2 Properties:

⋯ → H_{ℤ}^{k−1}(M) → A^{k−1}(M)/im(d) →^{a} Ĥ_{ℤ}^{k}(M) →^{v} H_{ℤ}^{k}(M) → …

is exact.

(1) Here a(α)(Z) := − ∫_{Z} α.

(2) R ∘ a = −d.

(3) v(ϕ) := [z] is given by dϕ̃ = z − R(ϕ), where ϕ̃ ∈ C^{k−1}(M, ℝ) extends ϕ and z ∈ Z^{k}(M, ℤ), R(ϕ) ∈ A^{k}(M)_{d=0} ⊂ C^{k}(M, ℝ).

0 → H_{ℝ/ℤ}^{k−1}(M) → Ĥ_{ℤ}^{k}(M) →^{R} A_{d=0}^{k}(M) → 0

is exact. Note that in general

H_{ℝ/ℤ}^{∗}(M) ≠ H_{ℝ}^{∗}(M)/H_{ℤ}^{∗}(M) .

1.2 Charactristic classes

1.2.1 A complex vector bundle E → M has natural Chern classes c_{2i}(E) ∈ H_{ℤ}^{2i}(M). Cheeger-Simons constructed natural lifts

ĉ_{2i}(E, ∇) ∈ Ĥ_{ℤ}^{2i}(M)

such that

R(ĉ_{t}(E)) = det(1 + \frac{t}{2πi}R^{∇}) .

If (E, ∇) is flat, then

ĉ_{2i}(E, ∇) ∈ H_{ℝ/ℤ}^{2i−1}(M) .

1.2.2 For a proper oriented submersion of dimension d

f : M → N

we have an integration

f_{∗} : Ĥ_{ℤ}^{k}(M) → Ĥ_{ℤ}^{k−d}(N) .

It is functorial and compatible with product and pull-back. (Dupont-Ljungman have a proof of the projection formula)

1.2.3 Ĥ_{ℤ}^{∗}(M) is a graded ring. R and v are multiplicative.

1.3 Generalizations

1.3.1 Can one define smooth extensions of other cohomology theories. What is their interpretations ?

1.3.2 Hopkins-Singer construct for any cohomology theory (and more general for any functor X with a transformation X → H_{R}^{∗} which is represented by a space X and a class z ∈ H_{R}^{∗}(X) ) a smooth version X̂.

1.3.3 For a cohomology theory it has similar properties as Ĥ_{ℤ}.

2 THE SMOOTH EXTENSION OF COMPLEX K-THEORY 4

1.3.4 They discuss the integration over the fibre. But there is no clear notion of a smooth orientation (equivalence relation on the orientation data).

There is no projection formula.

1.3.5 Bunke-Schick have a geometric construction of smooth extensions of bordism theories (integration and product is in fact much simpler than in integral cohomology) and an analytic construction of smooth complex K-theory.

2 The smooth extension of complex K-theory

2.1 The construction

2.1.1 Want most general class of cycles. This makes it easy to construct smooth classes. But it makes it complicated to verify well-definedness of operations.

2.1.2 Start with ch : K → H_{R}, where R := ℝ[[z]], deg(z) = −2. It is a multiplicative degree-preserving transformation of ℤ-graded cohomology theories.

A cycle for K̂(M) is a pair (E, α) of a geometric family E over M and a form α ∈ A_{R}(M). The data of a geometric family is:

(1) a proper submersion X → M

(2) a vertical orientation

(3) a vertical metric

(4) a connection

(5) a family of Dirac bundles along the fibres

A geometric family encodes all data needed to define a Bismut super connection and to do local index theory.

2.1.3 Form semi group of cycles (disjoint union of geometric families and sum of forms).

2.1.4 The equivalence relation: We set

(E, α) ∼ (E′, α′)

if there exists a taming (E ∪ −E′)_{t} such that

α − α′ = η((E ∪ −E′)_{t}) + im(d) .

A taming is a smoothing perturbation which makes the family of Dirac operators invertible.

Definition 2.1 K̂(M) is the semigroup of equivalence classes of cycles.

It is a group.

2.1.5 This constuction is similar in spirit to the constructions of Gillet-Soulet in the arithmetic case. Details are completely different, because we have much more cycles (G-S use finite-dimensional complexes of vector bundles, only). Furthermore, the notion of a K-orientation is not discussed in G-S.

2.1.6 Properties:

⋯ → K^{k−1}(M) → A_{R}^{k−1}(M)/im(d) →^{a} K̂^{k}(M) →^{v} K^{k}(M) → …

is exact. Here

(1) a(α) = [∅, α].

(2) v[E, α] = index(E), the family index of the associated Dirac operator

0 → K_{ℝ/ℤ}^{k−1}(M) → K̂^{k}(M) →^{R} A_{R,d=0}^{k}(M) → 0

is exact. Here (1) R([E, α]) := Ω(E) − dα,

(2) Ω(E) is the local index form.

(3) K_{ℝ/ℤ} is defined as the kernel of R. It is the ℝ/ℤ-K-theory introduced by Karoubi and Lott.

2.1.7 K̂ is a multiplicative.

2.1.8 Let f : M → N be a proper submersion which is K-oriented. There is a natural notion of a smooth K-orientation (there is a difference to the integral cohomology case, because A^{−1}(M) acts on the orientations non-trivially). For a smooth K-orientation we have f_{∗} : K̂^{k}(M) → K̂^{k−d}(N) with the usual properties. We can show the projection formula.

2.1.9 The restriction of f_{∗} to K_{ℝ/ℤ} is independent of the smooth data and coincides with the homotopy theoretic map. It was first constructed by Lott.

2.1.10 We have K̂^{0}(∗) ≅ ℤ and K̂^{−1}(∗) ≅ ℝ/ℤ.

2.1.11 We have

0 → ℝ/ℤ → K̂^{0}(S^{1}) → ℤ → 0

and

0 → C^{∞}(S^{1})/ℤ → K̂^{1}(S^{1}) → ℤ → 0 .

2.1.12 There is a trivial geometric family 1 based on M → M. If n is a point, then

f_{∗}([1, 0]) = η^{red}(M) .

The smooth K-orientation of M gives all data to define the Dirac operator of M.

2.2 The smooth Grothendieck-Riemann-Roch

2.2.1 Theorem 2.2 There is a multiplicative lift of the Chern character

ĉh : K̂(M) → Ĥ_{ℚ}(M) .

which is a rational isomorphism.

If E is given by a vector bundle with connection (E, ∇), then ĉh([E, 0]) = ĉh(E, ∇) (the Cheeger-Simons Chern character).

2.2.2 A smooth K-orientation provides a lift Â(f) ∈ Ĥ_{ℚ}^{0}(M) of the Â-class.

Theorem 2.3 The classical Grothendieck-Riemann-Roch theorem has the smooth refinement

\begin{matrix}K̂^{k}(M) & →^{ĉh} & Ĥ_{ℚ}^{k}(M) \\ f_{∗} ↓ & & f_{∗}(Â(f) ∪ …) \\ K̂^{k−d}(N) & →^{ĉh} & Ĥ_{ℚ}^{K−d}(N) \end{matrix} .

2.2.3 The smooth Grothendieck-Riemann-Roch theorem allows a rational calculation of η-forms.

2.2.4 Let us apply the theorem to N = ∗ and a family E given by a vector bundle (E, ∇).

η^{red}(D_{M} ⊗ E) ≡_{ℚ} ∫_{M}(Â(M) ∪ ĉh(E, ∇)) .

This was known to Cheeger-Simons.

2.3 Secondary classes

2.3.1 Let (E, ∇) be a flat unitary bundle of virtual dimension zero and E be corresponding family. Then we have

[E, 0] ∈ K_{ℝ/ℤ}^{0}(M) .

2.3.2 If N is a point, then f_{∗}([E, 0]) is the ρ-invariant.

2.4 Chern classes

2.4.1 It is not possible to define natural integral Chern classes ĉ_{i} : K̂(M) → Ĥ_{ℤ}^{i}(M) for all i.

In fact ĉ_{i} can be defined on the step of the Atiyah-Hirzebruch filtration F^{i}K̂(M) up to finitely many choices.

2.4.2 But For small i the integral Chern classes exist:

(1) ĉ_{1} : K̂^{1}(M) → Ĥ_{ℤ}^{1}(M) ≅ C^{∞}(M, U(1)) is the exponentiated η^{red}.

(2) ĉ_{2} : K̂^{0}(M) → Ĥ_{ℤ}^{2}(M) gives the isomorphism classes of the determinant line bundle with connection.

(3) ĉ_{3} : K̂^{1}(M) → Ĥ_{ℤ}^{3}(M) gives the isomorphism class of Lott’s geometric index gerbe.

2.4.3 An example: Let dim(f) = 2, x ∈ K̂^{0}(M), and assume that ĉ_{4}(x) is defined. Then the following gives a geometric calculation of the square of a determinant bundle (with connection and metric):

Theorem 2.4

2ĉ_{2}(f_{∗}(x)) = f_{∗}(ĉ_{4}(x))

2.5 Additional material

2.5.1 Equivariant version for finite groups exist.

2.5.2 Adams operations can be lifted

2.5.3 Mayer-Vietoris sequences:

⋯ → K_{ℝ/ℤ}^{∗−1}(U ∩ V) → K̂^{∗}(M) → K̂^{∗}(U) ⊕ K̂^{∗}(V) → K̂^{∗}(U ∩ V) → K(M) → ….

3 OTHER THEORIES 9 2.5.4 relative version 2.5.5 all known ℝ/ℤ-valued secondary invariants can be interpreted: Example: A family of ℤ/kℤ-manifolds is a geometric family W with boundary ∂W ≅ kE. We consider

x := [E, \frac{1}{k}Ω(W)] ∈ K̂(M) .

This class is flat. Assume that M is K-oriented and deg(x) + dim(M) ≡ 1(2). Proposition 2.5 f_{∗}(x) ∈ K̂_{ℝ/ℤ}(∗) ≅ ℝ/ℤ is the ℤ/kℤ-index of Freed-Melrose. 2.5.6 Let E be an even-dimensional geometric family over a point with an automorphism ϕ. We form the mapping torus T (E, ϕ) which is a geometric family over S^{1}. Let V := ker(D(E)) and ϕ^{±} the induced action of ϕ on V. Proposition 2.6 We have [T (E, ϕ), 0] ∈ K̂_{ℝ/ℤ}^{0}(S^{1}) ≅ ℝ/ℤ. In fact, this element is equal to [\frac{1}{2πi} log \frac{det(ϕ^{+})}{det(ϕ^{−})}]. Proof. Adiabatic limits.

3 Other theories

3.1 Bordism theories

We start with a genus, e.g. ϕ : MU → H_{ℝ}. Then we can define a smooth extension { \atop MU_{ϕ}}. 3.1.1 It has analogous properties as the theories already considered. 3.1.2 There is a natural notion of a smooth MU-orientation. { \atop MU_{ϕ}} is the universal smoothly MU-oriented theory.

3 OTHER THEORIES 10 3.1.3 If ϕ is given by Â, then there is a natural smooth K-orientation M̂U_{ϕ} → K̂. Theorem 3.1 (Smooth Connor Floyed) It induces an isomorphism

M̂U_{ϕ}(M) ⊗_{M̂U_{ϕ}(∗)} K̂(∗) ≅ K̂(M) .

Similar for Spin and Spin^{c} (smooth Hovey-Hopkins theorem)