Differential cohomology in geometry and analysis U. Bunke Universität Regensburg 19.9.2008/ DMV-Tagung 2008 U. Bunke (Universität Regensburg) 19.9.2008/ DMV-Tagung 2008 1 / 20
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Differential cohomology in geometry and analysis
U. Bunke
Universität Regensburg
19.9.2008/ DMV-Tagung 2008
U. Bunke (Universität Regensburg) 19.9.2008/ DMV-Tagung 2008 1 / 20
Theorem
e =1
1− e−z −1z
.
this power series starts with
U. Bunke (Universität Regensburg) 19.9.2008/ DMV-Tagung 2008 2 / 20
Theorem
e =1
1− e−z −1z
.
this power series starts with
12
+1
12z − 1
720z3 +
130240
z5 − 11209600
z7 +1
47900160z9 − . . . .
U. Bunke (Universität Regensburg) 19.9.2008/ DMV-Tagung 2008 2 / 20
Theorem
e =1
1− e−z −1z
.
this power series starts with
12
+1
12z − 1
720z3 +
130240
z5 − 11209600
z7 +1
47900160z9 − . . . .
U. Bunke (Universität Regensburg) 19.9.2008/ DMV-Tagung 2008 2 / 20
Theorem
e =1
1− e−z −1z
.
this power series starts with
12
+1
12z − 1
720z3 +
130240
z5 − 11209600
z7 +1
47900160z9 − . . . .
U. Bunke (Universität Regensburg) 19.9.2008/ DMV-Tagung 2008 2 / 20
1 Differentiable cohomology
2 e-invariant
3 Application of differentiable K-theory
U. Bunke (Universität Regensburg) 19.9.2008/ DMV-Tagung 2008 3 / 20
Notation
X , Y , . . . - smooth compact manifolds
h - generalized cohomology theory, h∗ := h(∗)Ω(X , h∗) := Ω(X )⊗Z h∗ smooth differential forms with coefficientsin h∗Ωd=0(X , h∗) ⊆ Ω(X , h∗) - closed forms
HdR(X , h∗) - cohomology of (Ω(X , h∗), d)
U. Bunke (Universität Regensburg) Differentiable cohomology 19.9.2008/ DMV-Tagung 2008 4 / 20
Notation
X , Y , . . . - smooth compact manifoldspossibly with boundary.
h - generalized cohomology theory, h∗ := h(∗)Ω(X , h∗) := Ω(X )⊗Z h∗ smooth differential forms with coefficientsin h∗Ωd=0(X , h∗) ⊆ Ω(X , h∗) - closed forms
HdR(X , h∗) - cohomology of (Ω(X , h∗), d)
U. Bunke (Universität Regensburg) Differentiable cohomology 19.9.2008/ DMV-Tagung 2008 4 / 20
Notation
X , Y , . . . - smooth compact manifolds
h - generalized cohomology theory, h∗ := h(∗)Ω(X , h∗) := Ω(X )⊗Z h∗ smooth differential forms with coefficientsin h∗Ωd=0(X , h∗) ⊆ Ω(X , h∗) - closed forms
HdR(X , h∗) - cohomology of (Ω(X , h∗), d)
U. Bunke (Universität Regensburg) Differentiable cohomology 19.9.2008/ DMV-Tagung 2008 4 / 20
Notation
X , Y , . . . - smooth compact manifolds
h - generalized cohomology theory, h∗ := h(∗)Examples: HZ, HQ, MU, , K
Ω(X , h∗) := Ω(X )⊗Z h∗ smooth differential forms with coefficientsin h∗Ωd=0(X , h∗) ⊆ Ω(X , h∗) - closed forms
HdR(X , h∗) - cohomology of (Ω(X , h∗), d)
U. Bunke (Universität Regensburg) Differentiable cohomology 19.9.2008/ DMV-Tagung 2008 4 / 20
Notation
X , Y , . . . - smooth compact manifolds
h - generalized cohomology theory, h∗ := h(∗)Ω(X , h∗) := Ω(X )⊗Z h∗ smooth differential forms with coefficientsin h∗Ωd=0(X , h∗) ⊆ Ω(X , h∗) - closed forms
HdR(X , h∗) - cohomology of (Ω(X , h∗), d)
U. Bunke (Universität Regensburg) Differentiable cohomology 19.9.2008/ DMV-Tagung 2008 4 / 20
Notation
X , Y , . . . - smooth compact manifolds
h - generalized cohomology theory, h∗ := h(∗)Ω(X , h∗) := Ω(X )⊗Z h∗ smooth differential forms with coefficientsin h∗Ωd=0(X , h∗) ⊆ Ω(X , h∗) - closed forms
HdR(X , h∗) - cohomology of (Ω(X , h∗), d)
U. Bunke (Universität Regensburg) Differentiable cohomology 19.9.2008/ DMV-Tagung 2008 4 / 20
Notation
X , Y , . . . - smooth compact manifolds
h - generalized cohomology theory, h∗ := h(∗)Ω(X , h∗) := Ω(X )⊗Z h∗ smooth differential forms with coefficientsin h∗Ωd=0(X , h∗) ⊆ Ω(X , h∗) - closed forms
HdR(X , h∗) - cohomology of (Ω(X , h∗), d)
U. Bunke (Universität Regensburg) Differentiable cohomology 19.9.2008/ DMV-Tagung 2008 4 / 20
Definition
A smooth extension of h is a contravariant functor h from compactsmooth manifolds to graded abelian groups which fits into the diagram
U. Bunke (Universität Regensburg) Differentiable cohomology 19.9.2008/ DMV-Tagung 2008 5 / 20
Definition
A smooth extension of h is a contravariant functor h from compactsmooth manifolds to graded abelian groups which fits into the diagram
h(X )
U. Bunke (Universität Regensburg) Differentiable cohomology 19.9.2008/ DMV-Tagung 2008 5 / 20
Definition
A smooth extension of h is a contravariant functor h from compactsmooth manifolds to graded abelian groups which fits into the diagram
Ωd=0(X , h∗)
h(X )
R99rrrrrrrrrr
I %% %%LLLLLLLLLL
h(X )
U. Bunke (Universität Regensburg) Differentiable cohomology 19.9.2008/ DMV-Tagung 2008 5 / 20
Definition
A smooth extension of h is a contravariant functor h from compactsmooth manifolds to graded abelian groups which fits into the diagram
Ωd=0(X , h∗)
h(X )
R99rrrrrrrrrr
I %% %%LLLLLLLLLLHdR(X , h∗)
h(X )
OO
U. Bunke (Universität Regensburg) Differentiable cohomology 19.9.2008/ DMV-Tagung 2008 5 / 20
Definition
A smooth extension of h is a contravariant functor h from compactsmooth manifolds to graded abelian groups which fits into the diagram
Ω(X , h∗)/im(d)
a&&NNNNNNNNNNN
d // Ωd=0(X , h∗)
h(X )
R99rrrrrrrrrr
I %% %%LLLLLLLLLLHdR(X , h∗)
h(X )
OO
U. Bunke (Universität Regensburg) Differentiable cohomology 19.9.2008/ DMV-Tagung 2008 5 / 20
Definition
A smooth extension of h is a contravariant functor h from compactsmooth manifolds to graded abelian groups which fits into the diagram
Ω(X , h∗)/h(X ) s
a&&MMMMMMMMMMM
d // Ωd=0(X , h∗)
h(X )
R99rrrrrrrrrr
I %% %%LLLLLLLLLLHdR(X , h∗)
h(X )
OO
U. Bunke (Universität Regensburg) Differentiable cohomology 19.9.2008/ DMV-Tagung 2008 5 / 20
Flat theory
Definition
We define hflat(X ) := ker(R : h(X ) → Ω(X , h∗)).
U. Bunke (Universität Regensburg) Differentiable cohomology 19.9.2008/ DMV-Tagung 2008 6 / 20
Flat theory
Definition
We define hflat(X ) := ker(R : h(X ) → Ω(X , h∗)).
Ω(X , h∗)/im(d)
a&&NNNNNNNNNNN
d // Ωd=0(X , h∗)
HdR(X , h∗)
OO
h(X )
R99rrrrrrrrrr
I %% %%KKKKKKKKKKHdR(X , h∗)
hflat(X )
+
88qqqqqqqqqqq
β// h(X )
OO
U. Bunke (Universität Regensburg) Differentiable cohomology 19.9.2008/ DMV-Tagung 2008 6 / 20
Why?
Chern-Simons invariants
Characteristic classes for flat vector bundles
Invariants of elements in stable homotopy groups
topological terms in σ-models
Configuration spaces of field theories with differential form fieldstrength
U. Bunke (Universität Regensburg) Differentiable cohomology 19.9.2008/ DMV-Tagung 2008 7 / 20
Why?
Differentiable cohomology provides a conceptual way to refinesecondary torsion invariants to R/Z-cohomology classes.
Chern-Simons invariants
Characteristic classes for flat vector bundles
Invariants of elements in stable homotopy groups
topological terms in σ-models
Configuration spaces of field theories with differential form fieldstrength
U. Bunke (Universität Regensburg) Differentiable cohomology 19.9.2008/ DMV-Tagung 2008 7 / 20
Why?
Differentiable cohomology provides a conceptual way to refinesecondary torsion invariants to R/Z-cohomology classes.
Chern-Simons invariants
Characteristic classes for flat vector bundles
Invariants of elements in stable homotopy groups
topological terms in σ-models
Configuration spaces of field theories with differential form fieldstrength
U. Bunke (Universität Regensburg) Differentiable cohomology 19.9.2008/ DMV-Tagung 2008 7 / 20
Why?
Differentiable cohomology provides a conceptual way to refinesecondary torsion invariants to R/Z-cohomology classes.
Chern-Simons invariants
Characteristic classes for flat vector bundles
Invariants of elements in stable homotopy groups
topological terms in σ-models
Configuration spaces of field theories with differential form fieldstrength
U. Bunke (Universität Regensburg) Differentiable cohomology 19.9.2008/ DMV-Tagung 2008 7 / 20
Why?
Differentiable cohomology provides a conceptual way to refinesecondary torsion invariants to R/Z-cohomology classes.
Chern-Simons invariants
Characteristic classes for flat vector bundles
Invariants of elements in stable homotopy groups
topological terms in σ-models
Configuration spaces of field theories with differential form fieldstrength
U. Bunke (Universität Regensburg) Differentiable cohomology 19.9.2008/ DMV-Tagung 2008 7 / 20
Why?
Differentiable cohomology provides a conceptual way to refinesecondary torsion invariants to R/Z-cohomology classes.
Chern-Simons invariants
Characteristic classes for flat vector bundles
Invariants of elements in stable homotopy groups
Other applications:
topological terms in σ-models
Configuration spaces of field theories with differential form fieldstrength
U. Bunke (Universität Regensburg) Differentiable cohomology 19.9.2008/ DMV-Tagung 2008 7 / 20
Why?
Differentiable cohomology provides a conceptual way to refinesecondary torsion invariants to R/Z-cohomology classes.
Chern-Simons invariants
Characteristic classes for flat vector bundles
Invariants of elements in stable homotopy groups
Other applications:
topological terms in σ-models
Configuration spaces of field theories with differential form fieldstrength
U. Bunke (Universität Regensburg) Differentiable cohomology 19.9.2008/ DMV-Tagung 2008 7 / 20
Why?
Differentiable cohomology provides a conceptual way to refinesecondary torsion invariants to R/Z-cohomology classes.
Chern-Simons invariants
Characteristic classes for flat vector bundles
Invariants of elements in stable homotopy groups
Other applications:
topological terms in σ-models
Configuration spaces of field theories with differential form fieldstrength
U. Bunke (Universität Regensburg) Differentiable cohomology 19.9.2008/ DMV-Tagung 2008 7 / 20
Natural questions
Existence of h?
Uniqueness of h?
What is hflat(X )?
Cup product?
Orientation and integration?
Riemann-Roch?
U. Bunke (Universität Regensburg) Differentiable cohomology 19.9.2008/ DMV-Tagung 2008 8 / 20
Natural questions
Existence of h?First example: HZ -Cheeger-Simons characters 1985K - Freed, Hopkins-Freed (geometric model, not complete) ∼ 2000,B. -Schick (analytic model) 2003S, MU and other bordism theories, B.-Schick (geometric model) ∼2004general h - Hopkins-Singer (homotopy theoretic model), 2005Landweber exact theories B.-Schick, based on MU, 2007
U. Bunke (Universität Regensburg) Differentiable cohomology 19.9.2008/ DMV-Tagung 2008 8 / 20
Natural questions
Existence of h?First example: HZ -Cheeger-Simons characters 1985K - Freed, Hopkins-Freed (geometric model, not complete) ∼ 2000,B. -Schick (analytic model) 2003S, MU and other bordism theories, B.-Schick (geometric model) ∼2004general h - Hopkins-Singer (homotopy theoretic model), 2005Landweber exact theories B.-Schick, based on MU, 2007
U. Bunke (Universität Regensburg) Differentiable cohomology 19.9.2008/ DMV-Tagung 2008 8 / 20
Natural questions
Existence of h?First example: HZ -Cheeger-Simons characters 1985K - Freed, Hopkins-Freed (geometric model, not complete) ∼ 2000,B. -Schick (analytic model) 2003S, MU and other bordism theories, B.-Schick (geometric model) ∼2004general h - Hopkins-Singer (homotopy theoretic model), 2005Landweber exact theories B.-Schick, based on MU, 2007
U. Bunke (Universität Regensburg) Differentiable cohomology 19.9.2008/ DMV-Tagung 2008 8 / 20
Natural questions
Existence of h?First example: HZ -Cheeger-Simons characters 1985K - Freed, Hopkins-Freed (geometric model, not complete) ∼ 2000,B. -Schick (analytic model) 2003S, MU and other bordism theories, B.-Schick (geometric model) ∼2004general h - Hopkins-Singer (homotopy theoretic model), 2005Landweber exact theories B.-Schick, based on MU, 2007
U. Bunke (Universität Regensburg) Differentiable cohomology 19.9.2008/ DMV-Tagung 2008 8 / 20
Natural questions
Existence of h?First example: HZ -Cheeger-Simons characters 1985K - Freed, Hopkins-Freed (geometric model, not complete) ∼ 2000,B. -Schick (analytic model) 2003S, MU and other bordism theories, B.-Schick (geometric model) ∼2004general h - Hopkins-Singer (homotopy theoretic model), 2005Landweber exact theories B.-Schick, based on MU, 2007
U. Bunke (Universität Regensburg) Differentiable cohomology 19.9.2008/ DMV-Tagung 2008 8 / 20
Natural questions
Existence of h?First example: HZ -Cheeger-Simons characters 1985K - Freed, Hopkins-Freed (geometric model, not complete) ∼ 2000,B. -Schick (analytic model) 2003S, MU and other bordism theories, B.-Schick (geometric model) ∼2004general h - Hopkins-Singer (homotopy theoretic model), 2005Landweber exact theories B.-Schick, based on MU, 2007
U. Bunke (Universität Regensburg) Differentiable cohomology 19.9.2008/ DMV-Tagung 2008 8 / 20
Natural questions
Existence of h?First example: HZ -Cheeger-Simons characters 1985K - Freed, Hopkins-Freed (geometric model, not complete) ∼ 2000,B. -Schick (analytic model) 2003S, MU and other bordism theories, B.-Schick (geometric model) ∼2004general h - Hopkins-Singer (homotopy theoretic model), 2005Landweber exact theories B.-Schick, based on MU, 2007
U. Bunke (Universität Regensburg) Differentiable cohomology 19.9.2008/ DMV-Tagung 2008 8 / 20
Natural questions
Existence of h?First example: HZ -Cheeger-Simons characters 1985K - Freed, Hopkins-Freed (geometric model, not complete) ∼ 2000,B. -Schick (analytic model) 2003S, MU and other bordism theories, B.-Schick (geometric model) ∼2004general h - Hopkins-Singer (homotopy theoretic model), 2005Landweber exact theories B.-Schick, based on MU, 2007
U. Bunke (Universität Regensburg) Differentiable cohomology 19.9.2008/ DMV-Tagung 2008 8 / 20
Natural questions
Existence of h?
Uniqueness of h?
What is hflat(X )?
Cup product?
Orientation and integration?
Riemann-Roch?
U. Bunke (Universität Regensburg) Differentiable cohomology 19.9.2008/ DMV-Tagung 2008 8 / 20
Natural questions
Uniqueness of h?
U. Bunke (Universität Regensburg) Differentiable cohomology 19.9.2008/ DMV-Tagung 2008 8 / 20
Natural questions
Uniqueness of h?ordinary cohomology theory (like HZ ) is unique- Wiethaup,Simons-Sullivan, 2006-7Differentiable extensions are not unique in general. K -theory is anexample.K is unique, if it is in addition compatible with suspension(Wiethaup).No clear picture in general.
U. Bunke (Universität Regensburg) Differentiable cohomology 19.9.2008/ DMV-Tagung 2008 8 / 20
Natural questions
Uniqueness of h?ordinary cohomology theory (like HZ ) is unique- Wiethaup,Simons-Sullivan, 2006-7Differentiable extensions are not unique in general. K -theory is anexample.K is unique, if it is in addition compatible with suspension(Wiethaup).No clear picture in general.
U. Bunke (Universität Regensburg) Differentiable cohomology 19.9.2008/ DMV-Tagung 2008 8 / 20
Natural questions
Uniqueness of h?ordinary cohomology theory (like HZ ) is unique- Wiethaup,Simons-Sullivan, 2006-7Differentiable extensions are not unique in general. K -theory is anexample.K is unique, if it is in addition compatible with suspension(Wiethaup).No clear picture in general.
U. Bunke (Universität Regensburg) Differentiable cohomology 19.9.2008/ DMV-Tagung 2008 8 / 20
Natural questions
Uniqueness of h?ordinary cohomology theory (like HZ ) is unique- Wiethaup,Simons-Sullivan, 2006-7Differentiable extensions are not unique in general. K -theory is anexample.K is unique, if it is in addition compatible with suspension(Wiethaup).No clear picture in general.
U. Bunke (Universität Regensburg) Differentiable cohomology 19.9.2008/ DMV-Tagung 2008 8 / 20
Natural questions
Uniqueness of h?ordinary cohomology theory (like HZ ) is unique- Wiethaup,Simons-Sullivan, 2006-7Differentiable extensions are not unique in general. K -theory is anexample.K is unique, if it is in addition compatible with suspension(Wiethaup).No clear picture in general.
U. Bunke (Universität Regensburg) Differentiable cohomology 19.9.2008/ DMV-Tagung 2008 8 / 20
Natural questions
Uniqueness of h?ordinary cohomology theory (like HZ ) is unique- Wiethaup,Simons-Sullivan, 2006-7Differentiable extensions are not unique in general. K -theory is anexample.K is unique, if it is in addition compatible with suspension(Wiethaup).No clear picture in general.
U. Bunke (Universität Regensburg) Differentiable cohomology 19.9.2008/ DMV-Tagung 2008 8 / 20
Natural questions
Uniqueness of h?ordinary cohomology theory (like HZ ) is unique- Wiethaup,Simons-Sullivan, 2006-7Differentiable extensions are not unique in general. K -theory is anexample.K is unique, if it is in addition compatible with suspension(Wiethaup).No clear picture in general.
U. Bunke (Universität Regensburg) Differentiable cohomology 19.9.2008/ DMV-Tagung 2008 8 / 20
Natural questions
Uniqueness of h?ordinary cohomology theory (like HZ ) is unique- Wiethaup,Simons-Sullivan, 2006-7Differentiable extensions are not unique in general. K -theory is anexample.K is unique, if it is in addition compatible with suspension(Wiethaup).No clear picture in general.
U. Bunke (Universität Regensburg) Differentiable cohomology 19.9.2008/ DMV-Tagung 2008 8 / 20
Natural questions
Existence of h?
Uniqueness of h?
What is hflat(X )?
Cup product?
Orientation and integration?
Riemann-Roch?
U. Bunke (Universität Regensburg) Differentiable cohomology 19.9.2008/ DMV-Tagung 2008 8 / 20
Natural questions
What is hflat(X )?
U. Bunke (Universität Regensburg) Differentiable cohomology 19.9.2008/ DMV-Tagung 2008 8 / 20
Natural questions
What is hflat(X )?Consequence of axioms:
X 7→ hflat(X ) := ker(R)
is homotopy invariant.natural candidate:This is clear by construction for the Hopkins-Singer example.
U. Bunke (Universität Regensburg) Differentiable cohomology 19.9.2008/ DMV-Tagung 2008 8 / 20
Natural questions
What is hflat(X )?Consequence of axioms:
X 7→ hflat(X ) := ker(R)
is homotopy invariant.natural candidate:This is clear by construction for the Hopkins-Singer example.
U. Bunke (Universität Regensburg) Differentiable cohomology 19.9.2008/ DMV-Tagung 2008 8 / 20
Natural questions
What is hflat(X )?Consequence of axioms:
X 7→ hflat(X ) := ker(R)
is homotopy invariant.natural candidate:fibre sequence of cohomology theories
→ h → hR → hR/Z →
This is clear by construction for the Hopkins-Singer example.
U. Bunke (Universität Regensburg) Differentiable cohomology 19.9.2008/ DMV-Tagung 2008 8 / 20
Natural questions
What is hflat(X )?Consequence of axioms:
X 7→ hflat(X ) := ker(R)
is homotopy invariant.natural candidate:fibre sequence of cohomology theories
→ h → hR → hR/Z →
In most examplesh−1
R/Z(X ) ∼= hflat(X )
so thatThis is clear by construction for the Hopkins-Singer example.
U. Bunke (Universität Regensburg) Differentiable cohomology 19.9.2008/ DMV-Tagung 2008 8 / 20
Natural questions
What is hflat(X )?Consequence of axioms:
X 7→ hflat(X ) := ker(R)
is homotopy invariant.natural candidate:fibre sequence of cohomology theories
→ h → hR → hR/Z →
In most examplesh−1
R/Z(X ) ∼= hflat(X )
so thatβ : hflat(X ) → h(X )
is Bockstein.This is clear by construction for the Hopkins-Singer example.
U. Bunke (Universität Regensburg) Differentiable cohomology 19.9.2008/ DMV-Tagung 2008 8 / 20
Natural questions
What is hflat(X )?Consequence of axioms:
X 7→ hflat(X ) := ker(R)
is homotopy invariant.natural candidate:fibre sequence of cohomology theories
→ h → hR → hR/Z →
In most examplesh−1
R/Z(X ) ∼= hflat(X )
so thatβ : hflat(X ) → h(X )
is Bockstein.This is clear by construction for the Hopkins-Singer example.
U. Bunke (Universität Regensburg) Differentiable cohomology 19.9.2008/ DMV-Tagung 2008 8 / 20
Natural questions
What is hflat(X )?Consequence of axioms:
X 7→ hflat(X ) := ker(R)
is homotopy invariant.natural candidate:fibre sequence of cohomology theories
→ h → hR → hR/Z →
In most examplesh−1
R/Z(X ) ∼= hflat(X )
so thatβ : hflat(X ) → h(X )
is Bockstein.This is clear by construction for the Hopkins-Singer example.
U. Bunke (Universität Regensburg) Differentiable cohomology 19.9.2008/ DMV-Tagung 2008 8 / 20
Natural questions
What is hflat(X )?Consequence of axioms:
X 7→ hflat(X ) := ker(R)
is homotopy invariant.natural candidate:fibre sequence of cohomology theories
→ h → hR → hR/Z →
In most examplesh−1
R/Z(X ) ∼= hflat(X )
so thatβ : hflat(X ) → h(X )
is Bockstein.This is clear by construction for the Hopkins-Singer example.
U. Bunke (Universität Regensburg) Differentiable cohomology 19.9.2008/ DMV-Tagung 2008 8 / 20
Natural questions
Existence of h?
Uniqueness of h?
What is hflat(X )?
Cup product?
Orientation and integration?
Riemann-Roch?
U. Bunke (Universität Regensburg) Differentiable cohomology 19.9.2008/ DMV-Tagung 2008 8 / 20
Natural questions
Cup product?Assume that h is multiplicative.Require hat I and R are homomorphisms of rings.Then we call h a multiplicative extension.HZ is multiplicative - Cheeger-SimonsK and bordism theories like S, MU have multiplicative extensions(B.-Schick)Extensions constructed using Landweber exactness from MU aremultiplicative.The general case is not clear.
U. Bunke (Universität Regensburg) Differentiable cohomology 19.9.2008/ DMV-Tagung 2008 8 / 20
Natural questions
Cup product?Assume that h is multiplicative.Require hat I and R are homomorphisms of rings.Then we call h a multiplicative extension.HZ is multiplicative - Cheeger-SimonsK and bordism theories like S, MU have multiplicative extensions(B.-Schick)Extensions constructed using Landweber exactness from MU aremultiplicative.The general case is not clear.
U. Bunke (Universität Regensburg) Differentiable cohomology 19.9.2008/ DMV-Tagung 2008 8 / 20
Natural questions
Cup product?Assume that h is multiplicative.Require hat I and R are homomorphisms of rings.Then we call h a multiplicative extension.HZ is multiplicative - Cheeger-SimonsK and bordism theories like S, MU have multiplicative extensions(B.-Schick)Extensions constructed using Landweber exactness from MU aremultiplicative.The general case is not clear.
U. Bunke (Universität Regensburg) Differentiable cohomology 19.9.2008/ DMV-Tagung 2008 8 / 20
Natural questions
Cup product?Assume that h is multiplicative.Require hat I and R are homomorphisms of rings.Then we call h a multiplicative extension.HZ is multiplicative - Cheeger-SimonsK and bordism theories like S, MU have multiplicative extensions(B.-Schick)Extensions constructed using Landweber exactness from MU aremultiplicative.The general case is not clear.
U. Bunke (Universität Regensburg) Differentiable cohomology 19.9.2008/ DMV-Tagung 2008 8 / 20
Natural questions
Cup product?Assume that h is multiplicative.Require hat I and R are homomorphisms of rings.Then we call h a multiplicative extension.HZ is multiplicative - Cheeger-SimonsK and bordism theories like S, MU have multiplicative extensions(B.-Schick)Extensions constructed using Landweber exactness from MU aremultiplicative.The general case is not clear.
U. Bunke (Universität Regensburg) Differentiable cohomology 19.9.2008/ DMV-Tagung 2008 8 / 20
Natural questions
Cup product?Assume that h is multiplicative.Require hat I and R are homomorphisms of rings.Then we call h a multiplicative extension.HZ is multiplicative - Cheeger-SimonsK and bordism theories like S, MU have multiplicative extensions(B.-Schick)Extensions constructed using Landweber exactness from MU aremultiplicative.The general case is not clear.
U. Bunke (Universität Regensburg) Differentiable cohomology 19.9.2008/ DMV-Tagung 2008 8 / 20
Natural questions
Cup product?Assume that h is multiplicative.Require hat I and R are homomorphisms of rings.Then we call h a multiplicative extension.HZ is multiplicative - Cheeger-SimonsK and bordism theories like S, MU have multiplicative extensions(B.-Schick)Extensions constructed using Landweber exactness from MU aremultiplicative.The general case is not clear.
U. Bunke (Universität Regensburg) Differentiable cohomology 19.9.2008/ DMV-Tagung 2008 8 / 20
Natural questions
Cup product?Assume that h is multiplicative.Require hat I and R are homomorphisms of rings.Then we call h a multiplicative extension.HZ is multiplicative - Cheeger-SimonsK and bordism theories like S, MU have multiplicative extensions(B.-Schick)Extensions constructed using Landweber exactness from MU aremultiplicative.The general case is not clear.
U. Bunke (Universität Regensburg) Differentiable cohomology 19.9.2008/ DMV-Tagung 2008 8 / 20
Natural questions
Cup product?Assume that h is multiplicative.Require hat I and R are homomorphisms of rings.Then we call h a multiplicative extension.HZ is multiplicative - Cheeger-SimonsK and bordism theories like S, MU have multiplicative extensions(B.-Schick)Extensions constructed using Landweber exactness from MU aremultiplicative.The general case is not clear.
U. Bunke (Universität Regensburg) Differentiable cohomology 19.9.2008/ DMV-Tagung 2008 8 / 20
Natural questions
Existence of h?
Uniqueness of h?
What is hflat(X )?
Cup product?
Orientation and integration?
Riemann-Roch?
U. Bunke (Universität Regensburg) Differentiable cohomology 19.9.2008/ DMV-Tagung 2008 8 / 20
Natural questions
Orientation and integration?f : X → Y proper submersion, h-multiplicative.
No additional structures needed for cHZ (Brylinski, Dupont-Ljungman,Gomi)The concept is developed for bordism theories S, MU (geometricconstruction).Landweber exact theories admit integration for MU-oriented maps(B.-Schick).Theory for K is developed and based on local index theory(B.-Schick).
U. Bunke (Universität Regensburg) Differentiable cohomology 19.9.2008/ DMV-Tagung 2008 8 / 20
Natural questions
Orientation and integration?f : X → Y proper submersion, h-multiplicative.A h-orientation of f is a choice h-Thom class of the stable normalbundle of f .
No additional structures needed for cHZ (Brylinski, Dupont-Ljungman,Gomi)The concept is developed for bordism theories S, MU (geometricconstruction).Landweber exact theories admit integration for MU-oriented maps(B.-Schick).Theory for K is developed and based on local index theory(B.-Schick).
U. Bunke (Universität Regensburg) Differentiable cohomology 19.9.2008/ DMV-Tagung 2008 8 / 20
Natural questions
Orientation and integration?f : X → Y proper submersion, h-multiplicative.A h-orientation of f is a choice h-Thom class of the stable normalbundle of f .In this case have integration f! : h(X ) → h(Y ).
No additional structures needed for cHZ (Brylinski, Dupont-Ljungman,Gomi)The concept is developed for bordism theories S, MU (geometricconstruction).Landweber exact theories admit integration for MU-oriented maps(B.-Schick).Theory for K is developed and based on local index theory(B.-Schick).
U. Bunke (Universität Regensburg) Differentiable cohomology 19.9.2008/ DMV-Tagung 2008 8 / 20
Natural questions
Orientation and integration?f : X → Y proper submersion, h-multiplicative.A h-orientation of f is a choice h-Thom class of the stable normalbundle of f .In this case have integration f! : h(X ) → h(Y ).Want a good concept of h-orientation and integrationf! : h(X ) → h(Y )
No additional structures needed for cHZ (Brylinski, Dupont-Ljungman,Gomi)The concept is developed for bordism theories S, MU (geometricconstruction).Landweber exact theories admit integration for MU-oriented maps(B.-Schick).Theory for K is developed and based on local index theory(B.-Schick).
U. Bunke (Universität Regensburg) Differentiable cohomology 19.9.2008/ DMV-Tagung 2008 8 / 20
Natural questions
Orientation and integration?f : X → Y proper submersion, h-multiplicative.A h-orientation of f is a choice h-Thom class of the stable normalbundle of f .In this case have integration f! : h(X ) → h(Y ).
Ω(X , h∗)/im(d)
RX/Y A(of )∧...
a // h(X )
f!
I //
R**
h(X )
f!
Ω(X , h∗)RX/Y A(of )∧...
Ω(Y , h∗)/im(d)
a// h(Y )
I//
R
44h(Y ) Ω(Y , h∗)
No additional structures needed for cHZ (Brylinski, Dupont-Ljungman,Gomi)The concept is developed for bordism theories S, MU (geometricconstruction).Landweber exact theories admit integration for MU-oriented maps(B.-Schick).Theory for K is developed and based on local index theory(B.-Schick).
U. Bunke (Universität Regensburg) Differentiable cohomology 19.9.2008/ DMV-Tagung 2008 8 / 20
Natural questions
Orientation and integration?f : X → Y proper submersion, h-multiplicative.A h-orientation of f is a choice h-Thom class of the stable normalbundle of f .In this case have integration f! : h(X ) → h(Y ).Want a good concept of h-orientation and integrationf! : h(X ) → h(Y )
No additional structures needed for cHZ (Brylinski, Dupont-Ljungman,Gomi)The concept is developed for bordism theories S, MU (geometricconstruction).Landweber exact theories admit integration for MU-oriented maps(B.-Schick).Theory for K is developed and based on local index theory(B.-Schick).
U. Bunke (Universität Regensburg) Differentiable cohomology 19.9.2008/ DMV-Tagung 2008 8 / 20
Natural questions
Orientation and integration?Want a good concept of h-orientation and integrationf! : h(X ) → h(Y )
No additional structures needed for cHZ (Brylinski, Dupont-Ljungman,Gomi)The concept is developed for bordism theories S, MU (geometricconstruction).Landweber exact theories admit integration for MU-oriented maps(B.-Schick).Theory for K is developed and based on local index theory(B.-Schick).
U. Bunke (Universität Regensburg) Differentiable cohomology 19.9.2008/ DMV-Tagung 2008 8 / 20
Natural questions
Orientation and integration?Want a good concept of h-orientation and integrationf! : h(X ) → h(Y )
No additional structures needed for cHZ (Brylinski, Dupont-Ljungman,Gomi)The concept is developed for bordism theories S, MU (geometricconstruction).Landweber exact theories admit integration for MU-oriented maps(B.-Schick).Theory for K is developed and based on local index theory(B.-Schick).
U. Bunke (Universität Regensburg) Differentiable cohomology 19.9.2008/ DMV-Tagung 2008 8 / 20
Natural questions
Orientation and integration?Want a good concept of h-orientation and integrationf! : h(X ) → h(Y )
No additional structures needed for cHZ (Brylinski, Dupont-Ljungman,Gomi)The concept is developed for bordism theories S, MU (geometricconstruction).Landweber exact theories admit integration for MU-oriented maps(B.-Schick).Theory for K is developed and based on local index theory(B.-Schick).
U. Bunke (Universität Regensburg) Differentiable cohomology 19.9.2008/ DMV-Tagung 2008 8 / 20
Natural questions
Existence of h?
Uniqueness of h?
What is hflat(X )?
Cup product?
Orientation and integration?
Riemann-Roch?
U. Bunke (Universität Regensburg) Differentiable cohomology 19.9.2008/ DMV-Tagung 2008 8 / 20
Natural questions
Riemann-Roch? Example: K.
U. Bunke (Universität Regensburg) Differentiable cohomology 19.9.2008/ DMV-Tagung 2008 8 / 20
Natural questions
Riemann-Roch? Example: K.f : X → Y proper submersion,
U. Bunke (Universität Regensburg) Differentiable cohomology 19.9.2008/ DMV-Tagung 2008 8 / 20
Natural questions
Riemann-Roch? Example: K.f : X → Y proper submersion,Let f be K-oriented.Riemann-Roch states that
K(X )
f!
ch // HQ(X )
f!(Td(T v f )∪... )
K(Y )
ch // HQ(Y )
commutes.
U. Bunke (Universität Regensburg) Differentiable cohomology 19.9.2008/ DMV-Tagung 2008 8 / 20
Natural questions
Riemann-Roch? Example: K.f : X → Y proper submersion,Let f be K-oriented.Differentiable Riemann-Roch states that
K(X )
f!
ch // HQ(X )
f!(cTd(T v f )∪... )
K(Y )
ch // HQ(Y )
commutes.
U. Bunke (Universität Regensburg) Differentiable cohomology 19.9.2008/ DMV-Tagung 2008 8 / 20
Natural questions
Riemann-Roch? Example: K.f : X → Y proper submersion,Let f be K-oriented.Differentiable Riemann-Roch states that
K(X )
f!
ch // HQ(X )
f!(cTd(T v f )∪... )
K(Y )
ch // HQ(Y )
commutes.
U. Bunke (Universität Regensburg) Differentiable cohomology 19.9.2008/ DMV-Tagung 2008 8 / 20
Natural questions
Riemann-Roch? Example: K.f : X → Y proper submersion,Let f be K-oriented.Differentiable Riemann-Roch states that
K(X )
f!
ch // HQ(X )
f!(cTd(T v f )∪... )
K(Y )
ch // HQ(Y )
commutes.
U. Bunke (Universität Regensburg) Differentiable cohomology 19.9.2008/ DMV-Tagung 2008 8 / 20
Natural questions
Riemann-Roch? Example: K.f : X → Y proper submersion,Let f be K-oriented.Differentiable Riemann-Roch states that
K(X )
f!
ch // HQ(X )
f!(cTd(T v f )∪... )
K(Y )
ch // HQ(Y )
commutes.
U. Bunke (Universität Regensburg) Differentiable cohomology 19.9.2008/ DMV-Tagung 2008 8 / 20
Natural questions
Riemann-Roch? Example: K.f : X → Y proper submersion,Let f be K-oriented.Differentiable Riemann-Roch states that
K(X )
f!
ch // HQ(X )
f!(cTd(T v f )∪... )
K(Y )
ch // HQ(Y )
commutes.
U. Bunke (Universität Regensburg) Differentiable cohomology 19.9.2008/ DMV-Tagung 2008 8 / 20
Natural questions
Riemann-Roch? Example: K.f : X → Y proper submersion,Let f be K-oriented.Differentiable Riemann-Roch states that
K(X )
f!
ch // HQ(X )
f!(cTd(T v f )∪... )
K(Y )
ch // HQ(Y )
commutes.
U. Bunke (Universität Regensburg) Differentiable cohomology 19.9.2008/ DMV-Tagung 2008 8 / 20
How?
There are various methods to construct smooth extensions:
Sheaf theory (Deligne cohomology) for HZDifferential characters HZ (Cheeger-Simons), K (Maghfoul, 2008).
Homotopy theory (Hopkins-Singer, 2005)
Cycles and relations (e.g. B. -Schick)
U. Bunke (Universität Regensburg) Differentiable cohomology 19.9.2008/ DMV-Tagung 2008 9 / 20
How?
There are various methods to construct smooth extensions:
Sheaf theory (Deligne cohomology) for HZDifferential characters HZ (Cheeger-Simons), K (Maghfoul, 2008).
Homotopy theory (Hopkins-Singer, 2005)
Cycles and relations (e.g. B. -Schick)
U. Bunke (Universität Regensburg) Differentiable cohomology 19.9.2008/ DMV-Tagung 2008 9 / 20
How?
There are various methods to construct smooth extensions:
Sheaf theory (Deligne cohomology) for HZDifferential characters HZ (Cheeger-Simons), K (Maghfoul, 2008).
Homotopy theory (Hopkins-Singer, 2005)
Cycles and relations (e.g. B. -Schick)
U. Bunke (Universität Regensburg) Differentiable cohomology 19.9.2008/ DMV-Tagung 2008 9 / 20
How?
There are various methods to construct smooth extensions:
Sheaf theory (Deligne cohomology) for HZDifferential characters HZ (Cheeger-Simons), K (Maghfoul, 2008).
Homotopy theory (Hopkins-Singer, 2005)
Cycles and relations (e.g. B. -Schick)
U. Bunke (Universität Regensburg) Differentiable cohomology 19.9.2008/ DMV-Tagung 2008 9 / 20
How?
There are various methods to construct smooth extensions:
Sheaf theory (Deligne cohomology) for HZDifferential characters HZ (Cheeger-Simons), K (Maghfoul, 2008).
Homotopy theory (Hopkins-Singer, 2005)
Cycles and relations (e.g. B. -Schick)
U. Bunke (Universität Regensburg) Differentiable cohomology 19.9.2008/ DMV-Tagung 2008 9 / 20
How?
There are various methods to construct smooth extensions:
Sheaf theory (Deligne cohomology) for HZDifferential characters HZ (Cheeger-Simons), K (Maghfoul, 2008).
Homotopy theory (Hopkins-Singer, 2005)
Cycles and relations (e.g. B. -Schick)
U. Bunke (Universität Regensburg) Differentiable cohomology 19.9.2008/ DMV-Tagung 2008 9 / 20
Stably framed manifolds
M manifold
TM stably framed
consider M ∼ M ′ if M and M ′ are framed bordant
[M] ∈ Ωfr - class of M in the group of framed bordism classes
Question: Is [M] trivial?
Pontrjagin-Thom: Ωfr ∼= πS. This group is complicated and onlypartially known.
U. Bunke (Universität Regensburg) e-invariant 19.9.2008/ DMV-Tagung 2008 10 / 20
Stably framed manifolds
M manifold
TM stably framed
consider M ∼ M ′ if M and M ′ are framed bordant
[M] ∈ Ωfr - class of M in the group of framed bordism classes
Question: Is [M] trivial?
Pontrjagin-Thom: Ωfr ∼= πS. This group is complicated and onlypartially known.
U. Bunke (Universität Regensburg) e-invariant 19.9.2008/ DMV-Tagung 2008 10 / 20
Stably framed manifolds
M manifold
TM stably framed
consider M ∼ M ′ if M and M ′ are framed bordant
[M] ∈ Ωfr - class of M in the group of framed bordism classes
Question: Is [M] trivial?
Pontrjagin-Thom: Ωfr ∼= πS. This group is complicated and onlypartially known.
U. Bunke (Universität Regensburg) e-invariant 19.9.2008/ DMV-Tagung 2008 10 / 20
Stably framed manifolds
M manifold
TM stably framedTM ⊕ Rk
M is trivialized for some large k .
consider M ∼ M ′ if M and M ′ are framed bordant
[M] ∈ Ωfr - class of M in the group of framed bordism classes
Question: Is [M] trivial?
Pontrjagin-Thom: Ωfr ∼= πS. This group is complicated and onlypartially known.
U. Bunke (Universität Regensburg) e-invariant 19.9.2008/ DMV-Tagung 2008 10 / 20
Stably framed manifolds
M manifold
TM stably framed
consider M ∼ M ′ if M and M ′ are framed bordant
[M] ∈ Ωfr - class of M in the group of framed bordism classes
Question: Is [M] trivial?
Pontrjagin-Thom: Ωfr ∼= πS. This group is complicated and onlypartially known.
U. Bunke (Universität Regensburg) e-invariant 19.9.2008/ DMV-Tagung 2008 10 / 20
Stably framed manifolds
M manifold
TM stably framed
consider M ∼ M ′ if M and M ′ are framed bordantThere exists a manifold W such that TW is stably framed and∂W ∼= M t −M ′
[M] ∈ Ωfr - class of M in the group of framed bordism classes
Question: Is [M] trivial?
Pontrjagin-Thom: Ωfr ∼= πS. This group is complicated and onlypartially known.
U. Bunke (Universität Regensburg) e-invariant 19.9.2008/ DMV-Tagung 2008 10 / 20
Stably framed manifolds
M manifold
TM stably framed
consider M ∼ M ′ if M and M ′ are framed bordant
[M] ∈ Ωfr - class of M in the group of framed bordism classes
Question: Is [M] trivial?
Pontrjagin-Thom: Ωfr ∼= πS. This group is complicated and onlypartially known.
U. Bunke (Universität Regensburg) e-invariant 19.9.2008/ DMV-Tagung 2008 10 / 20
Stably framed manifolds
M manifold
TM stably framed
consider M ∼ M ′ if M and M ′ are framed bordant
[M] ∈ Ωfr - class of M in the group of framed bordism classes
Question: Is [M] trivial?
Pontrjagin-Thom: Ωfr ∼= πS. This group is complicated and onlypartially known.
U. Bunke (Universität Regensburg) e-invariant 19.9.2008/ DMV-Tagung 2008 10 / 20
Stably framed manifolds
M manifold
TM stably framed
consider M ∼ M ′ if M and M ′ are framed bordant
[M] ∈ Ωfr - class of M in the group of framed bordism classes
Question: Is [M] trivial?
Pontrjagin-Thom: Ωfr ∼= πS. This group is complicated and onlypartially known.
U. Bunke (Universität Regensburg) e-invariant 19.9.2008/ DMV-Tagung 2008 10 / 20
Primary Invariants
Construct homomorphisms ε : Ωfr → A to known groups A and studyimage ε([M]) ∈ A.Ωfr are coefficients of a generalized homology theory represented byspectrum S.[M] corresponds to homotopy class f : Σdim(M)S → SIdea: map to simpler homology theories.Choices: HZ, K, MUuse unit ε : S → KProblem: The primary invariant vanishes for dim(M) > 0 since Ωfr
>0 istorsion (Serre) and K∗ is free.
∗ −2 −1 0 1 2 3 4 5 6 7Ωfr∗ 0 0 Z Z
2Z2
Z24 0 0 Z
2Z
240K∗ Z 0 Z 0 Z 0 Z 0 Z 0
U. Bunke (Universität Regensburg) e-invariant 19.9.2008/ DMV-Tagung 2008 11 / 20
Primary Invariants
Construct homomorphisms ε : Ωfr → A to known groups A and studyimage ε([M]) ∈ A.Ωfr are coefficients of a generalized homology theory represented byspectrum S.[M] corresponds to homotopy class f : Σdim(M)S → SIdea: map to simpler homology theories.Choices: HZ, K, MUuse unit ε : S → KProblem: The primary invariant vanishes for dim(M) > 0 since Ωfr
>0 istorsion (Serre) and K∗ is free.
∗ −2 −1 0 1 2 3 4 5 6 7Ωfr∗ 0 0 Z Z
2Z2
Z24 0 0 Z
2Z
240K∗ Z 0 Z 0 Z 0 Z 0 Z 0
U. Bunke (Universität Regensburg) e-invariant 19.9.2008/ DMV-Tagung 2008 11 / 20
Primary Invariants
Construct homomorphisms ε : Ωfr → A to known groups A and studyimage ε([M]) ∈ A.Ωfr are coefficients of a generalized homology theory represented byspectrum S.[M] corresponds to homotopy class f : Σdim(M)S → SIdea: map to simpler homology theories.Choices: HZ, K, MUuse unit ε : S → KProblem: The primary invariant vanishes for dim(M) > 0 since Ωfr
>0 istorsion (Serre) and K∗ is free.
∗ −2 −1 0 1 2 3 4 5 6 7Ωfr∗ 0 0 Z Z
2Z2
Z24 0 0 Z
2Z
240K∗ Z 0 Z 0 Z 0 Z 0 Z 0
U. Bunke (Universität Regensburg) e-invariant 19.9.2008/ DMV-Tagung 2008 11 / 20
Primary Invariants
Construct homomorphisms ε : Ωfr → A to known groups A and studyimage ε([M]) ∈ A.Ωfr are coefficients of a generalized homology theory represented byspectrum S.[M] corresponds to homotopy class f : Σdim(M)S → SIdea: map to simpler homology theories.Choices: HZ, K, MUuse unit ε : S → KProblem: The primary invariant vanishes for dim(M) > 0 since Ωfr
>0 istorsion (Serre) and K∗ is free.
∗ −2 −1 0 1 2 3 4 5 6 7Ωfr∗ 0 0 Z Z
2Z2
Z24 0 0 Z
2Z
240K∗ Z 0 Z 0 Z 0 Z 0 Z 0
U. Bunke (Universität Regensburg) e-invariant 19.9.2008/ DMV-Tagung 2008 11 / 20
Primary Invariants
Construct homomorphisms ε : Ωfr → A to known groups A and studyimage ε([M]) ∈ A.Ωfr are coefficients of a generalized homology theory represented byspectrum S.[M] corresponds to homotopy class f : Σdim(M)S → SIdea: map to simpler homology theories.Choices: HZ, K, MUuse unit ε : S → KProblem: The primary invariant vanishes for dim(M) > 0 since Ωfr
>0 istorsion (Serre) and K∗ is free.
∗ −2 −1 0 1 2 3 4 5 6 7Ωfr∗ 0 0 Z Z
2Z2
Z24 0 0 Z
2Z
240K∗ Z 0 Z 0 Z 0 Z 0 Z 0
U. Bunke (Universität Regensburg) e-invariant 19.9.2008/ DMV-Tagung 2008 11 / 20
Primary Invariants
Construct homomorphisms ε : Ωfr → A to known groups A and studyimage ε([M]) ∈ A.Ωfr are coefficients of a generalized homology theory represented byspectrum S.[M] corresponds to homotopy class f : Σdim(M)S → SIdea: map to simpler homology theories.Choices: HZ, K, MUuse unit ε : S → KProblem: The primary invariant vanishes for dim(M) > 0 since Ωfr
>0 istorsion (Serre) and K∗ is free.
∗ −2 −1 0 1 2 3 4 5 6 7Ωfr∗ 0 0 Z Z
2Z2
Z24 0 0 Z
2Z
240K∗ Z 0 Z 0 Z 0 Z 0 Z 0
U. Bunke (Universität Regensburg) e-invariant 19.9.2008/ DMV-Tagung 2008 11 / 20
Primary Invariants
Construct homomorphisms ε : Ωfr → A to known groups A and studyimage ε([M]) ∈ A.Ωfr are coefficients of a generalized homology theory represented byspectrum S.[M] corresponds to homotopy class f : Σdim(M)S → SIdea: map to simpler homology theories.Choices: HZ, K, MUuse unit ε : S → KProblem: The primary invariant vanishes for dim(M) > 0 since Ωfr
>0 istorsion (Serre) and K∗ is free.
∗ −2 −1 0 1 2 3 4 5 6 7Ωfr∗ 0 0 Z Z
2Z2
Z24 0 0 Z
2Z
240K∗ Z 0 Z 0 Z 0 Z 0 Z 0
U. Bunke (Universität Regensburg) e-invariant 19.9.2008/ DMV-Tagung 2008 11 / 20
Primary Invariants
Construct homomorphisms ε : Ωfr → A to known groups A and studyimage ε([M]) ∈ A.Ωfr are coefficients of a generalized homology theory represented byspectrum S.[M] corresponds to homotopy class f : Σdim(M)S → SIdea: map to simpler homology theories.Choices: HZ, K, MUuse unit ε : S → KProblem: The primary invariant vanishes for dim(M) > 0 since Ωfr
>0 istorsion (Serre) and K∗ is free.
∗ −2 −1 0 1 2 3 4 5 6 7Ωfr∗ 0 0 Z Z
2Z2
Z24 0 0 Z
2Z
240K∗ Z 0 Z 0 Z 0 Z 0 Z 0
U. Bunke (Universität Regensburg) e-invariant 19.9.2008/ DMV-Tagung 2008 11 / 20
Secondary invariants
Vanishing of primary invariant implies existence of lift in
Σ−1K
Σdim(M)S
f // S
ε
K
U. Bunke (Universität Regensburg) e-invariant 19.9.2008/ DMV-Tagung 2008 12 / 20
Secondary invariants
Vanishing of primary invariant implies existence of lift in
Σ−1K
Σdim(M)S
f //
0%%
S
ε
K
U. Bunke (Universität Regensburg) e-invariant 19.9.2008/ DMV-Tagung 2008 12 / 20
Secondary invariants
Vanishing of primary invariant implies existence of lift in
Σ−1K
Σdim(M)S
f //
f99
0%%
S
ε
K
U. Bunke (Universität Regensburg) e-invariant 19.9.2008/ DMV-Tagung 2008 12 / 20
Secondary invariants
Vanishing of primary invariant implies existence of lift in
Σ−1K
Σdim(M)S
f //
f99
0%%
S
ε
K
Problem: Lift f ∈ Kdim(M)+1 is not unique!
U. Bunke (Universität Regensburg) e-invariant 19.9.2008/ DMV-Tagung 2008 12 / 20
Secondary invariants
Vanishing of primary invariant implies existence of lift in
Σ−1K
Σdim(M)S
f //
f99
0%%
S
ε
K
Problem: Lift f ∈ Kdim(M)+1 is not unique!Problem: K∗ is as complicated as πS
∗
U. Bunke (Universität Regensburg) e-invariant 19.9.2008/ DMV-Tagung 2008 12 / 20
Secondary invariants
Vanishing of primary invariant implies existence of lift in
Σ−1K
Σdim(M)S
f //
f99
0%%
S
ε
K
Problem: Lift f ∈ Kdim(M)+1 is not unique!Problem: K∗ is as complicated as πS
∗
Need simpler targets!
U. Bunke (Universität Regensburg) e-invariant 19.9.2008/ DMV-Tagung 2008 12 / 20
R/Z-invariants
KR/Z,∗ is known
U. Bunke (Universität Regensburg) e-invariant 19.9.2008/ DMV-Tagung 2008 13 / 20
R/Z-invariants
KR/Z,∗ is known
KR/Z,ev∼= R/Z , KR/Z,odd
∼= 0
U. Bunke (Universität Regensburg) e-invariant 19.9.2008/ DMV-Tagung 2008 13 / 20
R/Z-invariants
KR/Z,∗ is known
KR/Z,ev∼= R/Z , KR/Z,odd
∼= 0
Use KR/Z as target!
U. Bunke (Universität Regensburg) e-invariant 19.9.2008/ DMV-Tagung 2008 13 / 20
R/Z-invariants
RationalizationS // SR
take homotopy cofibre
U. Bunke (Universität Regensburg) e-invariant 19.9.2008/ DMV-Tagung 2008 13 / 20
R/Z-invariants
Σ−1SR/Zδ // S // SR .
Relate with K -theory.
U. Bunke (Universität Regensburg) e-invariant 19.9.2008/ DMV-Tagung 2008 13 / 20
R/Z-invariants
Σ−1K
0
""Σ−1SR/Z
δ // S //
ε
SR
K
.
U. Bunke (Universität Regensburg) e-invariant 19.9.2008/ DMV-Tagung 2008 13 / 20
R/Z-invariants
Σ−1K
0
""
u
yyt tt
tt
Σ−1SR/Zδ // S //
ε
SR
K
.
observe existence and uniqueness of u!
U. Bunke (Universität Regensburg) e-invariant 19.9.2008/ DMV-Tagung 2008 13 / 20
R/Z-invariants
Σ−1K
0
""
u
yyss
ss
s
Σ−1SR/Z
εR/Z
δ // S
ε
// SR
εR
Σ−1KR/Z // K // KR
.
U. Bunke (Universität Regensburg) e-invariant 19.9.2008/ DMV-Tagung 2008 13 / 20
R/Z-invariants
Σdim(M)S
f
Σ−1K
0
$$
u
xxqq
q
Σ−1SR/Z
εR/Z
δ // S
ε
// SR
εR
Σ−1KR/Z // K // KR
.
U. Bunke (Universität Regensburg) e-invariant 19.9.2008/ DMV-Tagung 2008 13 / 20
R/Z-invariants
Σdim(M)S
f
e
Σ−1K
0
$$
u
xxqq
q
Σ−1SR/Z
εR/Z
δ // S
ε
// SR
εR
Σ−1KR/Z // K // KR
.
U. Bunke (Universität Regensburg) e-invariant 19.9.2008/ DMV-Tagung 2008 13 / 20
R/Z-invariants
Σdim(M)S
f
e
Σ−1K
0
$$
u
xxqq
q
Σ−1SR/Z
εR/Z
δ // S
ε
// SR
εR
Σ−1KR/Z // K // KR
.
Observe that e ∈ KR/Z,dim(M)+1 is well-defined.
U. Bunke (Universität Regensburg) e-invariant 19.9.2008/ DMV-Tagung 2008 13 / 20
A family version
Consider space B and stable cohomotopy class
f : ΣkB+ → S
U. Bunke (Universität Regensburg) e-invariant 19.9.2008/ DMV-Tagung 2008 14 / 20
A family version
Consider space B and stable cohomotopy class
f : ΣkB+ → S
Assume that primary invariant vanishes :
Σ−1K
ΣkB+
f //
0$$
S
ε
K
U. Bunke (Universität Regensburg) e-invariant 19.9.2008/ DMV-Tagung 2008 14 / 20
A family version
Consider space B and stable cohomotopy class
f : ΣkB+ → S
Have liftΣ−1K
ΣkB+
f //
f::
0$$
S
ε
K
U. Bunke (Universität Regensburg) e-invariant 19.9.2008/ DMV-Tagung 2008 14 / 20
A family version
Consider space B and stable cohomotopy class
f : ΣkB+ → S
Secondary invariant:
Σ−1K
εR/Zu// Σ−1KR/Z
ΣkB+
e
##
f //
f;;
0$$
S
ε
K
U. Bunke (Universität Regensburg) e-invariant 19.9.2008/ DMV-Tagung 2008 14 / 20
A family version
Consider space B and stable cohomotopy class
f : ΣkB+ → S
Secondary invariant:
Σ−1K
εR/Zu// Σ−1KR/Z
ΣkB+
e
##
f //
f;;
0$$
S
ε
K
e ∈K−k−1
R/Z (B)
S−k−1R (B)
is well-defined.
U. Bunke (Universität Regensburg) e-invariant 19.9.2008/ DMV-Tagung 2008 14 / 20
A family version
Consider space B and stable cohomotopy class
f : ΣkB+ → S
e ∈K−k−1
R/Z (B)
S−k−1R (B)
is well-defined.
Note that for k ≥ 0 we have
e ∈ K−k−1R/Z (B) .
U. Bunke (Universität Regensburg) e-invariant 19.9.2008/ DMV-Tagung 2008 14 / 20
Special Examples
Of particular interest is the following special case.
π : W → B - locally trivial fibre bundle
framing of vertical bundle T vπ := ker(dπ)
π : W → B with framing represents class
ΣkB+f→ S , k = dim(B)− dim(W )
more special case: π : W → B a G-principal bundle
U. Bunke (Universität Regensburg) e-invariant 19.9.2008/ DMV-Tagung 2008 15 / 20
Special Examples
Of particular interest is the following special case.
π : W → B - locally trivial fibre bundle
framing of vertical bundle T vπ := ker(dπ)
π : W → B with framing represents class
ΣkB+f→ S , k = dim(B)− dim(W )
more special case: π : W → B a G-principal bundle
U. Bunke (Universität Regensburg) e-invariant 19.9.2008/ DMV-Tagung 2008 15 / 20
Special Examples
Of particular interest is the following special case.
π : W → B - locally trivial fibre bundle
framing of vertical bundle T vπ := ker(dπ)
π : W → B with framing represents class
ΣkB+f→ S , k = dim(B)− dim(W )
more special case: π : W → B a G-principal bundle
U. Bunke (Universität Regensburg) e-invariant 19.9.2008/ DMV-Tagung 2008 15 / 20
Special Examples
Of particular interest is the following special case.
π : W → B - locally trivial fibre bundle
framing of vertical bundle T vπ := ker(dπ)
π : W → B with framing represents class
ΣkB+f→ S , k = dim(B)− dim(W )
more special case: π : W → B a G-principal bundle
U. Bunke (Universität Regensburg) e-invariant 19.9.2008/ DMV-Tagung 2008 15 / 20
Special Examples
Of particular interest is the following special case.
π : W → B - locally trivial fibre bundle
framing of vertical bundle T vπ := ker(dπ)
π : W → B with framing represents class
ΣkB+f→ S , k = dim(B)− dim(W )
more special case: π : W → B a G-principal bundlebasis of Lie(G) induces trivialization of T vπ by fundamental vectorfields
U. Bunke (Universität Regensburg) e-invariant 19.9.2008/ DMV-Tagung 2008 15 / 20
A secondary index theorem
π : W → B, T vπ framed, f ∈ [ΣkB+, S], e ∈ K−k−1R/Z (B)
Theorem (B.-Schick)
π has canonical K-orientation.
Defineπ!(1) ∈ K(B) .
Note that π!(1) is flat.Define
ean := π!(1) ∈ Kflat ,−k (B) ∼= K−k−1R/Z (B)
Theorem (Secondary index theorem)
ean = e
U. Bunke (Universität Regensburg) Application of differentiable K-theory 19.9.2008/ DMV-Tagung 2008 16 / 20
A secondary index theorem
π : W → B, T vπ framed, f ∈ [ΣkB+, S], e ∈ K−k−1R/Z (B)
Theorem (B.-Schick)
π has canonical K-orientation.
Defineπ!(1) ∈ K(B) .
Note that π!(1) is flat.Define
ean := π!(1) ∈ Kflat ,−k (B) ∼= K−k−1R/Z (B)
Theorem (Secondary index theorem)
ean = e
U. Bunke (Universität Regensburg) Application of differentiable K-theory 19.9.2008/ DMV-Tagung 2008 16 / 20
A secondary index theorem
π : W → B, T vπ framed, f ∈ [ΣkB+, S], e ∈ K−k−1R/Z (B)
Theorem (B.-Schick)
π has canonical K-orientation.
Defineπ!(1) ∈ K(B) .
Note that π!(1) is flat.Define
ean := π!(1) ∈ Kflat ,−k (B) ∼= K−k−1R/Z (B)
Theorem (Secondary index theorem)
ean = e
U. Bunke (Universität Regensburg) Application of differentiable K-theory 19.9.2008/ DMV-Tagung 2008 16 / 20
A secondary index theorem
π : W → B, T vπ framed, f ∈ [ΣkB+, S], e ∈ K−k−1R/Z (B)
Theorem (B.-Schick)
π has canonical K-orientation.
Defineπ!(1) ∈ K(B) .
Note that π!(1) is flat.Define
ean := π!(1) ∈ Kflat ,−k (B) ∼= K−k−1R/Z (B)
Theorem (Secondary index theorem)
ean = e
U. Bunke (Universität Regensburg) Application of differentiable K-theory 19.9.2008/ DMV-Tagung 2008 16 / 20
A secondary index theorem
π : W → B, T vπ framed, f ∈ [ΣkB+, S], e ∈ K−k−1R/Z (B)
Theorem (B.-Schick)
π has canonical K-orientation.
Defineπ!(1) ∈ K(B) .
Note that π!(1) is flat.Define
ean := π!(1) ∈ Kflat ,−k (B) ∼= K−k−1R/Z (B)
Theorem (Secondary index theorem)
ean = e
U. Bunke (Universität Regensburg) Application of differentiable K-theory 19.9.2008/ DMV-Tagung 2008 16 / 20
A secondary index theorem
π : W → B, T vπ framed, f ∈ [ΣkB+, S], e ∈ K−k−1R/Z (B)
Theorem (B.-Schick)
π has canonical K-orientation.
Defineπ!(1) ∈ K(B) .
Note that π!(1) is flat.Define
ean := π!(1) ∈ Kflat ,−k (B) ∼= K−k−1R/Z (B)
Theorem (Secondary index theorem)
ean = e
U. Bunke (Universität Regensburg) Application of differentiable K-theory 19.9.2008/ DMV-Tagung 2008 16 / 20
Bordism formula
Assume that q : V → B is a zero bordism of π : W → Bas K -oriented bundle.Can extend the smooth K -orientation of π over to q.
Theorem (bordism formula)
π!(1) = a(
∫V/B
Td)
This integral can often be evaluated!
U. Bunke (Universität Regensburg) Application of differentiable K-theory 19.9.2008/ DMV-Tagung 2008 17 / 20
Bordism formula
Assume that q : V → B is a zero bordism of π : W → Bas K -oriented bundle.Can extend the smooth K -orientation of π over to q.
Theorem (bordism formula)
π!(1) = a(
∫V/B
Td)
This integral can often be evaluated!
U. Bunke (Universität Regensburg) Application of differentiable K-theory 19.9.2008/ DMV-Tagung 2008 17 / 20
Bordism formula
Assume that q : V → B is a zero bordism of π : W → Bas K -oriented bundle.Can extend the smooth K -orientation of π over to q.
Theorem (bordism formula)
π!(1) = a(
∫V/B
Td)
This integral can often be evaluated!
U. Bunke (Universität Regensburg) Application of differentiable K-theory 19.9.2008/ DMV-Tagung 2008 17 / 20
Bordism formula
Assume that q : V → B is a zero bordism of π : W → Bas K -oriented bundle.Can extend the smooth K -orientation of π over to q.
Theorem (bordism formula)
π!(1) = a(
∫V/B
Td)
This integral can often be evaluated!
U. Bunke (Universität Regensburg) Application of differentiable K-theory 19.9.2008/ DMV-Tagung 2008 17 / 20
Principal bundles
π : W → B a G-principal bundleT ⊆ G maximal toruschoose U(1) ⊆ Tlet U(1) act on D2 in the standard way, ∂D2 ∼= U(1).q : V := W ×U(1) D2 → B is K -oriented zero bordism of π : W → B
TheoremIf rank(G) ≥ 2, then e = 0.
U. Bunke (Universität Regensburg) Application of differentiable K-theory 19.9.2008/ DMV-Tagung 2008 18 / 20
Principal bundles
π : W → B a G-principal bundleT ⊆ G maximal toruschoose U(1) ⊆ Tlet U(1) act on D2 in the standard way, ∂D2 ∼= U(1).q : V := W ×U(1) D2 → B is K -oriented zero bordism of π : W → B
TheoremIf rank(G) ≥ 2, then e = 0.
U. Bunke (Universität Regensburg) Application of differentiable K-theory 19.9.2008/ DMV-Tagung 2008 18 / 20
Principal bundles
π : W → B a G-principal bundleT ⊆ G maximal toruschoose U(1) ⊆ Tlet U(1) act on D2 in the standard way, ∂D2 ∼= U(1).q : V := W ×U(1) D2 → B is K -oriented zero bordism of π : W → B
TheoremIf rank(G) ≥ 2, then e = 0.
U. Bunke (Universität Regensburg) Application of differentiable K-theory 19.9.2008/ DMV-Tagung 2008 18 / 20
Principal bundles
π : W → B a G-principal bundleT ⊆ G maximal toruschoose U(1) ⊆ Tlet U(1) act on D2 in the standard way, ∂D2 ∼= U(1).q : V := W ×U(1) D2 → B is K -oriented zero bordism of π : W → B
TheoremIf rank(G) ≥ 2, then e = 0.
U. Bunke (Universität Regensburg) Application of differentiable K-theory 19.9.2008/ DMV-Tagung 2008 18 / 20
Principal bundles
π : W → B a G-principal bundleT ⊆ G maximal toruschoose U(1) ⊆ Tlet U(1) act on D2 in the standard way, ∂D2 ∼= U(1).q : V := W ×U(1) D2 → B is K -oriented zero bordism of π : W → B
TheoremIf rank(G) ≥ 2, then e = 0.
U. Bunke (Universität Regensburg) Application of differentiable K-theory 19.9.2008/ DMV-Tagung 2008 18 / 20
rank one case: U(1)
universal bundle W → BU(1)use Chern character
ch : KR → HRper
in order to identify
KR/Z(BU(1)) ∼=H(BU(1); R)
im(ch)
H(BU(1); R) ∼= R[[z]] , |z| = 2
Theorem ∫V/B
Td =1
1− e−z −1z
.
this power series starts with12
+1
12z − 1
720z3 +
130240
z5 − 11209600
z7 +1
47900160z9 − . . . .
U. Bunke (Universität Regensburg) Application of differentiable K-theory 19.9.2008/ DMV-Tagung 2008 19 / 20
rank one case: U(1)
universal bundle W → BU(1)use Chern character
ch : KR → HRper
in order to identify
KR/Z(BU(1)) ∼=H(BU(1); R)
im(ch)
H(BU(1); R) ∼= R[[z]] , |z| = 2
Theorem ∫V/B
Td =1
1− e−z −1z
.
this power series starts with12
+1
12z − 1
720z3 +
130240
z5 − 11209600
z7 +1
47900160z9 − . . . .
U. Bunke (Universität Regensburg) Application of differentiable K-theory 19.9.2008/ DMV-Tagung 2008 19 / 20
rank one case: U(1)
universal bundle W → BU(1)use Chern character
ch : KR → HRper
in order to identify
KR/Z(BU(1)) ∼=H(BU(1); R)
im(ch)
H(BU(1); R) ∼= R[[z]] , |z| = 2
Theorem ∫V/B
Td =1
1− e−z −1z
.
this power series starts with12
+1
12z − 1
720z3 +
130240
z5 − 11209600
z7 +1
47900160z9 − . . . .
U. Bunke (Universität Regensburg) Application of differentiable K-theory 19.9.2008/ DMV-Tagung 2008 19 / 20
rank one case: U(1)
universal bundle W → BU(1)use Chern character
ch : KR → HRper
in order to identify
KR/Z(BU(1)) ∼=H(BU(1); R)
im(ch)
H(BU(1); R) ∼= R[[z]] , |z| = 2
Theorem ∫V/B
Td =1
1− e−z −1z
.
this power series starts with12
+1
12z − 1
720z3 +
130240
z5 − 11209600
z7 +1
47900160z9 − . . . .
U. Bunke (Universität Regensburg) Application of differentiable K-theory 19.9.2008/ DMV-Tagung 2008 19 / 20
rank one case: U(1)
universal bundle W → BU(1)use Chern character
ch : KR → HRper
in order to identify
KR/Z(BU(1)) ∼=H(BU(1); R)
im(ch)
H(BU(1); R) ∼= R[[z]] , |z| = 2
Theorem ∫V/B
Td =1
1− e−z −1z
.
this power series starts with12
+1
12z − 1
720z3 +
130240
z5 − 11209600
z7 +1
47900160z9 − . . . .
U. Bunke (Universität Regensburg) Application of differentiable K-theory 19.9.2008/ DMV-Tagung 2008 19 / 20
Remarks
12
+1
12z − 1
720z3 +
130240
z5 − 11209600
z7 +1
47900160z9 − . . . .
first term 12 is Adams e invariant of S1
second term 112 shows that transfer in stable homotopy along the Hopf bundle
produces an element of order at least 12 in πS3∼= Z
24pairing e with higher dimensional primitive classes in K∗(BU(1)) reproducesresults by Miller and Knapp (late 70ties) on the transfer for U(1)-bundles
for higher rank, e.g. U(1)n must replace K by more complicated cohomologytheory, e.g. elliptic cohomologythis is work in progress with N. Naumann.
U. Bunke (Universität Regensburg) Application of differentiable K-theory 19.9.2008/ DMV-Tagung 2008 20 / 20
Remarks
12
+1
12z − 1
720z3 +
130240
z5 − 11209600
z7 +1
47900160z9 − . . . .
first term 12 is Adams e invariant of S1
second term 112 shows that transfer in stable homotopy along the Hopf bundle
produces an element of order at least 12 in πS3∼= Z
24pairing e with higher dimensional primitive classes in K∗(BU(1)) reproducesresults by Miller and Knapp (late 70ties) on the transfer for U(1)-bundles
for higher rank, e.g. U(1)n must replace K by more complicated cohomologytheory, e.g. elliptic cohomologythis is work in progress with N. Naumann.
U. Bunke (Universität Regensburg) Application of differentiable K-theory 19.9.2008/ DMV-Tagung 2008 20 / 20
Remarks
12
+1
12z − 1
720z3 +
130240
z5 − 11209600
z7 +1
47900160z9 − . . . .
first term 12 is Adams e invariant of S1
second term 112 shows that transfer in stable homotopy along the Hopf bundle
produces an element of order at least 12 in πS3∼= Z
24pairing e with higher dimensional primitive classes in K∗(BU(1)) reproducesresults by Miller and Knapp (late 70ties) on the transfer for U(1)-bundles
for higher rank, e.g. U(1)n must replace K by more complicated cohomologytheory, e.g. elliptic cohomologythis is work in progress with N. Naumann.
U. Bunke (Universität Regensburg) Application of differentiable K-theory 19.9.2008/ DMV-Tagung 2008 20 / 20
Remarks
12
+1
12z − 1
720z3 +
130240
z5 − 11209600
z7 +1
47900160z9 − . . . .
first term 12 is Adams e invariant of S1
second term 112 shows that transfer in stable homotopy along the Hopf bundle
produces an element of order at least 12 in πS3∼= Z
24pairing e with higher dimensional primitive classes in K∗(BU(1)) reproducesresults by Miller and Knapp (late 70ties) on the transfer for U(1)-bundles
for higher rank, e.g. U(1)n must replace K by more complicated cohomologytheory, e.g. elliptic cohomologythis is work in progress with N. Naumann.
U. Bunke (Universität Regensburg) Application of differentiable K-theory 19.9.2008/ DMV-Tagung 2008 20 / 20
Remarks
12
+1
12z − 1
720z3 +
130240
z5 − 11209600
z7 +1
47900160z9 − . . . .
first term 12 is Adams e invariant of S1
second term 112 shows that transfer in stable homotopy along the Hopf bundle
produces an element of order at least 12 in πS3∼= Z
24pairing e with higher dimensional primitive classes in K∗(BU(1)) reproducesresults by Miller and Knapp (late 70ties) on the transfer for U(1)-bundles
for higher rank, e.g. U(1)n must replace K by more complicated cohomologytheory, e.g. elliptic cohomologythis is work in progress with N. Naumann.
U. Bunke (Universität Regensburg) Application of differentiable K-theory 19.9.2008/ DMV-Tagung 2008 20 / 20
Remarks
12
+1
12z − 1
720z3 +
130240
z5 − 11209600
z7 +1
47900160z9 − . . . .
first term 12 is Adams e invariant of S1
second term 112 shows that transfer in stable homotopy along the Hopf bundle
produces an element of order at least 12 in πS3∼= Z
24pairing e with higher dimensional primitive classes in K∗(BU(1)) reproducesresults by Miller and Knapp (late 70ties) on the transfer for U(1)-bundles
for higher rank, e.g. U(1)n must replace K by more complicated cohomologytheory, e.g. elliptic cohomologythis is work in progress with N. Naumann.
U. Bunke (Universität Regensburg) Application of differentiable K-theory 19.9.2008/ DMV-Tagung 2008 20 / 20
Remarks
12
+1
12z − 1
720z3 +
130240
z5 − 11209600
z7 +1
47900160z9 − . . . .
first term 12 is Adams e invariant of S1
second term 112 shows that transfer in stable homotopy along the Hopf bundle
produces an element of order at least 12 in πS3∼= Z
24pairing e with higher dimensional primitive classes in K∗(BU(1)) reproducesresults by Miller and Knapp (late 70ties) on the transfer for U(1)-bundles
for higher rank, e.g. U(1)n must replace K by more complicated cohomologytheory, e.g. elliptic cohomologythis is work in progress with N. Naumann.
U. Bunke (Universität Regensburg) Application of differentiable K-theory 19.9.2008/ DMV-Tagung 2008 20 / 20
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