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The Borel and Cartan cohomology of a point

Example

Assume AC and let $T = R / Z$ act trivially on the one-point smooth manifold ${*}$ . Use the period-one tangent generator and its selected degree-two polynomial generator $u$ . In the chosen Borel model, $H_{T}^{*} ({*}; R) ≅ R [u]$ , with $∣ u ∣ = 2$ . The Cartan complex is $R [u]$ with zero differential, so its cohomology is also $R [u]$ under the selected coefficient normalization.

Example references

[Borel and Cartan definitions and cohomology of a point] Borel and Cartan equivariant cohomology for a compact torus

[Torus classifying-space coefficient ring] Real cohomology of the chosen torus classifying space

[Smooth differential forms] A smooth differential $k$ -form

Facts & Assumptions

Given: AC, the rank-one torus $T = R / Z$ , its trivial action on ${*}$ , and the degree-two normalization fixed in the A-page definition.

[A1]

Assume the Axiom of Choice, as required by the selected coefficient-ring and Cartan definitions. The Axiom of Choice

[F1]

The Borel construction and Cartan complex use the selected product Milnor model; the Cartan polynomial generators have degree two. Borel and Cartan equivariant cohomology for a compact torus

[F2]

The selected coefficient identification sends the period-one Lie-algebra dual generator to $u$ and gives $H^{*} (B T; R) ≅ R [u]$ . Real cohomology of the chosen torus classifying space

[F3]

Smooth differential $k$ -forms are sections of the exterior powers of the cotangent bundle. A smooth differential $k$ -form

Proof

1.1givenA1F1F2

Since the action on the point is trivial, $E T \times_{T} {*} = E T / T = B T$ by the Borel definition [F1]. Its singular cohomology ring is $R [u]$ by [F2].

1.2F1F3given

The tangent and cotangent spaces of a point are zero. Every real-valued function on the singleton is constant, so [F3] gives $Ω^{0} ({*}) = R$ ; for $k > 0$ , the $k$ th exterior power of its zero cotangent space is zero, hence $Ω^{k} ({*}) = 0$ . Every fundamental field is zero and the exterior derivative of a constant function is zero. Thus the Cartan complex is $R [u]$ with zero differential, whose cohomology ring is $R [u]$ .

2.1F2step 1.1step 1.2∎

Both presentations use the degree-two generator fixed in [F2], so the displayed rings agree under that normalization. The example asserts no comparison for other $T$ -manifolds.

The Borel and Cartan cohomology of a point

Example

Example references

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources