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Circle rotation on the complex projective line

Example

Assume AC. Let $T = R / Z$ act on $M = {C P}^{1}$ by $t \cdot [z_{0} : z_{1}] = [z_{0} : e^{2 π i t} z_{1}]$ . For $k \in {Q, R}$ , use the product Milnor Borel model and let $u = c_{1} (L_{χ}) \in H^{2} (B T; k)$ for $χ (t) = e^{2 π i t}$ . The fixed set is $F = {p_{+}, p_{-}}$ , where $p_{+} = [1 : 0]$ and $p_{-} = [0 : 1]$ ; the tangent weights are $+ 1$ at $p_{+}$ and $- 1$ at $p_{-}$ . If $S_{T} (k)$ is generated by nontrivial character classes and $A_{T} (k) = H^{*} (B T; k)$ , then restriction identifies

S_{T} (k)^{- 1} H_{T}^{*} ({C P}^{1}; k) ≅ S_{T} (k)^{- 1} A_{T} (k) \oplus S_{T} (k)^{- 1} A_{T} (k),

with the two summands given by evaluation at $p_{+}$ and $p_{-}$ .

Example references

[Compact-torus Borel localization theorem] Borel localization for compact torus actions

[Character classes and localization conventions] Character localization in Borel equivariant cohomology

[Finite-dimensional fixed loci and normal weights] Fixed loci and normal weights of compact torus actions

[Fixed-point restriction and normal weights] Atiyah–Bott, Section 3, printed pp. 7–9

Facts & Assumptions

Given: AC, the circle action on ${C P}^{1}$ , the coefficient field $k$ , and the product Milnor Borel model.

[A1]

Assume AC, as required by the character-localization and Borel localization suppliers. The Axiom of Choice

[F1]

The character localization definition gives $A_{T} (k) = k [u]$ , with $u$ the degree-two class of the weight-one character, and $- u$ is the class of the inverse character. Both coefficient cases are supplied under [A1]. Character localization in Borel equivariant cohomology

[F2]

The fixed-point and normal-weight conventions are those of the compact torus fixed-locus lemma. Fixed loci and normal weights of compact torus actions

[F3]

Under AC, for a smooth action of a product torus on a compact finite-dimensional Hausdorff second-countable smooth manifold without boundary, restriction to the fixed set becomes an isomorphism after inverting the nontrivial character classes. Borel localization for compact torus actions

Verification

Given: AC and the action and conventions in the Example.

1.1F1F2algebra

A point $[z_{0} : z_{1}]$ is fixed by all $t \in T$ only if one homogeneous coordinate is zero: if both were nonzero, equality $[z_{0} : e^{2 π i t} z_{1}] = [z_{0} : z_{1}]$ would force $e^{2 π i t} = 1$ for every $t$ . Thus the fixed points are $p_{+}$ and $p_{-}$ . In the local coordinate $w = z_{1} / z_{0}$ at $p_{+}$ , the action is $w \mapsto e^{2 π i t} w$ . In $v = z_{0} / z_{1}$ at $p_{-}$ , it is $v \mapsto e^{- 2 π i t} v$ . These give the stated weights and character classes $u$ and $- u$ , both inverted by $S_{T} (k)$ .

2.1A1F1F3step 1.1algebra∎

The standard projective line is a compact real two-dimensional Hausdorff second-countable smooth manifold without boundary, and the displayed chart actions are jointly smooth. Thus the hypotheses of [F3] hold under [A1]. Since $F$ is a pair of fixed points, $E T \times_{T} F = (B T \times {p_{+}}) ⨿ (B T \times {p_{-}})$ , so $H_{T}^{*} (F; k) = A_{T} (k) \oplus A_{T} (k)$ . Localizing this finite sum gives the displayed target, and [F3] identifies the restriction map with the asserted isomorphism.

Depends on

Dependency tree · 2 levels

Sources

M. F. Atiyah and R. Bott, The Moment Map and Equivariant Cohomology