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Rotation on one factor of a product with positive-dimensional fixed components

Example

Let $T = R / Z$ act on $M = {C P}^{1} \times {C P}^{1}$ by $t \cdot ([z_{0} : z_{1}], q) = ([z_{0} : e^{2 π i t} z_{1}], q)$ , and let $k \in {Q, R}$ . The fixed set is the disjoint union $F = ({p_{+}} \times {C P}^{1}) ⨿ ({p_{-}} \times {C P}^{1})$ , with $p_{+} = [1 : 0]$ and $p_{-} = [0 : 1]$ ; the normal weights are $+ 1$ and $- 1$ , respectively. Let $u \in H^{2} (B T; k)$ be the weight-one character class and $b \in H^{2} ({C P}^{1}; k)$ the generator with $b^{2} = 0$ . Then localized restriction identifies

S_{T} (k)^{- 1} H_{T}^{*} (M; k) ≅ (S_{T} (k)^{- 1} A_{T} (k) \otimes_{k} k [b] / (b^{2})) \oplus (S_{T} (k)^{- 1} A_{T} (k) \otimes_{k} k [b] / (b^{2})),

where the two summands are the Borel cohomology of the two positive-dimensional fixed components.

Example references

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

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

[Integral projective-space cohomology] Cohomology ring of infinite complex projective space

[Fixed-component restriction] Atiyah–Bott, Section 3, printed pp. 8–9

Facts & Assumptions

Given: The product action, the product Milnor model, and $k \in {Q, R}$ .

[F1]

The weight-one class is a polynomial generator $u$ , and the inverse character has class $- u$ ; the set $S_{T} (k)$ inverts both. The real coefficient interface still lacks its current Step 1 readiness record. Character localization in Borel equivariant cohomology

[F2]

$H^{*} ({C P}^{1}; k) = k [b] / (b^{2})$ , with $∣ b ∣ = 2$ , by the integral projective-space ring and natural coefficient change. Cohomology ring of infinite complex projective space Topological universal coefficient short exact sequence for cohomology

[F3]

The external product is a graded-ring isomorphism when one factor has finite-free homology in each degree; $B T$ has that property over $k$ . Cohomological Kunneth cross product is a ring isomorphism Rational coefficient ring and characters of a product torus classifying space

[F4]

Restriction to the fixed set becomes an isomorphism after character localization. Borel localization for compact torus actions

Verification

Given: The action and conventions in the Example.

1.1F1algebra

If both coordinates $z_{0}, z_{1}$ of the first factor are nonzero, its projective line cannot be fixed by every circle element, so the fixed points of that factor are exactly $p_{+}$ and $p_{-}$ ; the second factor is fixed pointwise. In the charts $w = z_{1} / z_{0}$ and $v = z_{0} / z_{1}$ , the normal weights are $+ 1$ and $- 1$ , so the corresponding character classes are $u$ and $- u$ , both invertible in $S_{T} (k)^{- 1} A_{T} (k)$ .

2.1F1F2F3F4algebra∎

The Borel construction of $F$ is the disjoint union of two copies of $B T \times {C P}^{1}$ . By [F2] and the ring Künneth isomorphism [F3], each component has ring $A_{T} (k) [b] / (b^{2})$ . As an $A_{T} (k)$ -module this is $A_{T} (k) \oplus A_{T} (k) b$ ; localizing its two coefficients gives $S_{T} (k)^{- 1} A_{T} (k) \otimes_{k} k [b] / (b^{2})$ . Applying [F4] proves the claimed localized restriction formula.

Rotation on one factor of a product with positive-dimensional fixed components

Example

Example references

Facts & Assumptions

Verification

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