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The fixed normal equivariant Euler class is a localized unit

Statement

Assume AC. Let $T = (R / Z)^{r}$ act smoothly on a compact finite-dimensional Hausdorff second-countable smooth manifold $M$ without boundary, and set $F = M^{T}$ . Give the normal bundle $ν_{F} = (T F)^{⊥}$ the componentwise $T$ -invariant complex orientation supplied by the fixed-locus normal-weight decomposition. Let $ν_{F, T} = E T \times_{T} ν_{F} \to B T \times F$ be its Borel vector bundle and let $e_{T} (ν_{F}) \in H_{T}^{*} (F; k) = H^{*} (B T \times F; k)$ , $k \in {Q, R}$ , be its zero-section Euler class. Then $e_{T} (ν_{F})$ is a unit in $S_{T} (k)^{- 1} H_{T}^{*} (F; k)$ . On each connected component $F_{0} \subseteq F$ , restriction to $B T \times {p}$ , $p \in F_{0}$ , is the product of the nontrivial character classes of the normal representation at $p$ , with multiplicity and the chosen complex orientation.

Statement references

[Normal-weight product for the equivariant Euler class] Atiyah–Bott, Section 3, equation (3.7), printed pp. 8–9

[Normal-bundle weight and Euler-class computation] Dwivedi et al., Section 9.9, Example 9.48, PDF pp. 85–86

Facts & Assumptions

Given: AC, the compact smooth torus action, its fixed set $F$ , and $k \in {Q, R}$ .

[A1]

Assume full AC, which supplies the Thom-class and partition-of-unity hypotheses below. The Axiom of Choice

[F1]

The fixed set $F$ is compact with finitely many components, and the normal representation splits into finitely many nontrivial character bundles with a $T$ -invariant complex orientation. Fixed loci and normal weights of compact torus actions

[F2]

The product Milnor Borel construction, coefficient ring $A_{T} (k)$ , character classes, and multiplicative set are as in the character-localization definition. Character localization in Borel equivariant cohomology

[F3]

There is an $N \geq 1$ with $H^{q} (F_{0}; k) = 0$ for $q \geq N$ , for every component $F_{0}$ , so the positive-degree ideal in each $H^{*} (F_{0}; k)$ has $N$ th power zero. Also $B T \times F_{0}$ is paracompact Hausdorff of CW type. Positive-degree cohomology of a compact manifold is nilpotent The fixed-locus Borel base is paracompact and of CW type

[F4]

Since $B T$ has finite-free homology in each degree over $k$ , the external product is a graded-ring isomorphism $A_{T} (k) \otimes_{k} H^{*} (F_{0}; k) ≅ H^{*} (B T \times F_{0}; k)$ . Rational coefficient ring and characters of a product torus classifying space Cohomological Kunneth cross product is a ring isomorphism

[F5]

The Milnor bundle $E T \to B T$ is numerable. The Borel normal vector bundle is locally trivial over the paracompact base $B T \times F_{0}$ , hence admits a subordinate numeration; its supplied complex orientation is an $R$ -orientation. The Thom isomorphism then applies, and the Euler class is the zero-section pullback of its normalized Thom class. Milnor's join model is a contractible free G-space Under choice and dependent choice, every open cover of a paracompact Hausdorff space admits a locally finite subordinate partition of unity Locally finite partitions of unity and subordination to an open cover Thom isomorphism for oriented vector bundles R-oriented vector bundle and orientation local system Euler class by zero-section pullback of the Thom class

[F6]

Euler classes are natural under pullback and multiply under ordered direct sums; for a complex line its Euler class is its first Chern class in the complex orientation. Naturality, orientation sign, and Whitney product for Euler classes Chern classes from the projective-bundle relation

Proof

Given: The data in the statement.

1.1A1F1F2F4F5

Fix a component $F_{0}$ . The action on $F_{0}$ is trivial, so its Borel base is $B T \times F_{0}$ . By [F5], the Borel normal bundle is numerable and oriented; hence its Thom and Euler classes exist. By [F4], identify $H_{T}^{*} (F_{0}; k)$ as the graded ring $A_{T} (k) \otimes_{k} H^{*} (F_{0}; k)$ .

2.1F1F2F5F6step 1.1

Fix $p \in F_{0}$ . Restriction of $e_{T} (ν_{F})$ along $B T \times {p} ↪ B T \times F_{0}$ is the Euler class of the associated normal representation bundle over $B T$ . By [F1], that representation is a sum of nontrivial complex character lines. The associated bundle is the direct sum of the corresponding $L_{χ}$ , with the stated multiplicities. Whitney multiplication in [F6] and $e (L_{χ}) = c_{1} (L_{χ})$ give the claimed product. Every factor is a generator of $S_{T} (k)$ .

3.1F3F4step 1.1step 2.1algebra

Under the Künneth decomposition of step 1.1 write $e_{T} (ν_{F}) = e_{0} + n$ , where $e_{0} \in A_{T} (k) \otimes H^{0} (F_{0}; k)$ is the component detected by restriction to $p$ , and $n \in A_{T} (k) \otimes H^{> 0} (F_{0}; k)$ . Step 2.1 shows $e_{0}$ is a product of generators of $S_{T} (k)$ , so it is invertible after localization. By [F3], the ideal $J = A_{T} (k) \otimes H^{> 0} (F_{0}; k)$ satisfies $J^{N} = 0$ ; therefore $z = e_{0}^{- 1} n$ is nilpotent, and

e_{T} (ν_{F})^{- 1} = e_{0}^{- 1} \sum_{j = 0}^{N - 1} (- z)^{j}

is a finite inverse.

4.1F1F2step 2.1step 3.1∎

There are finitely many components by [F1], so $H_{T}^{*} (F; k)$ and its localization split into the finite product of the component rings treated in step 3.1. A finite tuple of units is a unit componentwise. If $F = \emptyset$ , its cohomology is the zero ring and its unique element is a unit; if a normal component has rank zero, its product in step 2.1 is $1$ . These cases include $r = 0$ , trivial actions, and empty $M$ .

The fixed normal equivariant Euler class is a localized unit

Statement

Statement references

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources