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The total L-class of complex projective space is a power of $x / \tanh x$

Statement

Assume AC. Let $γ \to {C P}^{n}$ be the tautological complex line and $y = c_{1} (γ^{*}) \in H^{2} ({C P}^{n}; Z)$ the standard generator with $H^{*} ({C P}^{n}; Z) = Z [y] / (y^{n + 1})$ and $⟨ y^{n}, [{C P}^{n}] ⟩ = 1$ (The tangent bundle of complex projective space and its Pontryagin classes), and give ${C P}^{n}$ its complex orientation. Then in $H^{4 *} ({C P}^{n}; Q)$ , $L (T {C P}^{n}) = Q (y)^{n + 1} = (\frac{y}{\tanh y})^{n + 1} .$ Its degree- $4 j$ component is $c_{j} y^{2 j}$ with $c_{j} = \sum_{i_{1} + \dots + i_{n + 1} = j} q_{2 i_{1}} \dots q_{2 i_{n + 1}} = [y^{2 j}] (y / \tanh y)^{n + 1}$ , the multinomial coefficient sum; for $j = 1$ this is $(n + 1) q_{2}$ .

Facts & Assumptions

Given: AC, the tautological line $γ \to {C P}^{n}$ , the generator $y = c_{1} (γ^{*})$ , and the complex orientation.

[F1]

The total L-class of a smooth manifold is $L (M) = L (T M) \in {\hat{H}}^{4 *} (M; Q)$ , with degree- $4 j$ component $L_{j} (T M)$ , and $L$ of a bundle is a polynomial in its Pontryagin classes (The total L-class and the L-genus of a smooth manifold, The Hirzebruch L-polynomials and the total L-class of a real vector bundle).

[F2]

$L$ is stable, multiplicative and natural, and for a complex line bundle $ℓ$ with $c_{1} (ℓ) = t$ the underlying real L-class is $L (ℓ_{R}) = Q (t) = t / \tanh t$ (The L-polynomials are well defined and form a multiplicative, natural and stable sequence).

[F3]

The in-run supplier establishes the complex bundle isomorphism $\underline{C} \oplus T {C P}^{n} ≅ (γ^{*})^{\oplus (n + 1)}$ and $p (T {C P}^{n}) = (1 + y^{2})^{n + 1}$ (The tangent bundle of complex projective space and its Pontryagin classes).

[F4]

The projective tangent-bundle supplier gives $H^{*} ({C P}^{n}; Z) = Z [y] / (y^{n + 1})$ for $y = c_{1} (γ^{*})$ and $⟨ y^{n}, [{C P}^{n}] ⟩ = 1$ (The tangent bundle of complex projective space and its Pontryagin classes). Rational coefficient change gives the same truncated ring over $Q$ : the integral groups are finite free, and the universal coefficient sequence identifies both coefficient groups with duals of the integral homology free quotients. Thus $H^{4 j} ({C P}^{n}; Q) = Q y^{2 j}$ for $2 j \leq n$ and is zero otherwise (Topological universal coefficient short exact sequence for cohomology).

[F5]

For a complex line bundle, the Euler class of the underlying oriented real bundle is the first Chern class (Top Chern class equals Euler class of the underlying real bundle, Euler class by zero-section pullback of the Thom class, Naturality, orientation sign, and Whitney product for Euler classes).

[F6]

$Q (u) = \sum_{j \geq 0} q_{2 j} u^{2 j}$ with $q_{0} = 1$ (The Hirzebruch L-polynomials and the total L-class of a real vector bundle).

Proof

technique · direct; split the stable tangent bundle and apply the rank-two evaluation

1.1givenF1F2F3

Step [F3] gives a complex bundle isomorphism $\underline{C} \oplus T {C P}^{n} ≅ (γ^{*})^{\oplus (n + 1)}$ . Since $L$ is computed from Pontryagin classes and is therefore unchanged under bundle isomorphism, and since $L$ is stable, $L (T {C P}^{n}) = L (T {C P}^{n} \oplus \underline{C}) = L ((γ^{*})^{\oplus (n + 1)})$ in the completed ring of ${C P}^{n}$ .

1.2givenF2F4F5

The dual tautological bundle $γ^{*}$ is a complex line bundle with $c_{1} (γ^{*}) = y$ by [F4], so by [F5] its underlying oriented real bundle has Euler class $y$ and the complex-line clause of [F2] gives $L (γ_{R}^{*}) = Q (y) = y / \tanh y$ .

2.1step 1.1step 1.2F2

By multiplicativity in [F2], applied $(n + 1)$ times to the Whitney sum of copies of $γ_{R}^{*}$ , $L ((γ^{*})^{\oplus (n + 1)}) = L (γ^{*})^{n + 1} = Q (y)^{n + 1} = (y / \tanh y)^{n + 1}$ .

3.1step 2.1F4F6algebra∎

Degree components: writing $Q (y)^{n + 1} = (\sum_{i \geq 0} q_{2 i} y^{2 i})^{n + 1}$ and extracting coefficients first in the formal indeterminate $y$ and then reducing modulo $y^{n + 1}$ by [F4], the degree- $4 j$ component is the multinomial sum $c_{j} y^{2 j}$ with $c_{j} = \sum_{i_{1} + \dots + i_{n + 1} = j} q_{2 i_{1}} \dots q_{2 i_{n + 1}} = [y^{2 j}] Q (y)^{n + 1}$ , because a product of $n + 1$ factors of weights $4 i_{a}$ has weight $4 j$ exactly when $i_{1} + \dots + i_{n + 1} = j$ ; the case $j = 1$ gives $c_{1} = (n + 1) q_{2}$ , and components with $2 j > n$ vanish by [F4] and the convention on Pontryagin classes. Hence $L (T {C P}^{n}) = Q (y)^{n + 1}$ with the displayed components.

Depends on

Used by

The L-genus of complex projective space of even complex dimension is one Lemma

Dependency tree · two levels

71 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

John Milnor and James Stasheff, Characteristic Classes (re-typeset scan; original pagination) (standard reference, not scraped)
Tom Weston, An Introduction to Cobordism Theory (lecture notes, Stanford) (standard reference, not scraped)
Daniel S. Freed, Bordism: Old and New (lecture notes, UT Austin, Fall 2012) (standard reference, not scraped)

The total L-class of complex projective space is a power of x/tanh⁡xx/tanhx