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

Statement

Assume AC, inherited from the L-class and projective-space characteristic-class suppliers. For every $k \geq 0$ , with the complex orientation of ${C P}^{2 k}$ , $L [{C P}^{2 k}] = ⟨ L_{k} (T {C P}^{2 k}), [{C P}^{2 k}] ⟩ = 1.$

Facts & Assumptions

Given: AC, the integer $k \geq 0$ , and the complex orientation of ${C P}^{2 k}$ .

[F1]

$L (T {C P}^{2 k}) = Q (y)^{2 k + 1} = (y / \tanh y)^{2 k + 1}$ in $H^{4 *} ({C P}^{2 k}; Q)$ , where $y = c_{1} (γ^{*})$ is the standard generator (The total L-class of complex projective space is a power of $x / \tanh x$ ).

[F2]

The in-run supplier gives $H^{*} ({C P}^{n}; Z) = Z [y] / (y^{n + 1})$ , $⟨ y^{n}, [{C P}^{n}] ⟩ = 1$ , and $p (T {C P}^{n}) = (1 + y^{2})^{n + 1}$ (The tangent bundle of complex projective space and its Pontryagin classes).

[F3]

For every $m \geq 0$ , $[z^{m}] (z / \tanh z)^{m + 1} = 1$ when $m$ is even and $0$ when $m$ is odd (The coefficient identity $[z^{2 k}] (z / \tanh z)^{2 k + 1} = 1$ for every $k$ ).

[F4]

The L-genus in dimension $4 k$ is $L [M] = ⟨ L_{k} (T M), [M] ⟩$ , the degree- $4 k$ evaluation of the total L-class under the Kronecker pairing, and $L (T {C P}^{2 k})$ has degree- $4 k$ component lying in $H^{4 k} ({C P}^{2 k}; Q) = Q y^{2 k}$ (The total L-class and the L-genus of a smooth manifold, Kronecker evaluation pairing, The Hirzebruch L-polynomials and the total L-class of a real vector bundle).

Proof

technique · direct; reduce the genus to a single power-series coefficient

1.1givenF1F2F3

By [F1], $L (T {C P}^{2 k}) = Q (y)^{2 k + 1} = (y / \tanh y)^{2 k + 1}$ , whose degree- $4 k$ component is $c y^{2 k}$ with $c = [y^{2 k}] (y / \tanh y)^{2 k + 1}$ , since $H^{4 k} ({C P}^{2 k}; Q)$ is one-dimensional spanned by $y^{2 k}$ by [F2]. Taking $m = 2 k$ in [F3] gives $c = 1$ .

2.1step 1.1F2F4

Evaluating: $L [{C P}^{2 k}] = ⟨ c y^{2 k}, [{C P}^{2 k}] ⟩ = c ⟨ y^{2 k}, [{C P}^{2 k}] ⟩ = c = 1$ , using $⟨ y^{2 k}, [{C P}^{2 k}] ⟩ = 1$ from [F2] and the $Q$ -linearity of the Kronecker pairing [F4]. For $k = 0$ the manifold is a point with $y \in H^{2} ({C P}^{0}) = 0$ , the total class is $Q (0) = 1$ , and the evaluation on $[p t]$ is $1$ , the same computation with an empty product.

3.1step 1.1step 2.1given∎

Steps 1.1 and 2.1 compute $L [{C P}^{2 k}] = ⟨ L_{k} (T {C P}^{2 k}), [{C P}^{2 k}] ⟩ = 1$ for every $k \geq 0$ , as asserted; AC is used only through the inherited L-class and projective-space suppliers.

Depends on

Used by

Dependency tree · two levels

51 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)