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First Chern class of a complex bundle equals that of its determinant

Statement

Assume AC. If $E \to B$ is a numerable complex vector bundle of positive finite rank over a path-connected paracompact Hausdorff CGWH space $B$ of CW homotopy type, then $c_{1} (E) = c_{1} (\det_{C} E)$ in $H^{2} (B; Z)$ , where $\det_{C} E = Λ^{{rank}_{C} E} E$ .

Statement references

Complex splitting principle with integral injective pullback

Naturality, normalization, and Whitney sum for Chern classes

Facts & Assumptions

Given: AC and a numerable complex vector bundle $E \to B$ of rank $n \geq 1$ on the stated base.

[A1]

Full AC is the choice-function principle of The Axiom of Choice; it is inherited by the projective-bundle and Chern-class suppliers below.

[F1]

For a numerable complex bundle on a path-connected paracompact Hausdorff CW complex, the flag pullback splits $q^{*} E = L_{1} \oplus \dots \oplus L_{n}$ and $q^{*}$ is injective on integral cohomology (Complex splitting principle with integral injective pullback).

[F2]

Pulling a numerable bundle back along a continuous map preserves numerability. For a bundle over a CW model, its flag bundle is an iterated projective bundle with compact CW fibers ${C P}^{r}$ ; the general compact-fiber statement makes every stage paracompact Hausdorff CGWH of CW type. A homotopy equivalence induces an isomorphism on singular cohomology (Complex flag bundle and Chern roots, Compact-fibre bundle totals preserve paracompactness, and CW type under CW-type hypotheses, Homotopic maps induce equal maps in singular cohomology).

[F3]

The projective-bundle definition gives a unique monic relation for each numerable complex bundle over a path-connected paracompact Hausdorff CGWH base of CW type. For a pullback, the projective charts identify $P (f^{*} V)$ with the base-change $B^{'} \times_{B} P (V)$ and identify its tautological line with the pulled-back line. The Thom definition of its Euler class and naturality of Thom classes then pull back the tautological class and the defining relation; coefficient uniqueness gives Chern-class naturality on these bases. On a CW base the total Chern class is multiplicative under Whitney sums (Chern classes from the projective-bundle relation, Naturality, normalization, and Whitney sum for Chern classes).

[F4]

On a line bundle, the first Chern class is its coefficient in the projective-bundle relation (Chern classes from the projective-bundle relation).

[F5]

First Chern class is additive under line-bundle tensor product: $c_{1} (L \otimes M) = c_{1} (L) + c_{1} (M)$ (First Chern class of tensor, dual, and conjugate lines).

Proof

technique · splitting principle

1.1F3constructalgebra

We record the pullback naturality used in this proof, including the case of a CW-type target. Let $b : B^{'} \to B$ be continuous between path-connected paracompact Hausdorff CGWH bases of CW type, and let $V \to B$ be numerable of positive rank. In projective charts, $P (b^{*} V)$ is $B^{'} \times_{B} P (V)$ ; the induced map $\tilde{b} : P (b^{*} V) \to P (V)$ pulls the tautological line of $V$ back to that of $b^{*} V$ . Its oriented Euler class $x$ is therefore natural under $\tilde{b}$ . Pull back the defining monic relation for $V$ along $\tilde{b}$ . The result is a monic relation for $x_{b^{*} V}$ with coefficients $b^{*} c_{i} (V)$ ; uniqueness in the projective-bundle theorem on $B^{'}$ gives $c_{i} (b^{*} V) = b^{*} c_{i} (V)$ .

1.2A1F1given

Choose a path-connected CW complex $K$ and a homotopy equivalence $f : K \to B$ , and put $E_{K} = f^{*} E$ . The pulled-back numeration makes $E_{K}$ a numerable complex bundle over the CW base $K$ . Apply [F1] to its flag bundle $q : F = Fl (E_{K}) \to K$ . It gives a splitting $q^{*} E_{K} = ⨁_{j = 1}^{n} L_{j}$ and injectivity of $q^{*} : H^{2} (K; Z) \to H^{2} (F; Z)$ .

2.1F2step 1.2

Each projective stage in the flag tower has fiber ${C P}^{r}$ , a compact finite CW complex. The general compact-fiber conclusion in [F2] makes $F$ paracompact Hausdorff CGWH of CW type. The flag fiber $U (n) / T^{n}$ is connected and $K$ is path connected, so $F$ is path connected. Choose a path-connected CW model $h : W \to F$ that is a homotopy equivalence.

3.1F2F3F5step 1.2step 2.1

Let $V = q^{*} E_{K}$ . By step 1.1, naturality on CW-type bases gives $h^{*} c_{1} (V) = c_{1} (h^{*} V)$ and $h^{*} c_{1} (\det_{C} V) = c_{1} (h^{*} \det_{C} V)$ . On the CW base $W$ , the splitting in step 1.2, Whitney multiplicativity, and line-bundle tensor additivity yield $c_{1} (h^{*} V) = \sum_{j = 1}^{n} c_{1} (h^{*} L_{j}) = c_{1} ⁣ (⨂_{j = 1}^{n} h^{*} L_{j}) = c_{1} (h^{*} \det_{C} V) .$ Here $h^{*} \det_{C} V ≅ \det_{C} (h^{*} V) ≅ ⨂_{j} h^{*} L_{j}$ . Since $h^{*}$ is an isomorphism by [F2], this proves $c_{1} (V) = c_{1} (\det_{C} V)$ on $F$ . The injectivity of $q^{*}$ from step 1.2 then gives $c_{1} (E_{K}) = c_{1} (\det_{C} E_{K})$ on $K$ .

4.1F2F3step 3.1∎

Apply step 1.1 to $f : K \to B$ : $c_{1} (E_{K}) = f^{*} c_{1} (E)$ , and exterior powers commute with pullback, so $c_{1} (\det_{C} E_{K}) = f^{*} c_{1} (\det_{C} E)$ . The map $f^{*}$ is an isomorphism by homotopy invariance [F2]. The equality on $K$ from step 3.1 therefore implies $c_{1} (E) = c_{1} (\det_{C} E)$ on $B$ .

Depends on

Dependency tree · 2 levels

Sources

Allen Hatcher, Vector Bundles & K-Theory