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The complex orientation of the underlying real bundle

Statement

Assume AC. Let $E \to B$ be a numerable complex rank- $n$ vector bundle over a CW complex and let $E_{R}$ be its underlying real bundle.

$E_{R}$ carries a canonical integral orientation, the complex orientation: on a local complex frame $(v_{1}, \dots, v_{n})$ the ordered real frame $(v_{1}, i v_{1}, \dots, v_{n}, i v_{n})$ is positive. The orientation is independent of the complex frame used to define it, is natural under pullback, and is preserved by complex-linear bundle isomorphisms.
For numerable complex bundles $E, F$ over $B$ , the complex orientation of $(E \oplus F)_{R}$ is the ordered direct-sum orientation of the complex orientations of $E_{R}$ and $F_{R}$ .
Let $V$ be a numerable real bundle of rank $2 n$ over $B$ with an integral orientation $o$ , and let $φ : V_{C} \to V \oplus V$ be the canonical real-linear isomorphism $v \otimes (a + i b) \mapsto (a v, b v)$ . Then $φ$ carries the complex orientation of $(V_{C})_{R}$ to $(- 1)^{n}$ times the product orientation $o \oplus o$ ; consequently $e ((V_{C})_{R}) = (- 1)^{n} e (V)^{2}$ in $H^{4 n} (B; Z)$ .

The rank-zero case is included: $E_{R}$ is the zero bundle with its canonical orientation and $e (0_{B}) = 1$ .

Facts & Assumptions

[A1]

The Axiom of Choice is assumed, exactly as inherited by the numerability, Thom and Euler-class suppliers used below (The Axiom of Choice).

[F1]

The underlying real bundle $E_{R}$ is obtained by regarding the complex transition matrices as real-linear; the construction commutes with pullback and with direct sums (Whitney sum, tensor, dual, Hom, and exterior-power bundles). Complexification, passage to the underlying real bundle and finite direct sums use the same trivializing cover, so a partition of unity numerating that cover also numerates each resulting bundle.

[F2]

For positive rank, an orientation of a real bundle is a continuous choice of one of the two fiber orientations and is determined by positive local frames; the zero vector space and every rank-zero bundle have one canonical orientation (Oriented real bundles and oriented frame bundles).

[F3]

Bundles over a common cover are glued from their transition cocycles, and the cocycle determines the bundle up to canonical isomorphism (Vector bundles are glued from transition cocycles).

[F4]

For $R$ -oriented numerable real bundles the Euler class is natural under orientation-preserving pullback, negates under orientation reversal over $Z$ , and multiplies over ordered direct sums (Naturality, orientation sign, and Whitney product for Euler classes).

[F5]

Every endomorphism of a finite-dimensional complex vector space is upper triangularisable (Every finite-dimensional endomorphism over an algebraically closed field is triangularisable).

[F6]

The determinant of a block upper triangular real matrix is the product of the determinants of its diagonal blocks (The characteristic polynomial of a block upper- or lower-triangular matrix is the product of the characteristic polynomials of its diagonal blocks).

[F7]

The Euler class of a rank-zero bundle is the unit $e (0_{B}) = 1$ (Euler class by zero-section pullback of the Thom class).

Proof

technique · direct

Given: AC, numerable complex bundles over $B$ as in statements 1 and 2, a numerable oriented real rank- $2 n$ bundle $V \to B$ as in statement 3, and local complex frames $(v_{1}, \dots, v_{n})$ and $(w_{1}, \dots, w_{n})$ over a common open set.

1.1F1F2given

The list $(v_{1}, i v_{1}, \dots, v_{n}, i v_{n})$ is a real basis of each fiber: since the $v_{j}$ form a complex basis, $\sum_{j} (a_{j} v_{j} + b_{j} i v_{j}) = \sum_{j} (a_{j} + i b_{j}) v_{j}$ vanishes only when all $a_{j} + i b_{j} = 0$ , that is, all $a_{j} = b_{j} = 0$ . Hence the list orients the fibers of $E_{R}$ over the chart, and by [F2] these local data are the candidate local orientations.

2.1F5F6step 1.1algebra

Compatibility on overlaps. Let $A \in {GL}_{n} (C)$ be the complex change-of-frame matrix $w_{j} = \sum_{i} A_{i j} v_{i}$ . In the real bases of step 1.1 the change-of-frame matrix is the realification $R_{A}$ obtained by replacing every complex entry by its $2 \times 2$ real block; realification is multiplicative in the sense $R_{A B} = R_{A} R_{B}$ , because it is the matrix of the same complex-linear map read in real coordinates. By [F5] choose a complex basis in which $A$ is upper triangular; then $R_{A}$ is block upper triangular with diagonal blocks $(\begin{array}{cc} a_{j} & - b_{j} \\ b_{j} & a_{j} \end{array})$ for the diagonal entries $a_{j} + i b_{j}$ of $A$ . By [F6] its determinant is the product of the block determinants $\prod_{j} (a_{j}^{2} + b_{j}^{2}) = ∣ \det_{C} A ∣^{2} > 0$ . A positive determinant means the two ordered real frames induce the same orientation, and multiplicativity of realification reduces every frame pair to this comparison.

2.2F1givenalgebra

Complexification. Let $(e_{1}, \dots, e_{2 n})$ be an oriented real basis of a fiber of $V$ . The vectors $e_{1}, \dots, e_{2 n}$ are a complex basis of the complexification, so by step 1.1 the complex orientation of $(V_{C})_{R}$ is represented by the ordered real basis $e_{1}, i e_{1}, e_{2}, i e_{2}, \dots, e_{2 n}, i e_{2 n}$ . Under the isomorphism $φ$ of statement 3 this list becomes $(e_{1}, 0), (0, e_{1}), (e_{2}, 0), (0, e_{2}), \dots, (e_{2 n}, 0), (0, e_{2 n})$ , while the product orientation $o \oplus o$ is represented by the blocked list $(e_{1}, 0), \dots, (e_{2 n}, 0), (0, e_{1}), \dots, (0, e_{2 n})$ . Passing from the interleaved list to the blocked list is the shuffle of two length- $2 n$ blocks; its inversion number is $0 + 1 + \dots + (2 n - 1) = 2 n (2 n - 1) / 2 = n (2 n - 1)$ , so the orientation sign is $(- 1)^{n (2 n - 1)} = (- 1)^{n}$ .

3.1F1F3step 2.1

Hence the local orientations of steps 1.1 and 2.1 agree on every overlap of a complex linear atlas, and [F3] glues them into a global integral orientation of $E_{R}$ , the complex orientation. The same determinant computation with $A$ the transition function of a pullback chart gives naturality under pullback, and with $A$ the matrix of a complex-linear isomorphism it gives invariance under complex-linear bundle isomorphisms.

3.2F7step 1.1

For rank $n = 0$ the frame list of step 1.1 is empty and the determinant computation of step 2.1 is vacuous, so the zero bundle carries its canonical orientation; [F7] supplies $e (0_{B}) = 1$ for use below.

3.3F1F4step 2.2

Taking Euler classes. The numeration of $V$ also numerates $V_{C}$ , $(V_{C})_{R}$ and $V \oplus V$ by [F1], so every Euler class in this step lies in the scope of [F4]. If two orientations of a real bundle differ by a sign $ε = \pm 1$ on positive frames, their Euler classes differ by the same $ε$ by the orientation-sign clause of [F4]; for the ordered direct sum $V \oplus V$ the Whitney product clause of [F4] gives $e (V \oplus V) = e (V) e (V)$ . Therefore $e ((V_{C})_{R}) = (- 1)^{n} e (V \oplus V) = (- 1)^{n} e (V)^{2}$ , which is statement 3.

4.1F1F2step 1.1step 3.2

Direct sums. A local complex frame of $E \oplus F$ is the concatenation of a complex frame $(v_{1}, \dots, v_{m})$ of $E$ and a complex frame $(w_{1}, \dots, w_{k})$ of $F$ , so the real frame of step 1.1 is $(v_{1}, i v_{1}, \dots, v_{m}, i v_{m}, w_{1}, i w_{1}, \dots, w_{k}, i w_{k})$ , exactly the ordered direct-sum frame of the complex-oriented summands $E_{R}$ and $F_{R}$ . By [F2] the two orientations coincide, so statement 2 holds, and the rank-zero case is step 3.2.

5.1A1F4F7step 3.2step 4.1step 2.2∎

Boundary cases. Rank zero is step 3.2. In statement 2, step 4.1 says that the complex orientation on $0 \oplus F$ is the ordered sum of the canonical orientation on $0$ and the complex orientation on $F$ ; only after applying the Whitney product formula [F4] and $e (0) = 1$ from [F7] does one obtain $e (0 \oplus F) = 1 \cdot e (F)$ . In statement 3 with $n = 0$ , both sides are the unit. For a complex line, $n = 1$ in step 2.1 gives $∣ \det_{C} A ∣^{2} > 0$ directly, and for $V$ of rank $2$ step 2.2 has inversion number $2 \cdot 1 / 2 = 1$ and sign $(- 1)^{1} = - 1$ . The argument uses no choice beyond the inherited numerability data recorded in [A1].

Source notes

The orientation convention $(v, i v)$ on a complex line and its determinant computation are the standard ones of Milnor-Stasheff, Lemma 14.1, and the $(- 1)^{n}$ comparison of the complex orientation of $V_{C}$ with the product orientation of $V \oplus V$ is Hatcher, Vector Bundles & K-Theory section 3.2, proof of Proposition 3.15(b), printed pp. 94-96 ("n(2n-1) transpositions, so a sign (-1)^n").

The complex orientation of the underlying real bundle

Statement

Facts & Assumptions

Proof

Source notes

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