The splitting principle is a methodological tool in algebraic topology and algebraic geometry that allows one to reduce statements about vector bundles to statements about direct sums of line bundles. It asserts that for any complex (or real) vector bundle $E$ over a base space $B$, there exists a space $f\colon B' \to B$ (often constructed as a flag bundle associated with $E$) such that the pull‑back bundle $f^{}E$ splits as a direct sum of line bundles: $$ f^{}E \cong L_{1}\oplus L_{2}\oplus\cdots\oplus L_{r}, $$ where $r$ is the rank of $E$. Moreover, the induced map $f^{}\colon H^{}(B) \to H^{*}(B')$ is injective in cohomology, so any cohomological identity valid for split bundles and which is natural with respect to pull‑back holds for the original bundle.
Formal Statement
Let $E \to B$ be a rank‑$r$ complex vector bundle. Define the complete flag bundle $$ \operatorname{Fl}(E) = { (b, V_{1}\subset V_{2}\subset\cdots\subset V_{r}=E_{b}) \mid \dim V_{i}=i}, $$ with projection $\pi\colon \operatorname{Fl}(E) \to B$. The tautological subbundles $\mathcal{S}{i}$ on $\operatorname{Fl}(E)$ satisfy $$ \pi^{*}E \cong \mathcal{S}{1}\oplus (\mathcal{S}{2}/\mathcal{S}{1})\oplus\cdots\oplus (\mathcal{S}{r}/\mathcal{S}{r-1}), $$ each quotient being a line bundle. The map $\pi^{}\colon H^{}(B) \to H^{*}(\operatorname{Fl}(E))$ is injective, establishing the principle.
Applications
- Computation of Characteristic Classes: By reducing to line bundles, the total Chern class $c(E) = \prod_{i=1}^{r}(1 + x_{i})$ can be expressed in terms of the first Chern classes $x_{i}=c_{1}(L_{i})$. This facilitates calculations of Chern numbers and other characteristic invariants.
- Proofs of Algebraic Identities: Many identities in the theory of characteristic classes (e.g., the Whitney sum formula, the relation between Chern and Pontryagin classes) are proved by applying the splitting principle, verifying the identity for sums of line bundles, and then invoking naturality.
- K‑Theory: In complex topological $K$-theory, the splitting principle underlies the description of the representation ring of the unitary group and the definition of λ‑operations.
- Schubert Calculus: The flag bundle construction is a central object in Schubert calculus, where the splitting principle connects the geometry of flags with cohomological computations.
Historical Context
The concept was formalized in the mid‑20th century within the development of characteristic class theory. Notable early references include works by Chern (1946) on characteristic classes, and later expositions by Hirzebruch, Milnor, and Stasheff in Characteristic Classes (1974). The principle is sometimes attributed to the “splitting principle for Chern classes” and has become a standard component of graduate‑level textbooks on algebraic topology and geometry.
Variants
- Real Splitting Principle: For real vector bundles, one typically works with Stiefel‑Whitney or Pontryagin classes, and the associated flag bundle is built from orthogonal flags. The resulting line bundles are real line bundles, and the injectivity of the induced map on cohomology holds with $\mathbb{Z}_2$ coefficients for Stiefel‑Whitney classes.
- Algebraic‑Geometric Version: In the category of algebraic varieties, the principle is applied using the projectivization $\mathbb{P}(E)$ and successive projective bundles to achieve a filtration by line bundles in the Chow ring.
References
- H. B. Miller, Algebraic Topology: An Introduction, Springer, 1994.
- A. Borel and J.-P. Serre, “Groupes de Lie et classes caractéristiques,” Bulletin de la Société Mathématique de France, 86 (1958): 453–517.
- J. Milnor and J. Stasheff, Characteristic Classes, Annals of Mathematics Studies, No. 76, Princeton University Press, 1974.
- R. Fulton, Intersection Theory, Springer, 1998 – Chapter 3 discusses the splitting principle in the context of Chow rings.
The splitting principle remains a foundational technique for simplifying problems involving vector bundles by allowing arguments to be carried out in the more tractable setting of line bundles.
