Whitehead's Integral Formula

• Whitehead's integral formula is a method that computes the Hopf invariant by integrating a wedge product of a primitive form and a pulled-back volume form.
• It establishes a clear connection between algebraic topology and differential analysis, generalizing linking numbers to higher dimensions.
• The approach provides both analytic and simplicial algorithms, with a computational complexity of O(fₙ³) for triangulated sphere mappings.

Whitehead’s integral formula expresses the Hopf invariant of a continuous or simplicial map $f \colon S^{2n-1} \to S^n$ as a single integral involving differential forms. The formula creates a direct computational and conceptual connection between algebraic topology (notably, the computation of the Hopf invariant) and the analysis of differential forms. Whitehead’s approach generalizes the classical notion of linking numbers to higher dimensions and provides both an analytic and a combinatorial (simplicial) computational method (Musin et al., 2 Dec 2025).

1. Cohomological and Linking-Number Definitions

Let $f \colon S^{2n-1} \to S^n$ be a continuous map, with regular values $x, y \in S^n$, $x \ne y$. The preimages $f^{-1}(x)$ and $f^{-1}(y)$ are disjoint $(n-1)$-dimensional submanifolds of $S^{2n-1}$. Their linking number defines the classical Hopf invariant: $$H(f) = \mathrm{lk}\bigl(f^{-1}(x), f^{-1}(y)\bigr) \in \mathbb{Z}.$$ This value is independent of the choice of $x, y$.

Equivalently, in de Rham cohomology, take a normalized volume form $\omega \in \Omega^n(S^n)$ with $\int_{S^n} \omega = 1$. The pullback $f^*\omega \in \Omega^n(S^{2n-1})$ is closed and exact, since $H^n_{\mathrm{dR}}(S^{2n-1}) = 0$. Therefore, there exists a global $(n-1)$-form $\theta$ on $S^{2n-1}$ such that $\mathrm{d}\theta = f^*\omega$. The Hopf invariant is then captured by Whitehead’s formula.

2. Whitehead’s Integral Formula: Derivation and Statement

The integral characterization of the Hopf invariant, as established in de Rham cohomology, is as follows: $H(f) = \int_{S^{2n-1}} \theta \wedge f^*\omega,$ where: $$\begin{cases} \omega \in \Omega^n(S^n), \ \int_{S^n} \omega = 1, \ f^*\omega \in \Omega^n(S^{2n-1}), \ \mathrm{d}\theta = f^*\omega. \end{cases}$$ Here, $S^{2n-1}$ carries its standard orientation, $\omega$ is a normalized volume form on $S^n$, and $\theta$ is a global primitive for $f^*\omega$.

The wedge product $\theta \wedge f^*\omega$ is a closed $(2n-1)$-form, since by calculation: $$\mathrm{d}(\theta \wedge f^*\omega) = \mathrm{d}\theta \wedge f^*\omega - \theta \wedge \mathrm{d}(f^*\omega) = f^*\omega \wedge f^*\omega = 0.$$ Integrating this over the fundamental cycle of $S^{2n-1}$ produces the integer-valued Hopf invariant.

3. Simplicial and Cochain Formulation

For a simplicial approach, a triangulation $T_1$ of $S^{2n-1}$ and $T_2$ of $S^n$ determines a simplicial map via a vertex labeling $L\colon V(T_1) \to V(T_2)$. The generator of $H^n(S^n; \mathbb{Z})$ is represented by an integral $n$-cocycle $\omega \in C^n(T_2; \mathbb{Z})$. The pullback $f^*\omega \in C^n(T_1; \mathbb{Z})$ is then exact, given $H^n(S^{2n-1}; \mathbb{Z}) = 0$, admitting an $(n-1)$-cochain $\theta \in C^{n-1}(T_1; \mathbb{Q})$ with $\delta\theta = f^*\omega$.

The cup-product $\theta \smile f^*\omega$ yields a cocycle in $C^{2n-1}(T_1)$ and evaluates—when paired with the fundamental cycle—to the integer Hopf invariant: $$H(f) = \langle \theta \smile f^*\omega, \Delta \rangle.$$

4. Hypotheses for Validity

Whitehead’s formula requires the following conditions:

• The map $f: S^{2n-1} \to S^n$ must be at least $C^1$ (for de Rham theory) or simplicial (for cochain computations).
• Both spheres must be oriented.
• Automatically, $H^n(S^{2n-1}) = 0$, ensuring $f^*\omega$ is exact.
• One must select a volume form $\omega$ with $\int_{S^n} \omega = 1$.

5. Algorithmic Computation for Simplicial Mappings

Given a simplicial map $f$ between triangulated spheres, an explicit algorithm for computing $H(f)$ is as follows:

1. Initialization: Fix a distinguished oriented $n$-simplex $\bar{\sigma}$ in the standard $(n+2)$-vertex triangulation $S^n_{n+2}$. Define the integral cocycle $\omega(\sigma)$ accordingly.
2. Pullback: For each $n$-simplex $\sigma = [v_0, \ldots, v_n]$ in $T_1$, set $$f^*\omega(\sigma) = \omega([\ell(v_0), \ldots, \ell(v_n)])$$.
3. Cochain Solution: For each oriented $(n-1)$-simplex $\tau$, introduce a rational unknown $x_\tau = \theta(\tau)$. For each $n$-simplex $\sigma$, the coboundary equation is

$$(\delta\theta)(\sigma) = \sum_{i=0}^{n} (-1)^i \, x_{[v_0, \ldots, \widehat{v_i}, \ldots, v_n]} = f^*\omega(\sigma).$$

This gives a linear system over $\mathbb{Q}$, which is solvable due to exactness.

1. Cup-Product Pairing: Calculate the cup product $(\theta \smile f^*\omega)(\sigma)$ for each $(2n-1)$-simplex:

$$(\theta \smile f^*\omega)(\sigma) = \theta([u_0, \ldots, u_{n-1}]) \cdot f^*\omega([u_{n-1}, \ldots, u_{2n-1}]).$$

2. Summation: The Hopf invariant is computed as the sum over all $(2n-1)$-simplices, weighted by orientation:

$$H(f) = \sum_{\sigma} \varepsilon_\sigma (\theta \smile f^*\omega)(\sigma) \in \mathbb{Z}.$$

The computational complexity is dominated by Gaussian elimination on the coboundary system for the $n$-simplices, scaling as $O(f_n^3)$, where $f_n$ is the number of $n$-simplices (Musin et al., 2 Dec 2025).

6. Illustrative Cases and Consequences

Table: Special Cases of Whitehead's Integral Formula

Value of $n$	Domain/Target	Interpretation	Hopf Invariant $H(f)$
$1$	$S^1 \to S^1$	Degree of map; classical winding number	$H(f) = \deg(f)$
$2$	$S^3 \to S^2$	Classical Hopf invariant; Hopf fibration	$H(f) = 1$ for the Hopf fibration
odd $n > 1$	$S^{2n-1} \to S^n$	All maps are null-homotopic	$H(f) = 0$

For $n=1$, the formula recovers the winding number as the degree. For $n=2$, the Hopf fibration's Hopf invariant is $1$, computed either integrally or via the simplicial algorithm for minimal triangulations. When $n$ is odd and greater than $1$, the groups $\pi_{2n-1}(S^n)$ vanish, so $H(f) = 0$ for all such maps.

7. Synthesis and Impact

Whitehead’s integral formula provides a bridge between topological invariants and analytic (differential form) constructions. The formula generalizes the interpretation of the Hopf invariant in terms of linking numbers and extends it to a robust computational framework via its cochain manifestation for simplicial maps. The algorithmic realization delivers a practical $O(f_n^3)$ procedure to compute $H(f)$ for any simplicial map $S^{2n-1} \to S^n$, unifying classical theory and computational techniques for homotopy-invariant computation (Musin et al., 2 Dec 2025).