Dual Stiefel–Whitney Class Overview

• Dual Stiefel–Whitney Class is a characteristic cohomology class that encapsulates embedding and immersion obstructions in real vector bundles.
• Its optimal nonvanishing is determined by the function ˆa(n), which uses the binary expansion of n to link algebraic invariants with manifold embeddability.
• Explicit constructions via real Bott manifolds demonstrate how these classes refine classical results by providing sharp obstructions in embedding and immersion theories.

The theory of dual Stiefel–Whitney classes is a central topic in the paper of characteristic classes of real vector bundles, especially in the context of differentiable manifolds. These dual classes play a critical role in obstruction theory, embedding and immersion problems, and the topology of real Bott manifolds. For orientable manifolds, the existence and explicit construction of dual Stiefel–Whitney classes in the highest possible grading has deep implications for both geometry and combinatorics, especially relating to the connection between binary invariants and embedding dimensions.

1. Definition and Geometric Significance

The dual Stiefel–Whitney class $_i(M) \in H^i(M; \mathbb{Z}/2)$ of a compact differentiable n-manifold $M$ is a characteristic cohomology class constructed dually to the usual total Stiefel–Whitney class. Specifically, for an embedding or immersion problem, the nonvanishing of $_i(M)$ is a direct obstruction:

• If $_i(M) \ne 0$, the manifold $M$ cannot be embedded into $\mathbb{R}^{n+i}$, nor immersed into $\mathbb{R}^{n+i-1}$.

This property links dual Stiefel–Whitney classes closely to the theory of embedding and immersion of manifolds, echoing the interplay between characteristic classes and geometry established in classical algebraic topology.

2. Optimal Nonvanishing: Main Theorem and Best Possible Grading

The optimal existence of nonzero dual Stiefel–Whitney classes is determined by a grading function $\hat{a}(n)$ constructed from the binary expansion of the manifold's dimension $n$. The sharp theorem is as follows:

• For any orientable $n$-manifold $M$, $_i(M) = 0$ for all $i > n - \hat{a}(n)$, and there exists (constructively, for all $n \not\equiv 0 \pmod{4}$) an orientable manifold with $_{n-\hat{a}(n)}(M) \neq 0$ (Davis, 31 Jul 2025).

The function $\hat{a}(n)$ is defined according to the congruence class of $n$ (modulo 4) and its binary expansion:

• If $n \equiv 1 \pmod{4}$, then $\hat{a}(n)$ equals the number of 1's in the binary expansion of $n$;
• Otherwise, $\hat{a}(n)$ is one more than that number.

This grading is optimal: it is not possible to have a nontrivial dual Stiefel–Whitney class in a strictly higher degree, and orientable manifolds achieving equality have the maximal embedding (or immersion) obstructions compatible with their dimension and orientability.

3. Explicit Constructions via Real Bott Manifolds

Real Bott manifolds $B_n$ are constructed via iterated projectivizations titled Bott towers,

$B_n \to B_{n-1} \to \dots \to B_1 \to B_0 = \{*\},$

with each stage $B_j$ the projectivization of a sum of a line bundle (with nontrivial first Stiefel–Whitney class) and the trivial bundle. Their cohomology rings are generated by $x_1,\dots, x_n$ in $H^1(B_n; \mathbb{Z}/2)$ with the relations: $x_j^2 = \sum_{i=1}^{j-1} a_{i, j}\, x_i x_j,\,\,\,\, x_1^2=0\quad\text{with %%%%26%%%% strictly upper-triangular.}$ A Bott manifold is orientable if every row of the matrix $A$ has even parity.

For $n \equiv 1 \pmod{4}$, explicit choices of $A$ (e.g., $x_n^2 = (x_1+\cdots + x_{n-2})x_n$ and $x_i^2 = x_{i-1} x_i$ for $2 \le i < n$) yield orientable $n$-manifolds $B_n$ with

$_{\phantom{a}n-\hat{a}(n)}(B_n) \ne 0.$

For $n \equiv 2,3 \pmod{4}$, product constructions $B_{n-1} \times S^1$ and $B_{n-2} \times S^1 \times S^1$ ensure analogous nonvanishing in the optimal grading (Davis, 31 Jul 2025).

4. Role of the Function $\hat{a}(n)$ and Obstruction Theory

The function $\hat{a}(n)$,

$$\hat{a}(n) = \begin{cases} \mathrm{number\ of\ 1's\ in\ the\ binary\ expansion\ of}\ n, & n \equiv 1 \pmod{4} \ \mathrm{number\ of\ 1's} + 1, & n \not\equiv 1 \pmod{4} \end{cases}$$

specifies the highest possible grading with nontrivial dual Stiefel–Whitney class. The connection between $\hat{a}(n)$ and the algebraic structure of real Bott manifolds derives from the action of Steenrod operations and the anti-automorphism $\chi$ of the Steenrod algebra. Specifically, for a nonzero class $z$ in $H^{(n)}(B_n)$ (with $(n)$ the number of 1's), the calculation

$\chi Sq^{n-\hat{a}(n)}(z) \neq 0$

furnishes the desired nonvanishing, and Poincaré duality then identifies the corresponding dual class.

This mechanism directly constrains the possible embedding and immersion dimensions for $M$. Nonvanishing $_i(M)$ obstructs embeddings into $\mathbb{R}^{n+i}$, thus generalizing classical relations between Stiefel–Whitney numbers and embeddability.

5. Applications and Implications: Embedding, Immersion, and Topological Invariants

The existence of orientable manifolds with the maximally graded nontrivial dual Stiefel–Whitney class yields new understanding in geometric topology:

• The vanishing theorem (for degrees above $n-\hat{a}(n)$) and explicit constructions (attaining equality) refine classical embedding and immersion results for orientable manifolds.
• Real Bott manifolds serve as concrete test cases for the realization of these obstructions.
• These results tie the combinatorics of the binary digit expansion of $n$ to deep geometric and algebraic properties, notably in the optimality of dual Stiefel–Whitney invariants.
• The explicit calculations, employing the properties of Steenrod operations, the Whitney sum formula, and product decompositions, provide a roadmap for constructing further examples or classifying manifolds with given embedding obstructions.

The analysis also clarifies that, outside of the cases $n \equiv 0 \pmod{4}$ (where the problem remains subtle, except for $n$ a power of $2$), the explicit examples via Bott manifolds fully resolve the existence question for orientable manifolds with nontrivial dual Stiefel–Whitney class in the largest allowable grading.

6. Broader Context and Open Questions

While real Bott manifolds deliver the necessary examples for $n \not\equiv 0 \pmod{4}$, the case $n \equiv 0 \pmod{4}$ is not resolved in general, except in specific instances (e.g., $n$ a $2$-power). Attempts with generalized Dold manifolds indicate limitations, and the problem of fully classifying which orientable $n$-manifolds admit nonvanishing dual classes in degree $n-\hat{a}(n)$ for all $n$ remains partially open.

Dual Stiefel–Whitney classes, by obstructing high-dimensional embeddings and immersions, continue to serve as foundational invariants in both topology and combinatorics. Their explicit realizations offer both theoretical and computational footholds for contemporary research at the interface of manifold topology, combinatorial geometry, and characteristic class theory.