Is the Chern Number a Topological Invariant?

Understanding whether the Chern number is a topological invariant involves a deep dive into the intricacies of manifold theory and complex structures. In this article, we will explore the dependency of the Chern number on the manifold's structure and consider different notions of complex structures. We will conclude with specific examples and implications for mathematicians working in topology and differential geometry.

The Dependency of Chern Number on Structural Details

It's important to clarify that the Chern number, a topological invariant in complex geometry and fiber bundles, is not solely dependent on the topology of the underlying manifold. Let's first consider the complex structure on a manifold. The Chern number of an almost complex manifold can depend on more than just the smooth structure, indicating that it is not a topological invariant in the strictest sense. This means that two topologically identical manifolds can have different Chern numbers.

The question of whether the Chern number is a topological invariant is indeed complex and requires precise definitions. There are several notions of complex structures to consider:

Complex Analytic Manifolds: These are manifolds equipped with a complex structure that is compatible with the smooth structure. While the Chern number for a complex analytic manifold is a topological invariant, it still depends on the complex structure. Manifolds with a Complex Structure on the Tangent Bundle: Here, the tangent bundle is given a complex structure, which can lead to different Chern numbers even for manifolds with the same topology. Manifolds with Stable Tangent or Normal Bundles: These bundles can have complex structures, and their Chern numbers may vary independently of the underlying smooth structure.

It is generally acknowledged that none of these notions of complex structures ensure that the Chern number is solely dependent on the underlying smooth structure, let alone the topology.

Examples and Implications: The Stable Tangent Bundle Case

The stable tangent bundle case provides a concrete example where the Chern number is not a topological invariant. For a given smooth manifold, it can accommodate numerous complex structures with distinct Chern numbers.

For instance, consider a sphere of dimension (4n-2). This manifold can possess infinitely many complex structures, each characterized by a different Chern number (c_{2n-1}). This means that even if we focus on topologically identical manifolds, the choice of complex structure can significantly alter the Chern number.

This phenomenon is crucial for mathematicians working in differential geometry and topology. Understanding the interplay between the complex structure and the Chern number is essential for advancing the field. It opens up new avenues for studying manifolds and their properties, revealing how subtle changes in the complex structure can lead to drastically different Chern numbers.

Conclusion

In conclusion, the Chern number is not a topological invariant. Its value depends on the specific complex structure given to the manifold, rather than just the topology. This insight is vital for mathematicians and researchers in differential geometry and topology, as it provides a deeper understanding of the relationship between complex structures and topological invariants.

For readers interested in further exploration, consider diving into the literature on complex structures and fiber bundles. Key concepts include Chern classes, Chern numbers, and the classification of complex manifolds. This area of study is rich with opportunities for new discoveries and applications in mathematics.