Atiyah-Hirzebruch Spectral Sequence

Definition

The Atiyah-Hirzebruch spectral sequence is a powerful tool in algebraic topology that provides a way to compute the K-theory of a space from its cohomology. It connects the geometry of vector bundles to topological invariants, allowing for the classification of vector bundles through the lens of K-theory and characteristic classes.

5 Must Know Facts For Your Next Test

1. The Atiyah-Hirzebruch spectral sequence arises when considering the cohomology of a space with coefficients in K-theory, capturing information about vector bundles over that space.
2. This spectral sequence is particularly useful for computing K-groups, as it allows one to systematically derive K-theory from more elementary cohomological data.
3. The E2-page of the spectral sequence involves the cohomology groups of the base space, while the differentials encode information about how these groups interact with the structure of vector bundles.
4. The convergence of this spectral sequence is crucial for obtaining results about the relationships between various types of K-groups and cohomology theories.
5. It plays a significant role in studying characteristic classes, revealing deeper connections between algebraic topology and differential geometry.

Review Questions

• How does the Atiyah-Hirzebruch spectral sequence facilitate the computation of K-groups from cohomology?
  • The Atiyah-Hirzebruch spectral sequence serves as a bridge between cohomology and K-theory by starting with the cohomology of a space and producing a sequence that converges to the K-groups. The initial stages of the sequence involve the E2-page, which is derived from cohomology groups with coefficients in K-theory. As one progresses through the differentials, the structure and interactions of these cohomological invariants reveal how they contribute to K-groups, allowing for systematic computation.
• Discuss the significance of the Thom Isomorphism theorem in relation to the Atiyah-Hirzebruch spectral sequence.
  • The Thom Isomorphism theorem is significant because it establishes an isomorphism between the K-theory of a space and its cohomology, which directly informs the structure of the Atiyah-Hirzebruch spectral sequence. This relationship allows us to understand how vector bundles can be classified through their associated characteristic classes and enables the spectral sequence to effectively connect these ideas. Consequently, insights gained from one area enrich our understanding of the other, highlighting their interdependence.
• Evaluate how the Atiyah-Hirzebruch spectral sequence impacts the study of characteristic classes and their applications in various fields like arithmetic geometry.
  • The Atiyah-Hirzebruch spectral sequence profoundly impacts our understanding of characteristic classes by providing a systematic method to derive them from K-theory. This connection enhances our ability to compute these classes and explore their implications across diverse areas such as arithmetic geometry. For instance, characteristic classes can give insights into the properties of vector bundles over algebraic varieties, linking geometric intuition with algebraic structures, and thereby fostering applications that bridge topology and number theory.

Related terms

K-Theory: A branch of mathematics that studies vector bundles and their classifications, providing a way to associate algebraic objects to topological spaces.

Cohomology: A mathematical concept that provides a way to classify topological spaces using algebraic invariants, often used in conjunction with homology.

Thom Isomorphism: A result in algebraic topology that establishes an isomorphism between the K-theory of a space and its cohomology, specifically related to vector bundles.