Atiyah-Hirzebruch spectral sequence

Definition

The Atiyah-Hirzebruch spectral sequence is a powerful tool in algebraic topology and algebraic K-theory that provides a way to compute the K-groups of a space by relating them to the homology of that space. This sequence connects various mathematical concepts, allowing for deeper insights and computations, particularly in the study of vector bundles and characteristic classes.

5 Must Know Facts For Your Next Test

1. The Atiyah-Hirzebruch spectral sequence is specifically constructed to relate the homology groups of a space with its K-groups, facilitating easier calculations.
2. This spectral sequence arises from the filtration of a topological space via a CW complex structure, connecting algebraic properties with topological features.
3. Convergence of the spectral sequence provides significant results in computations of K-theory, especially in terms of understanding how vector bundles behave over spaces.
4. The first page of the spectral sequence typically involves homology groups, while subsequent pages refine these calculations until they converge on the desired K-groups.
5. Applications of the Atiyah-Hirzebruch spectral sequence include its use in proving results such as the Bott periodicity theorem and understanding the structure of K-groups in various contexts.

Review Questions

• How does the Atiyah-Hirzebruch spectral sequence connect homology groups with K-groups in algebraic K-theory?
  • The Atiyah-Hirzebruch spectral sequence organizes computations by filtering a topological space into CW complexes, which allows for the analysis of its homology groups. These homology groups serve as the initial data on the first page of the spectral sequence. As this sequence converges, it refines these calculations and reveals insights into the associated K-groups, effectively bridging topological and algebraic concepts.
• Discuss the implications of Bott periodicity within the context of the Atiyah-Hirzebruch spectral sequence and its applications.
  • Bott periodicity is crucial to understanding the behavior of K-theory, as it implies that K-groups exhibit periodic properties when viewed through the lens of vector bundles. The Atiyah-Hirzebruch spectral sequence incorporates this periodicity by showing how vector bundles can be related across different dimensions. This connection simplifies computations and allows mathematicians to predict relationships between various K-groups based on their periodic nature.
• Evaluate how the applications of the Atiyah-Hirzebruch spectral sequence extend beyond pure algebraic K-theory to other areas of mathematics.
  • The Atiyah-Hirzebruch spectral sequence is not only a cornerstone in algebraic K-theory but also plays an important role in algebraic topology and characteristic classes. Its ability to compute K-groups effectively has implications for other mathematical fields like differential geometry, where it helps analyze vector bundles over manifolds. Additionally, its connections to homological algebra and representation theory allow it to influence various branches, showcasing its significance in broader mathematical landscapes.

Related terms

K-theory: A branch of mathematics that studies vector bundles and their classifications using algebraic methods, often represented through K-groups.

Spectral sequence: A computational tool that helps organize and compute homology groups, consisting of a sequence of pages that converge to a desired object.

Bott periodicity: A theorem stating that the K-theory groups exhibit periodicity, specifically in relation to the stable rank of vector bundles over complex manifolds.