Atiyah-Hirzebruch Spectral Sequence

Definition

The Atiyah-Hirzebruch spectral sequence is a powerful tool in algebraic topology that provides a systematic way to compute generalized cohomology theories, particularly K-theory. It arises from the study of vector bundles and their associated characteristic classes, leading to a connection between topology and algebraic structures. This spectral sequence is especially useful in computing higher K-theories, revealing deep insights into the structure of vector bundles over topological spaces.

5 Must Know Facts For Your Next Test

1. The Atiyah-Hirzebruch spectral sequence arises from the filtration of the total space of a fibration, allowing computations in K-theory to be reduced to simpler components.
2. This spectral sequence provides a way to relate stable homotopy groups with cohomology theories, making it a bridge between algebraic topology and algebraic geometry.
3. It converges to the group of global sections of a certain sheaf associated with a fibration, giving rise to explicit computations in higher K-theory.
4. The first page of the spectral sequence consists of the homology groups of the space associated with the fibers of the fibration, providing a starting point for computations.
5. The spectral sequence is named after Michael Atiyah and Friedrich Hirzebruch, who developed it in the context of understanding the topological properties of complex vector bundles.

Review Questions

• How does the Atiyah-Hirzebruch spectral sequence facilitate computations in higher K-theory?
  • The Atiyah-Hirzebruch spectral sequence allows for computations in higher K-theory by breaking down complex vector bundles into simpler components via filtration. It organizes the information from the total space of a fibration and provides a systematic method for relating these components to K-theory classes. As a result, this spectral sequence serves as a crucial tool for understanding relationships between different topological spaces and their associated vector bundles.
• Discuss the significance of characteristic classes within the framework of the Atiyah-Hirzebruch spectral sequence.
  • Characteristic classes play a critical role in the Atiyah-Hirzebruch spectral sequence by serving as invariants that represent vector bundles in cohomology theories. Through this spectral sequence, one can relate these classes to various cohomology groups, allowing for insights into the topology of vector bundles. The relationship between characteristic classes and K-theory helps reveal deeper structures within algebraic topology and highlights how these algebraic invariants can inform geometric properties.
• Evaluate the impact of the Atiyah-Hirzebruch spectral sequence on modern algebraic topology and its applications.
  • The Atiyah-Hirzebruch spectral sequence has had a profound impact on modern algebraic topology by providing a robust framework for computing invariants related to vector bundles and their higher K-theories. Its applications extend beyond pure mathematics into areas like string theory and theoretical physics, where understanding the topological properties of spaces is essential. The ability to compute and relate different cohomology theories through this spectral sequence has led to new discoveries and deeper insights across various fields, illustrating its lasting significance in contemporary mathematics.

Related terms

K-theory: A branch of mathematics that studies vector bundles and their classes through the lens of algebraic topology, providing invariants that classify bundles up to stable isomorphism.

Cohomology: A mathematical concept that assigns algebraic invariants to topological spaces, helping in the classification of spaces based on their topological properties.

Characteristic Classes: Invariants associated with vector bundles that provide information about their geometry and topology, allowing for the comparison of different bundles.