insight - Scientific Computing - # Equivariant Cohomology

The Calculation of the C2-Equivariant Ordinary Cohomology of the Classifying Space BT^2 with Extended Grading

Q: How does the structure of the C2-equivariant ordinary cohomology of BT^n change as n increases beyond 2?

While the provided context focuses on the case for $n = 2$, extrapolating to higher values of $n$ suggests a significant increase in complexity for the C2-equivariant ordinary cohomology of $BT^n$. Here's a breakdown of the anticipated changes: Exponential growth in generators: For $BT^2$, we have eight generators related to Euler classes ($cω_i$, $cχω_i$, $cω_i⊗ω_j$, $cχω_i⊗ω_j$) and four generators corresponding to fixed-set components ($ζ_{ij}$). As $n$ increases, the number of possible tensor products of the tautological bundles grows exponentially, leading to a similar increase in the number of Euler class generators. Additionally, the fixed-set $(BT^n)^{C_2}$ will have $2^n$ components, necessitating an equal number of $ζ$ generators. Increasingly intricate relations: The relations defining the ideal $I$ in the cohomology ring become more numerous and complex. For $BT^2$, we already observe relations intertwining the $ζ_{ij}$ with Euler classes of tensor products. This intertwining is expected to escalate with higher $n$ as we incorporate Euler classes of more complicated tensor product bundles. Challenges in finding a concise closed form: Expressing the cohomology ring in a compact, closed form like Theorem A becomes increasingly difficult. The relations will likely involve intricate interactions between various generators, making it challenging to find a manageable presentation. Computational burden: The computational effort required to determine the cohomology ring and its basis grows substantially. The diamond lemma approach, while effective, becomes significantly more tedious with a larger set of generators and relations. In summary, while the general principles outlined in the provided context would still apply, calculating the C2-equivariant ordinary cohomology of $BT^n$ for $n > 2$ becomes significantly more challenging due to the explosion in the number of generators and the increasing complexity of the relations.

Q: Could the use of alternative cohomology theories, such as K-theory or cobordism, potentially simplify the calculation or reveal different aspects of the structure of BT^2?

Yes, employing alternative cohomology theories like K-theory or cobordism could offer a different perspective and potentially simplify certain aspects of the calculation or highlight different structural features of $BT^2$. Here's a comparison: K-Theory: Potential Simplifications: K-theory is known for its simpler behavior with respect to tensor products. The lack of a formal group law in equivariant ordinary cohomology complicates the handling of tensor products, as seen in the need for $cω_1⊗ω_2$. Equivariant K-theory, with its direct sum representation of tensor products, might offer a more streamlined approach. Revealing Different Structures: K-theory could reveal information about the representation theory of $C_2$ acting on the fibers of vector bundles over $BT^2$. This information is not directly captured by ordinary cohomology. Cobordism: Geometric Insight: Equivariant cobordism provides a more geometric perspective, relating the cohomology of $BT^2$ to the classification of manifolds with $C_2$-actions. This could offer insights into the geometric significance of the generators and relations. Computational Challenges: Cobordism calculations can be quite involved, and it's not guaranteed that they would be simpler than ordinary cohomology in this specific case. Overall: While K-theory and cobordism might provide valuable insights and potential simplifications, they also come with their own computational challenges. The choice of the "best" cohomology theory depends on the specific aspects of $BT^2$ one wishes to study and the trade-off between computational complexity and the richness of the information obtained.

Core Concepts

This paper calculates the C2-equivariant ordinary cohomology of BT^2, the classifying space for C2-equivariant complex 2-plane bundles, using an extended grading to capture a more natural set of generators, including the Euler class of the tensor product of line bundles.

Abstract

Bibliographic Information: Costenoble, S. R., & Hudson, T. (2024). THE C2-EQUIVARIANT ORDINARY COHOMOLOGY OF BT 2. arXiv preprint arXiv:2411.06470.

Research Objective: This research paper aims to calculate and analyze the C2-equivariant ordinary cohomology of BT^2, the classifying space for C2-equivariant complex 2-plane bundles, using an extended grading system. The authors seek to understand the structure of this cohomology ring and its relationship to the cohomologies of related spaces, such as BT^1 and BU(2).

Methodology: The authors employ techniques from algebraic topology, specifically equivariant cohomology theory. They utilize an extended grading based on the representation ring RO(ΠBT^2) to capture a more comprehensive set of characteristic classes. The computation involves analyzing the restriction of the cohomology to fixed points and comparing it to the cohomology of related spaces.

Key Findings: The paper presents a complete calculation of the C2-equivariant ordinary cohomology of BT^2 with extended grading. The authors identify a set of generators and relations for this cohomology ring, highlighting the crucial role of the Euler class of the tensor product of line bundles. They demonstrate that the cohomology of BT^2 is a free module over the cohomology of a point and provide explicit bases for this module structure.

Main Conclusions: The study reveals that the relationship between the equivariant cohomologies of BT^2 and BU(2) is more intricate than in the nonequivariant case. While the cohomology of BU(2) can be identified with the symmetric part of the cohomology of BT^2 under certain restrictions, the authors show that the equivariant cohomology of BT^2 is not free over the cohomology of BU(2).

Significance: This research contributes significantly to the field of equivariant homotopy theory by providing a detailed analysis of the cohomology of a fundamental classifying space. The explicit calculations and structural insights gained from this study have implications for understanding equivariant characteristic classes and their applications in other areas of mathematics and physics.

Limitations and Future Research: The authors acknowledge that the extended grading system used in their calculation, while necessary for capturing the essential features of the cohomology, leads to a more complex description than in the nonequivariant case. Future research could explore alternative approaches to simplifying the presentation of these results or investigate the cohomology of BT^n for higher values of n. Additionally, the authors' findings on the relationship between the cohomologies of BT^2 and BU(2) suggest further avenues for investigating the equivariant topology of these and related classifying spaces.

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The $C_2$-equivariant ordinary cohomology of $BT^2$

by Steven R. Co... at arxiv.org 11-12-2024

https://arxiv.org/pdf/2411.06470.pdf

The $C_2$-equivariant ordinary cohomology of $BT^2$

Deeper Inquiries

How does the structure of the C2-equivariant ordinary cohomology of BT^n change as n increases beyond 2?

While the provided context focuses on the case for $n = 2$, extrapolating to higher values of $n$ suggests a significant increase in complexity for the C2-equivariant ordinary cohomology of $BT^n$.
Here's a breakdown of the anticipated changes:

Exponential growth in generators: For $BT^2$, we have eight generators related to Euler classes ($cω_i$, $cχω_i$, $cω_i⊗ω_j$, $cχω_i⊗ω_j$) and four generators corresponding to fixed-set components ($ζ_{ij}$). As $n$ increases, the number of possible tensor products of the tautological bundles grows exponentially, leading to a similar increase in the number of Euler class generators. Additionally, the fixed-set $(BT^n)^{C_2}$ will have $2^n$ components, necessitating an equal number of $ζ$ generators.

Increasingly intricate relations: The relations defining the ideal $I$ in the cohomology ring become more numerous and complex.  For $BT^2$, we already observe relations intertwining the $ζ_{ij}$ with Euler classes of tensor products. This intertwining is expected to escalate with higher $n$ as we incorporate Euler classes of more complicated tensor product bundles.

Challenges in finding a concise closed form:  Expressing the cohomology ring in a compact, closed form like Theorem A becomes increasingly difficult. The relations will likely involve intricate interactions between various generators, making it challenging to find a manageable presentation.

Computational burden:  The computational effort required to determine the cohomology ring and its basis grows substantially. The diamond lemma approach, while effective, becomes significantly more tedious with a larger set of generators and relations.
In summary, while the general principles outlined in the provided context would still apply, calculating the C2-equivariant ordinary cohomology of $BT^n$ for $n > 2$ becomes significantly more challenging due to the explosion in the number of generators and the increasing complexity of the relations.

Could the use of alternative cohomology theories, such as K-theory or cobordism, potentially simplify the calculation or reveal different aspects of the structure of BT^2?

Yes, employing alternative cohomology theories like K-theory or cobordism could offer a different perspective and potentially simplify certain aspects of the calculation or highlight different structural features of $BT^2$. Here's a comparison:
K-Theory:

Potential Simplifications: K-theory is known for its simpler behavior with respect to tensor products. The lack of a formal group law in equivariant ordinary cohomology complicates the handling of tensor products, as seen in the need for $cω_1⊗ω_2$.  Equivariant K-theory, with its direct sum representation of tensor products, might offer a more streamlined approach.
Revealing Different Structures: K-theory could reveal information about the representation theory of $C_2$ acting on the fibers of vector bundles over $BT^2$. This information is not directly captured by ordinary cohomology.
Cobordism:

Geometric Insight: Equivariant cobordism provides a more geometric perspective, relating the cohomology of $BT^2$ to the classification of manifolds with $C_2$-actions. This could offer insights into the geometric significance of the generators and relations.
Computational Challenges: Cobordism calculations can be quite involved, and it's not guaranteed that they would be simpler than ordinary cohomology in this specific case.
Overall:
While K-theory and cobordism might provide valuable insights and potential simplifications, they also come with their own computational challenges. The choice of the "best" cohomology theory depends on the specific aspects of $BT^2$ one wishes to study and the trade-off between computational complexity and the richness of the information obtained.

What are the implications of the lack of a formal group law in equivariant ordinary cohomology for the study of geometric objects with symmetries?

The absence of a formal group law in equivariant ordinary cohomology has profound implications for studying geometric objects with symmetries. It signifies a departure from the familiar additive behavior of characteristic classes observed in the non-equivariant setting and introduces significant complexities:

Limited Computational Tools: The Whitney sum formula, a cornerstone in non-equivariant characteristic class computations, relies on the additive formal group law. Its absence in the equivariant setting restricts our ability to directly compute characteristic classes of direct sums of vector bundles.

Increased Complexity of Relations: As seen in the context, the relations between Euler classes of tensor products become more intricate. This complexity arises because we can no longer express the Euler class of a tensor product simply as a sum of Euler classes of the factors.

Obscured Geometric Information: The lack of a formal group law can obscure geometric information encoded in the cohomology ring.  Relationships between characteristic classes that are straightforward in the non-equivariant case become more complex and harder to interpret geometrically.

Need for Alternative Approaches:  The absence of a formal group law necessitates the development of alternative techniques for studying equivariant characteristic classes. These might include:

Using different cohomology theories: As discussed earlier, K-theory or cobordism might offer more manageable frameworks for certain computations.
Developing specialized tools: New computational tools and techniques tailored to the equivariant setting are needed to overcome the limitations imposed by the lack of a formal group law.

Deeper Understanding of Symmetries: While posing challenges, the lack of a formal group law also reflects the richer structure inherent in equivariant cohomology. It underscores the importance of carefully considering the role of symmetries when studying geometric objects and motivates the search for new ways to extract geometric and topological information from equivariant characteristic classes.
In conclusion, the absence of a formal group law in equivariant ordinary cohomology presents both challenges and opportunities. It demands new approaches and a deeper understanding of the interplay between geometry, topology, and symmetry.