Pontryagin Class: Evaluating \(Ap_1\) On 5-Manifolds

Let's dive into the fascinating world of manifolds, cohomology, and characteristic classes! Specifically, we're going to explore the evaluation of ${Ap_1}$ with the Pontryagin class on a 5-manifold. This topic sits at the intersection of differential geometry, algebraic topology, and geometric topology, making it a rich area for mathematical investigation. This article aims to break down the key concepts and address the question of whether ${\int_{M^5} Ap_1 = 0 \mod 3}$, where ${A}$ is a ${\mathbb{Z}/3}$-valued 1-cochain on a closed 5-manifold ${M^5}$.

Pontryagin Classes: A Quick Overview

First, let's get everyone on the same page about Pontryagin classes. Pontryagin classes are characteristic classes that are used to study the topology of real vector bundles. Specifically, they are defined for vector bundles over smooth manifolds and are particularly useful in understanding the structure of the tangent bundle. These classes are cohomology classes that provide information about how the manifold is 'twisted' or curved. The Pontryagin classes ${p_i}$ are defined as:

${ p_i = (-1)^i c_{2i}(E \otimes \mathbb{C}) \in H^{4i}(M; \mathbb{Z}) }$

where ${c_{2i}}$ are the Chern classes of the complexified bundle ${E \otimes \mathbb{C}}$. For our purposes, we're primarily interested in the first Pontryagin class, ${p_1}$, which lives in ${H^4(M; \mathbb{Z})}$.

The Role of ${A}$: A ${\mathbb{Z}/3}$-Valued 1-cochain

Now, let's introduce ${A}$, which is a ${\mathbb{Z}/3}$-valued 1-cochain. A 1-cochain can be thought of as a linear functional that assigns values (in this case, from ${\mathbb{Z}/3}$) to 1-chains (which are essentially paths or curves) on the manifold. The cochain ${A}$ provides a way to probe the manifold using these 1-dimensional objects. When we consider ${Ap_1}$, we're essentially pairing this 1-cochain with the first Pontryagin class. Since ${p_1 \in H^4(M; \mathbb{Z})}$, the product ${Ap_1}$ will live in a cohomology group of degree 5. Now guys, what does this all mean when we integrate over a 5-manifold?

Evaluating ${\int_{M^5} Ap_1}$

The central question is whether ${\int_{M^5} Ap_1 = 0 \mod 3}$. This integral represents the evaluation of the 5-dimensional cohomology class ${Ap_1}$ over the fundamental cycle of the closed 5-manifold ${M^5}$. To tackle this, we need to consider the properties of Pontryagin classes and the implications of working with ${\mathbb{Z}/3}$ coefficients. We need to investigate whether the specific properties of ${p_1}$ and the cochain ${A}$ force this integral to be divisible by 3.

Consider the case where ${A}$ is a coboundary, i.e., ${A = d\alpha}$ for some 0-cochain ${\alpha}$. Then, we have:

${ \int_{M^5} Ap_1 = \int_{M^5} (d\alpha) p_1 = \int_{M^5} d(\alpha p_1) - \int_{M^5} \alpha (dp_1) }$

By Stokes' theorem, ${\int_{M^5} d(\alpha p_1) = 0}$ since ${M^5}$ is closed (i.e., has no boundary). Also, since ${p_1}$ is a 4-dimensional cohomology class, ${dp_1 = 0}$ (because the exterior derivative increases the degree by one, and there are no non-trivial 5-forms to map to). Therefore,

${ \int_{M^5} Ap_1 = 0 }$

However, this argument only holds if ${A}$ is a coboundary. If ${A}$ is not a coboundary, then we need to delve deeper into the properties of ${p_1}$ and the manifold ${M^5}$. Specifically, we must explore how torsion in the cohomology of ${M^5}$ interacts with the Pontryagin class and the cochain ${A}$.

Diving Deeper: Torsion and Characteristic Classes

The presence of torsion in the cohomology of ${M^5}$ can significantly impact the evaluation of characteristic classes. Torsion elements are cohomology classes that, when multiplied by some integer, become zero. In our case, working modulo 3 suggests that we should be particularly attentive to 3-torsion in the cohomology of ${M^5}$. If ${M^5}$ has 3-torsion, it means there exist cohomology classes ${x}$ such that ${3x = 0}$, but ${x \neq 0}$.

The interaction between torsion classes and characteristic classes is a subtle and complex area. In general, the evaluation of characteristic classes on manifolds with torsion involves more sophisticated techniques from algebraic topology, such as the use of spectral sequences and the analysis of the Bockstein homomorphism.

To prove ${\int_{M^5} Ap_1 = 0 \mod 3}$, one potential approach would be to use the fact that ${p_1}$ is an integral class and to understand how it reduces modulo 3. The Bockstein homomorphism ${\beta: H^i(M; \mathbb{Z}/3) \to H^{i+1}(M; \mathbb{Z}/3)}$ associated with the short exact sequence

${ 0 \to \mathbb{Z} \xrightarrow{\times 3} \mathbb{Z} \to \mathbb{Z}/3 \to 0 }$

can provide insights into the 3-torsion in the integral cohomology. Specifically, if ${Ap_1}$ is the reduction modulo 3 of an integral class, then its integral over ${M^5}$ must be divisible by 3. However, proving that ${Ap_1}$ has this property requires a more detailed analysis of the relationship between ${A}$, ${p_1}$, and the Bockstein homomorphism.

Lemma 1: Is It True?

So, let's circle back to the original question: Is it true that ${\int_{M^5} Ap_1 = 0 \mod 3}$ when ${A}$ is a ${\mathbb{Z}/3}$-valued 1-cochain on a closed 5-manifold ${M^5}$?

Based on the above discussion, the answer is not immediately obvious. While the integral vanishes when ${A}$ is a coboundary, the general case requires a deeper understanding of the interaction between ${A}$, ${p_1}$, and the torsion in the cohomology of ${M^5}$. Proving this assertion would likely involve:

1. Understanding the properties of the first Pontryagin class ${p_1}$ modulo 3: How does ${p_1}$ behave when reduced to ${\mathbb{Z}/3}$ coefficients?
2. Analyzing the Bockstein homomorphism: How does the Bockstein homomorphism relate ${A}$ and ${p_1}$ in ${\mathbb{Z}/3}$ cohomology?
3. Using specific properties of 5-manifolds: Are there particular characteristics of 5-manifolds that simplify the analysis?

Without further information or assumptions, it's challenging to provide a definitive answer. However, the analysis above provides a framework for approaching this problem and highlights the key concepts and tools needed to tackle it.

Conclusion

Evaluating ${Ap_1}$ with the Pontryagin class on a 5-manifold is a complex problem that touches on fundamental aspects of algebraic and geometric topology. While we've explored the key concepts and provided a roadmap for addressing the question of whether ${\int_{M^5} Ap_1 = 0 \mod 3}$, a complete answer requires a deeper dive into the properties of Pontryagin classes, torsion in cohomology, and the Bockstein homomorphism. Nevertheless, understanding these concepts provides a solid foundation for further exploration in this fascinating area of mathematics. Keep exploring, you guys!