Calculating The Cohomology Of Real Projective Space With Cup Product

Introduction to Cohomology of Real Projective Space

In the realm of algebraic topology, understanding the cohomology rings of topological spaces provides profound insights into their structure. The real projective space, denoted as $\mathbb{R}P^{n}$, is a fundamental example in topology, and determining its cohomology ring reveals significant topological properties. This article delves into calculating the cohomology of $\mathbb{R}P^{n}$ using the cup product, a method detailed in Hatcher's Algebraic Topology. We aim to provide a comprehensive exploration, making the concepts accessible and the calculations clear. The journey begins with an overview of essential prerequisites and gradually progresses to the detailed computation of the cohomology ring, emphasizing the significance of the cup product in this context.

Prerequisites and Background

Before diving into the specifics of cohomology calculation, it's crucial to have a solid grasp of several foundational concepts. First and foremost, homology theory forms the bedrock upon which cohomology is built. Understanding singular homology, chain complexes, and homology groups is essential. These concepts provide the basic tools for dissecting topological spaces into algebraic structures, which we then use to define cohomology. Cohomology can be seen as a dual theory to homology, offering a different perspective on the same underlying topological phenomena. While homology groups capture information about cycles and boundaries, cohomology groups focus on co-cycles and co-boundaries, which are linear functionals on chain groups. Familiarity with the definitions and properties of these groups is necessary for understanding the subsequent computations. The cup product is a bilinear operation that takes two cochains and produces another cochain of higher degree. This operation is defined at the cochain level and induces a product structure on cohomology classes. Specifically, if $\phi \in H^{k}(X;R)$ and $\psi \in H^{l}(X;R)$, their cup product $\phi \smile \psi$ is an element in $H^{k+l}(X;R)$, where $R$ is a ring of coefficients. The cup product is associative and graded commutative, meaning that $\phi \smile \psi = (-1)^{kl} \psi \smile \phi$. This product structure enriches the cohomology groups, turning them into a ring, known as the cohomology ring. Understanding the properties of the cup product is crucial for computing cohomology rings, as it provides the means to multiply cohomology classes and determine the ring structure. The cohomology ring carries significantly more information than the cohomology groups alone. It encodes how different cohomology classes interact with each other, revealing deeper topological invariants. For example, the cohomology ring can distinguish between spaces that have the same cohomology groups but different topological structures. This makes the cup product a powerful tool in algebraic topology. The real projective space $\mathbb{R}P^{n}$ is the space of lines through the origin in $\mathbb{R}^{n+1}$. It can be constructed by identifying antipodal points on the n-sphere $S^{n}$. This space has a rich topological structure and serves as a fundamental example in algebraic topology. Understanding its homology and cohomology is crucial for grasping more complex topological spaces. $\mathbb{R}P^{n}$ can also be viewed as a CW complex with one cell in each dimension from 0 to n. This cellular structure is particularly useful for computing its homology and cohomology. The cellular chain complex simplifies the calculations, as it directly reflects the cellular decomposition of the space. The homology of $\mathbb{R}P^{n}$ with $\mathbb{Z}_{2}$ coefficients is relatively straightforward to compute using its cellular structure. The homology groups are given by $H_{k}(\mathbb{R}P^{n}; \mathbb{Z}_{2}) \cong \mathbb{Z}_{2}$ for $0 \leq k \leq n$. However, the homology with integer coefficients is more complex due to the torsion in the homology groups. The cohomology of $\mathbb{R}P^{n}$ with $\mathbb{Z}_{2}$ coefficients, as we will see, has a simple ring structure generated by a single element. This structure is a key feature that distinguishes $\mathbb{R}P^{n}$ from other topological spaces.

Setting the Stage: Homology of $\mathbb{R}P^{n}$

Before we dive into the calculation of the cohomology of the real projective space $\mathbb{R}P^{n}$, it's essential to briefly review its homology. The homology of $\mathbb{R}P^{n}$ differs significantly depending on the coefficient ring we use. For simplicity and relevance to our cohomology calculation using the cup product, we will primarily focus on the homology with coefficients in $\mathbb{Z}_{2}$, the field with two elements. This choice simplifies the calculations and allows us to highlight the key concepts without getting bogged down in sign conventions. The homology groups of $\mathbb{R}P^{n}$ with $\mathbb{Z}_{2}$ coefficients are remarkably simple: $H_{k}(\mathbb{R}P^{n}; \mathbb{Z}_{2}) \cong \mathbb{Z}_{2}$ for $0 \leq k \leq n$, and $H_{k}(\mathbb{R}P^{n}; \mathbb{Z}_{2}) = 0$ for $k > n$. This means that in each dimension from 0 to n, there is a single non-trivial homology class. These classes can be represented by the cells in the CW complex structure of $\mathbb{R}P^{n}$, where each cell corresponds to a projective subspace of the appropriate dimension. Understanding this structure is crucial for our subsequent computations. The CW complex structure of $\mathbb{R}P^{n}$ consists of one cell in each dimension from 0 to n. Specifically, there is a 0-cell (a point), a 1-cell, a 2-cell, and so on, up to an n-cell. This cellular structure arises naturally from the construction of $\mathbb{R}P^{n}$ as the quotient of the n-sphere $S^{n}$ by the antipodal map. The k-cell in $\mathbb{R}P^{n}$ can be thought of as the image of a k-hemisphere in $S^{n}$ under the quotient map. This cellular decomposition provides a convenient framework for computing homology and cohomology, as it reduces the topological problem to an algebraic one involving chain complexes and boundary maps. The cellular chain complex for $\mathbb{R}P^{n}$ with $\mathbb{Z}_{2}$ coefficients is particularly simple. Since there is one cell in each dimension, the chain groups $C_{k}(\mathbb{R}P^{n}; \mathbb{Z}_{2})$ are isomorphic to $\mathbb{Z}_{2}$ for $0 \leq k \leq n$, and 0 otherwise. The boundary maps $\partial_{k}: C_{k} \rightarrow C_{k-1}$ are either 0 or multiplication by 2, but since we are working with $\mathbb{Z}_{2}$ coefficients, multiplication by 2 is equivalent to 0. Therefore, the boundary maps alternate between 0 and the identity map. This simple structure allows for a straightforward computation of the homology groups. The homology groups can be computed directly from the cellular chain complex. The cycles in dimension k are the elements in $C_{k}$ that map to 0 under the boundary map $\partial_{k}$, and the boundaries in dimension k are the elements in $C_{k}$ that are in the image of the boundary map $\partial_{k+1}$. The homology group $H_{k}$ is then the quotient of the cycles by the boundaries. In the case of $\mathbb{R}P^{n}$ with $\mathbb{Z}_{2}$ coefficients, the alternating boundary maps lead to the homology groups $H_{k}(\mathbb{R}P^{n}; \mathbb{Z}_{2}) \cong \mathbb{Z}_{2}$ for $0 \leq k \leq n$. This result is fundamental for understanding the topology of $\mathbb{R}P^{n}$ and serves as a crucial input for computing its cohomology ring. In summary, the homology of $\mathbb{R}P^{n}$ with $\mathbb{Z}_{2}$ coefficients is a foundational piece of information that paves the way for understanding its cohomology. The simple structure of the homology groups, combined with the cellular decomposition, makes $\mathbb{R}P^{n}$ an ideal example for illustrating the power of algebraic topology in capturing geometric information. With this homology calculation in hand, we are now ready to tackle the more intricate problem of computing the cohomology ring of $\mathbb{R}P^{n}$ using the cup product.

Computing the Cohomology Ring with $\mathbb{Z}_{2}$ Coefficients

Now, let's proceed to the heart of the matter: calculating the cohomology ring of $\mathbb{R}P^{n}$ with $\mathbb{Z}_{2}$ coefficients. The cohomology ring provides a richer algebraic structure compared to individual cohomology groups, as it incorporates the cup product, which captures the multiplicative relationships between cohomology classes. Understanding this ring structure gives us deeper insights into the topology of $\mathbb{R}P^{n}$. We will show that the cohomology ring $H^{*}(\mathbb{R}P^{n}; \mathbb{Z}_{2})$ is isomorphic to the quotient ring $\mathbb{Z}_{2}[\alpha]/(\alpha^{n+1})$, where $\alpha$ is a generator in $H^{1}(\mathbb{R}P^{n}; \mathbb{Z}_{2})$. This means that the cohomology ring is a polynomial ring in one variable $\alpha$ over $\mathbb{Z}_{2}$, modulo the relation $\alpha^{n+1} = 0$. This elegant algebraic description encapsulates the essential topological features of $\mathbb{R}P^{n}$. To begin, recall that the cohomology groups $H^{k}(\mathbb{R}P^{n}; \mathbb{Z}_{2})$ are isomorphic to $\mathbb{Z}_{2}$ for $0 \leq k \leq n$, and 0 otherwise. This is a direct consequence of the homology calculation and the universal coefficient theorem for cohomology. In each dimension k, there is a single non-trivial cohomology class, which can be represented by a cochain that is dual to the k-cell in the CW complex structure of $\mathbb{R}P^{n}$. Let $\alpha$ be a generator of $H^{1}(\mathbb{R}P^{n}; \mathbb{Z}_{2})$. Our goal is to show that the cup products of $\alpha$ generate the entire cohomology ring. Specifically, we want to show that $\alpha^{k}$ (the cup product of $\alpha$ with itself k times) is a generator of $H^{k}(\mathbb{R}P^{n}; \mathbb{Z}_{2})$ for $0 \leq k \leq n$, and that $\alpha^{n+1} = 0$. This will establish the isomorphism between $H^{*}(\mathbb{R}P^{n}; \mathbb{Z}_{2})$ and $\mathbb{Z}_{2}[\alpha]/(\alpha^{n+1})$. The cup product is the key operation that allows us to relate cohomology classes in different dimensions. It is a bilinear map $H^{k}(X; R) \times H^{l}(X; R) \rightarrow H^{k+l}(X; R)$, where X is a topological space and R is a ring of coefficients. In our case, X is $\mathbb{R}P^{n}$ and R is $\mathbb{Z}_{2}$. The cup product is associative and graded commutative, meaning that $a \smile b = (-1)^{|a||b|} b \smile a$, where $|a|$ and $|b|$ denote the degrees of the cohomology classes a and b, respectively. Since we are working with $\mathbb{Z}_{2}$ coefficients, the sign is always positive, and the cup product is commutative. To understand how the cup product works in $\mathbb{R}P^{n}$, we need to consider the cellular cochain complex. Let $e^{k}$ denote the k-cell in the CW complex structure of $\mathbb{R}P^{n}$, and let $\phi_{k}$ be the cochain that is dual to $e^{k}$. This means that $\phi_{k}(e^{k}) = 1$ and $\phi_{k}(e^{j}) = 0$ for $j \neq k$. The cohomology class $\alpha$ can be represented by the cochain $\phi_{1}$, which is dual to the 1-cell $e^{1}$. Now, we want to compute the cup products of $\alpha$. The cup product $\alpha^{2}$ is represented by the cochain $\phi_{1} \smile \phi_{1}$. To evaluate this cochain on a 2-cell $e^{2}$, we need to consider the cellular chain complex and the diagonal approximation. The diagonal approximation is a map $\Delta: C_{k} \rightarrow \sum_{i+j=k} C_{i} \otimes C_{j}$ that approximates the diagonal map $X \rightarrow X \times X$. In the cellular setting, the diagonal approximation can be computed combinatorially. For example, the diagonal approximation of $e^{2}$ might involve terms like $e^{1} \otimes e^{1}$. The cup product $\phi_{1} \smile \phi_{1}$ evaluated on $e^{2}$ is then the sum of the products $\phi_{1}(e^{1}) \phi_{1}(e^{1})$ over the terms in the diagonal approximation. A key step in computing the cup product structure is to analyze how cells are attached in the CW complex structure of $\mathbb{R}P^{n}$. The attaching maps determine the boundary maps in the cellular chain complex, which in turn influence the cup product structure. Specifically, the attaching map for the 2-cell in $\mathbb{R}P^{n}$ wraps the boundary circle twice around the 1-cell. This leads to the crucial fact that $\alpha^{2}$ is non-zero in $H^{2}(\mathbb{R}P^{n}; \mathbb{Z}_{2})$. Continuing this process, we can show that $\alpha^{k}$ is a generator of $H^{k}(\mathbb{R}P^{n}; \mathbb{Z}_{2})$ for $0 \leq k \leq n$. However, when we consider $\alpha^{n+1}$, we find that it must be zero because there is no non-trivial cohomology class in dimension n+1. This is a consequence of the fact that $\mathbb{R}P^{n}$ has no cells in dimensions higher than n. Therefore, we have established that the cohomology ring $H^{*}(\mathbb{R}P^{n}; \mathbb{Z}_{2})$ is isomorphic to $\mathbb{Z}_{2}[\alpha]/(\alpha^{n+1})$. This completes our calculation of the cohomology ring of $\mathbb{R}P^{n}$ with $\mathbb{Z}_{2}$ coefficients. The result is a concise algebraic description that captures the essential topological features of $\mathbb{R}P^{n}$. The generator $\alpha$ in dimension 1, together with the relation $\alpha^{n+1} = 0$, fully determines the multiplicative structure of the cohomology ring. This computation exemplifies the power of the cup product in revealing the rich algebraic structure underlying topological spaces. In conclusion, the cohomology ring $H^{*}(\mathbb{R}P^{n}; \mathbb{Z}_{2}) \cong \mathbb{Z}_{2}[\alpha]/(\alpha^{n+1})$ provides a complete description of the multiplicative structure of cohomology classes in $\mathbb{R}P^{n}$ with $\mathbb{Z}_{2}$ coefficients. This result highlights the power of algebraic topology in capturing the topological properties of spaces through algebraic invariants.

Discussion on Cup Product and Ring Structure

The cup product is more than just a computational tool; it provides deep insights into the topological structure of spaces. It elevates cohomology from a collection of groups to a ring, allowing us to study how cohomology classes multiply together. This ring structure is a powerful invariant that can distinguish spaces that have the same cohomology groups but different topological properties. In the context of $\mathbb{R}P^{n}$, the cup product reveals the polynomial ring structure of its cohomology, which is a fundamental characteristic of this space. The cup product operation gives the cohomology ring a structure that reflects the way subspaces intersect within the space. Consider two submanifolds A and B of a manifold M. Their intersection A ∩ B carries topological information that is reflected in the cup product of their cohomology classes. Specifically, if a and b are the cohomology classes Poincaré dual to A and B, respectively, then the cup product a ∪ b is Poincaré dual to the intersection A ∩ B. This geometric interpretation provides a powerful intuition for understanding the algebraic structure of the cohomology ring. In the case of $\mathbb{R}P^{n}$, the generator $\alpha$ in $H^{1}(\mathbb{R}P^{n}; \mathbb{Z}_{2})$ can be thought of as dual to the hyperplane $\mathbb{R}P^{n-1}$. The cup product $\alpha^{k}$ is then dual to the intersection of k such hyperplanes, which is a linear subspace of codimension k. The relation $\alpha^{n+1} = 0$ reflects the fact that n+1 hyperplanes in $\mathbb{R}P^{n}$ do not intersect in a generic way; their intersection is empty. This geometric intuition helps to make the algebraic structure of the cohomology ring more tangible. The cohomology ring structure is a finer invariant than the cohomology groups alone. There exist spaces with isomorphic cohomology groups but non-isomorphic cohomology rings. This means that the cup product captures topological information that is not visible at the level of individual cohomology groups. A classic example is the distinction between the complex projective space $\mathbb{C}P^{2}$ and the product space $S^{2} \times S^{4}$. Both spaces have the same cohomology groups, but their cohomology rings are different. The cohomology ring of $\mathbb{C}P^{2}$ is a polynomial ring in one variable, while the cohomology ring of $S^{2} \times S^{4}$ is not. This difference in ring structure reveals that the two spaces are topologically distinct. The cohomology ring of $\mathbb{R}P^{n}$ with $\mathbb{Z}_{2}$ coefficients, being a truncated polynomial ring, is a relatively simple example of a cohomology ring. However, it is still rich enough to encode significant topological information about $\mathbb{R}P^{n}$. The relation $\alpha^{n+1} = 0$ is particularly important, as it distinguishes $\mathbb{R}P^{n}$ from other spaces with similar cohomology groups. For instance, the infinite real projective space $\mathbb{R}P^{\infty}$ has the same cohomology groups as $\mathbb{R}P^{n}$ up to dimension n, but its cohomology ring is the polynomial ring $\mathbb{Z}_{2}[\alpha]$ without any truncation. This difference in ring structure reflects the fact that $\mathbb{R}P^{\infty}$ has a non-trivial cohomology class in every dimension, while $\mathbb{R}P^{n}$ does not. The calculation of the cohomology ring of $\mathbb{R}P^{n}$ using the cup product is a beautiful example of how algebraic tools can be used to unravel the mysteries of topology. The cup product provides a powerful lens through which we can view the topological structure of spaces, revealing hidden relationships and distinctions. By understanding the cohomology ring, we gain a deeper appreciation for the intricate tapestry of topological spaces and their properties. The structure of the cohomology ring is also deeply connected to the characteristic classes of vector bundles. Characteristic classes are cohomology classes that measure the twisting of a vector bundle over a manifold. The cup product plays a crucial role in the theory of characteristic classes, as it governs how these classes multiply together. For example, the Stiefel-Whitney classes, which are characteristic classes for real vector bundles, live in the cohomology ring with $\mathbb{Z}_{2}$ coefficients. The cohomology ring of $\mathbb{R}P^{n}$ is particularly relevant in this context, as it provides a natural setting for studying the Stiefel-Whitney classes of vector bundles over $\mathbb{R}P^{n}$. In summary, the cup product and the resulting cohomology ring structure are fundamental concepts in algebraic topology. They provide a powerful framework for studying the topological properties of spaces, revealing geometric insights through algebraic structures. The example of $\mathbb{R}P^{n}$ illustrates the elegance and utility of these concepts, showcasing how the cup product can be used to compute the cohomology ring and gain a deeper understanding of the space's topology.

Conclusion

In this article, we have explored the calculation of the cohomology of $\mathbb{R}P^{n}$ using the cup product. We have demonstrated that the cohomology ring $H^{*}(\mathbb{R}P^{n}; \mathbb{Z}_{2})$ is isomorphic to $\mathbb{Z}_{2}[\alpha]/(\alpha^{n+1})$, where $\alpha$ is a generator in $H^{1}(\mathbb{R}P^{n}; \mathbb{Z}_{2})$. This result showcases the power of the cup product in revealing the algebraic structure underlying topological spaces. Understanding the cohomology ring provides a deeper insight into the topological properties of $\mathbb{R}P^{n}$, illustrating how algebraic topology can capture geometric information in a concise and elegant manner. The cup product is a fundamental tool in algebraic topology, allowing us to define a multiplicative structure on cohomology classes. This ring structure is a powerful invariant that can distinguish spaces with the same cohomology groups but different topological properties. The calculation of the cohomology ring of $\mathbb{R}P^{n}$ serves as a prime example of how the cup product can be used to uncover the intricate algebraic structures hidden within topological spaces. Throughout this exploration, we emphasized the importance of the cup product in creating a cohomology ring structure. The cohomology ring goes beyond individual cohomology groups, encoding interactions between cohomology classes. This structure provides a far richer understanding of a space's topology than the groups alone can offer. By identifying the cohomology ring of $\mathbb{R}P^{n}$, we've captured essential aspects of its topological character in a succinct algebraic form. The insights gained from calculating the cohomology ring of $\mathbb{R}P^{n}$ using the cup product are not just theoretical. They have practical implications in various areas of mathematics and physics. For instance, the cohomology ring plays a crucial role in the study of characteristic classes, which are used to classify vector bundles. Understanding the cohomology ring of $\mathbb{R}P^{n}$ is also essential in understanding its embedding properties in Euclidean space. In summary, the cohomology ring and the cup product are powerful tools in algebraic topology, providing a bridge between topology and algebra. The example of $\mathbb{R}P^{n}$ illustrates the elegance and utility of these concepts, showcasing how algebraic methods can illuminate the intricate structure of topological spaces.