Long Exact Sequence of Cohomology

Introduction

Cohomology theories provide a powerful tool for studying the topological properties of spaces by translating algebraic problems into homological ones. Among the most fundamental constructions in cohomology is the long exact sequence of cohomology, a sequence that relates the cohomology groups of a space to those of its subspaces and quotient spaces. This sequence is derived from short exact sequences of sheaves and encapsulates the relationship between cohomology groups through the lens of exactness. The long exact sequence of cohomology is a cornerstone of algebraic topology, enabling the computation of cohomology groups for complex spaces using known data from simpler ones.

The long exact sequence of cohomology is particularly significant because it allows for the comparison of cohomology groups across different spaces, even when the spaces are not homotopy equivalent. This sequence is deeply connected to the concept of derived functors, which arise from the failure of certain functors to be exact. In particular, the long exact sequence of cohomology is constructed using the derived functors of the global section functor, which is a fundamental operation in sheaf theory. The sequence is also closely tied to the study of fibrations and fiber sequences, providing a structured way to analyze the homological properties of spaces.

Construction and Derivation

The long exact sequence of cohomology is constructed from a short exact sequence of sheaves, which is a sequence of sheaves $ \mathcal{F} \rightarrow \mathcal{G} \rightarrow \mathcal{H} $ satisfying $ \mathcal{F} \rightarrow \mathcal{G} \rightarrow \mathcal{H} \rightarrow 0 $. This sequence is then used to derive the long exact sequence of cohomology groups. The key idea is that cohomology groups are invariant under homotopy equivalence, and the long exact sequence captures the interplay between the cohomology of the domain and the codomain of the short exact sequence.

The derivation of the long exact sequence relies on the fact that cohomology groups are computed using the global sections of sheaves. For a short exact sequence of sheaves, the cohomology groups of the three sheaves are related through the long exact sequence. Specifically, if $ \mathcal{F} \rightarrow \mathcal{G} \rightarrow \mathcal{H} \rightarrow 0 $ is a short exact sequence, then the long exact sequence of cohomology is given by:
$$ \cdots \rightarrow H^n(\mathcal{F}) \rightarrow H^n(\mathcal{G}) \rightarrow H^n(\mathcal{H}) \rightarrow H^{n+1}(\mathcal{F}) \rightarrow \cdots $$
This sequence is exact, meaning that the image of each map is equal to the kernel of the next. The exactness of this sequence is a direct consequence of the properties of cohomology functors, which are left-exact but not necessarily right-exact. The long exact sequence thus provides a way to relate the cohomology of the three sheaves in a precise and structured manner.

Properties and Examples

The long exact sequence of cohomology is a powerful tool for computing cohomology groups, especially when the cohomology of a space is difficult to determine directly. One of the key properties of this sequence is its compatibility with homotopy equivalence, which ensures that cohomology groups of homotopy equivalent spaces are isomorphic. This property is crucial in algebraic topology, as it allows for the comparison of cohomology groups across different spaces.

A classic example of the long exact sequence of cohomology is the Mayer-Vietoris sequence, which splits a space into two open subsets and relates their cohomology groups to the cohomology of the entire space. The Mayer-Vietoris sequence is derived from a short exact sequence of sheaves that describes the overlap of the two subsets. This sequence provides a way to compute the cohomology of a space by breaking it down into simpler components.

Another important example is the long exact sequence associated with a fiber bundle. If $ \pi: E \rightarrow B $ is a fiber bundle with fiber $ F $, then the long exact sequence of cohomology for the bundle is given by:
$$ \cdots \rightarrow H^n(B) \rightarrow H^n(E) \rightarrow H^n(F) \rightarrow H^{n+1}(B) \rightarrow \cdots $$
This sequence is particularly useful in studying the cohomology of spaces that are fiber bundles, as it allows for the computation of cohomology groups of the total space $ E $ using the cohomology of the base space $ B $ and the fiber $ F $.

Applications in Algebraic Topology

The long exact sequence of cohomology has numerous applications in algebraic topology, ranging from the computation of cohomology groups to the study of homotopy theory and homotopy equivalence. One of the most significant applications is the use of the sequence to determine the cohomology groups of a space by comparing them to the cohomology groups of simpler spaces. This is particularly useful in cases where the cohomology of a space is difficult to compute directly.

The long exact sequence also plays a crucial role in the study of homotopy groups. By relating the cohomology of a space to the cohomology of its subspaces, the sequence provides a way to compute homotopy groups of spaces that are not directly accessible. Additionally, the sequence is used in the construction of spectral sequences, which are powerful tools for computing cohomology groups in complex situations.

Another important application is the study of cohomology theories in different cohomological dimensions. The long exact sequence of cohomology allows for the comparison of cohomology groups in different dimensions, providing a way to analyze the structure of a space in terms of its cohomological properties. This is particularly useful in the study of spaces with non-trivial cohomology, where the sequence provides a structured way to analyze the relationships between different cohomology groups.

Conclusion

The long exact sequence of cohomology is a fundamental concept in algebraic topology, providing a powerful tool for analyzing the cohomological properties of spaces. By relating the cohomology groups of a space to those of its subspaces and quotient spaces, the sequence enables the computation of cohomology groups for complex spaces using known data from simpler ones. The construction of the sequence from short exact sequences of sheaves and the properties of cohomology functors ensure that the sequence is exact and provides a structured way to analyze the homological properties of spaces.

The long exact sequence of cohomology has numerous applications in algebraic topology, including the computation of cohomology groups, the study of fiber bundles, and the construction of spectral sequences. These applications highlight the importance of the sequence in understanding the topological structure of spaces and its relationship to other cohomological theories. The sequence also plays a crucial role in the study of homotopy equivalence and homotopy groups, providing a way to analyze the cohomological properties of spaces in a structured and systematic manner.

In conclusion, the long exact sequence of