8.3 Homotopy fiber sequences

Homotopy fiber sequences are a powerful tool in algebraic topology. They link the fiber, total space, and base space of a fibration, revealing connections between their homotopy groups. This allows us to compute complex homotopy groups using simpler spaces in the sequence.

The long exact sequence of homotopy groups is key to understanding fiber sequences. It relates the homotopy groups of the spaces involved, letting us calculate one in terms of the others. This "zig-zag" method is crucial for unraveling the topology of spaces.

Homotopy Fiber Sequences

Definition and Role

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• A homotopy fiber sequence is a sequence of topological spaces and continuous maps that arises from a fibration
• The sequence consists of the fiber, the total space, and the base space of the fibration, along with connecting maps between them
• Homotopy fiber sequences provide a powerful tool for studying the relationship between the homotopy groups of the involved spaces
  • They allow for the computation of homotopy groups of a space by relating them to the homotopy groups of simpler spaces in the sequence (long exact sequence of homotopy groups)
  • Example: Computing the homotopy groups of a complex projective space using the Hopf fibration
• Homotopy fiber sequences play a crucial role in understanding the structure and properties of topological spaces from a homotopy-theoretic perspective
  • They reveal important connections between spaces and their homotopy invariants
  • Example: Studying the topology of sphere bundles using homotopy fiber sequences

Long Exact Sequence of Homotopy Groups

• Associated with every homotopy fiber sequence is a long exact sequence of homotopy groups
• The long exact sequence relates the homotopy groups of the fiber, total space, and base space
  • It provides a systematic way to compute the homotopy groups of one space in terms of the others
  • The connecting homomorphisms in the sequence encode important information about the relationships between the spaces
• The exactness of the sequence means that the image of each map is equal to the kernel of the next map
  • This property allows for the "zig-zag" computation of homotopy groups
  • Example: Using the long exact sequence to compute the fundamental group of the circle

Constructing Fiber Sequences

Fibrations and Fibers

• Given a fibration $p: E \to B$, the homotopy fiber sequence is constructed by considering the fiber $F$ of the fibration, which is the preimage of a point in the base space $B$
  • The fiber $F$ captures the "vertical" structure sitting above each point in the base space
  • The total space $E$ is obtained by "gluing" these fibers together in a continuous way
• The sequence is written as $F \to E \to B$, where the first map is the inclusion of the fiber into the total space, and the second map is the fibration $p$
  • The maps in the sequence are continuous and satisfy certain properties related to homotopy lifting
  • Example: The Hopf fibration $S^1 \to S^3 \to S^2$ is a prototypical example of a homotopy fiber sequence

Constructing the Sequence

• The construction of homotopy fiber sequences relies on the lifting properties of fibrations and the ability to relate the homotopy groups of the involved spaces
• The connecting maps in the sequence are obtained by considering the long exact sequence of homotopy groups associated with the fibration
  • These maps relate the homotopy groups of the fiber, total space, and base space
  • The connecting homomorphisms provide a means to compute one homotopy group in terms of the others
• In some cases, the homotopy fiber sequence can be extended to the left or right by considering the homotopy groups of the base space and the fiber, respectively
  • This extension allows for a more comprehensive understanding of the relationships between the spaces
  • Example: Extending the Hopf fibration sequence to the left to include the higher homotopy groups of the sphere

Fiber Sequences vs Fibrations

Relationship between Fiber Sequences and Fibrations

• Homotopy fiber sequences arise naturally from fibrations, which are continuous maps satisfying the homotopy lifting property
  • Fibrations capture the idea of a "parameterized family" of spaces over a base space
  • The homotopy lifting property ensures that paths in the base space can be lifted to paths in the total space
• The fiber of a fibration can be viewed as the "vertical" structure sitting above each point in the base space, and the total space is obtained by "gluing" these fibers together
  • The fibers may vary continuously over the base space, forming a fiber bundle
  • Example: The Möbius strip as a non-trivial bundle over the circle
• Understanding the interplay between homotopy fiber sequences and fibrations is essential for studying the topology of fiber bundles and other related structures
  • Fibrations provide a geometric perspective on homotopy fiber sequences
  • Homotopy fiber sequences capture the algebraic and homotopy-theoretic aspects of fibrations

Long Exact Sequence and Fibrations

• The long exact sequence of homotopy groups associated with a fibration provides a powerful tool for studying the relationship between the homotopy groups of the fiber, total space, and base space
  • It relates the homotopy groups of the spaces involved in the fibration
  • The connecting homomorphisms in the sequence encode important information about the fibration
• The connecting homomorphisms in the long exact sequence relate the homotopy groups of the fiber and the base space, allowing for the computation of one in terms of the other
  • This relationship is crucial for understanding the topology of the spaces involved in the fibration
  • Example: Using the long exact sequence to study the topology of principal bundles

Applications of Fiber Sequences

Computing Homotopy Groups

• Homotopy fiber sequences can be used to compute the homotopy groups of a space by breaking it down into simpler spaces whose homotopy groups are known or easier to calculate
  • The long exact sequence of homotopy groups associated with a homotopy fiber sequence provides a systematic way to relate the homotopy groups of the involved spaces
  • By studying the properties of the maps in the homotopy fiber sequence, one can deduce information about the homotopy groups of the spaces and their relationships
• Example: Computing the homotopy groups of a complex projective space using the Hopf fibration
  • The Hopf fibration $S^1 \to S^3 \to S^2$ gives rise to a homotopy fiber sequence
  • By exploiting the long exact sequence and the known homotopy groups of spheres, one can compute the homotopy groups of complex projective spaces

Proving Theorems and Studying Manifolds

• Homotopy fiber sequences can be used to prove important results in algebraic topology, such as the Hurewicz theorem and the Whitehead theorem
  • These theorems relate the homotopy groups of a space to its homology groups and provide criteria for homotopy equivalence
  • Homotopy fiber sequences offer a powerful framework for proving such results
• In applications, homotopy fiber sequences can be employed to study the topology of manifolds, classify principal bundles, and investigate the structure of classifying spaces
  • Manifolds can be studied by considering their associated fiber bundles and the corresponding homotopy fiber sequences
  • Principal bundles can be classified using the homotopy groups of their classifying spaces, which can be analyzed using homotopy fiber sequences
• Example: Studying the topology of loop spaces and their relationship to classifying spaces using homotopy fiber sequences
  • The loop space of a topological space has a rich homotopy-theoretic structure
  • Homotopy fiber sequences provide a means to relate the topology of a space to its loop space and classifying space