🔢Algebraic Topology Unit 8 – Fibrations and Homotopy Fiber Sequences

Key Concepts and Definitions

• Fibration is a map $p: E \to B$ satisfying the homotopy lifting property for all spaces $X$
• Fiber of a fibration $p: E \to B$ over a point $b \in B$ is the preimage $p^{-1}(b)$
• Homotopy fiber of a map $f: X \to Y$ over a point $y \in Y$ is the fiber product $X \times_Y Y^I$ where $Y^I$ is the path space of $Y$
  • Homotopy fiber is denoted as $hofib_f(y)$ or simply $hofib(f)$
• Fiber sequence is a sequence of spaces and maps $F \to E \xrightarrow{p} B$ where $p$ is a fibration and $F$ is the fiber of $p$
• Long exact sequence of homotopy groups is an infinite exact sequence relating the homotopy groups of spaces in a fiber sequence
• Pullback is a universal construction that captures the idea of "restricting" a map to a subspace
• Homotopy equivalence is a continuous map between topological spaces that has a homotopy inverse

Historical Context and Motivation

• Fibrations were introduced by Serre in the 1950s as a generalization of fiber bundles
  • Fiber bundles are locally trivial fibrations with extra structure (transition functions)
• Motivation came from studying the topology of function spaces and loop spaces
• Fibrations provide a way to study the global structure of a space by understanding its local structure (fibers)
• Homotopy fiber and fiber sequences were introduced to study the homotopy theory of maps between spaces
  • Allows for comparison of homotopy types of fibers and total spaces
• Long exact sequence of homotopy groups was a powerful tool for computing homotopy groups of spaces
• Applications in algebraic topology include the study of classifying spaces, spectral sequences, and obstruction theory

Fibrations: Properties and Examples

• Fibrations are characterized by the homotopy lifting property
  • For any homotopy $h: X \times I \to B$ and lift $\tilde{h}_0: X \to E$ of $h_0$, there exists a homotopy $\tilde{h}: X \times I \to E$ lifting $h$
• Examples of fibrations include projection maps from product spaces, covering spaces, and fiber bundles
• Fibrations are preserved under pullbacks and composition
• Fibers of a fibration are homotopy equivalent if the base space is path-connected
• Homotopy fibers of homotopic maps are homotopy equivalent
• Fibrations can be characterized by the existence of a path lifting function
• Fibrations with contractible fibers are homotopy equivalences

Homotopy Fiber and Fiber Sequences

• Homotopy fiber of a map $f: X \to Y$ is the fiber product $X \times_Y Y^I$
  • Points in the homotopy fiber are pairs $(x, \gamma)$ where $x \in X$ and $\gamma$ is a path in $Y$ from $f(x)$ to a fixed point $y \in Y$
• Homotopy fibers measure the failure of a map to be a fibration
  • A map is a fibration if and only if its homotopy fibers are homotopy equivalent to its actual fibers
• Fiber sequences are sequences of spaces and maps $F \to E \xrightarrow{p} B$ where $p$ is a fibration and $F$ is the fiber of $p$
• Fiber sequences can be extended to the left by taking the homotopy fiber of the previous map
  • Leads to long exact sequences of homotopy groups
• Puppe sequence is a fiber sequence that extends a given map $f: X \to Y$ to the right by taking homotopy fibers and suspensions

Long Exact Sequence of Homotopy Groups

• A fiber sequence $F \to E \xrightarrow{p} B$ induces a long exact sequence of homotopy groups
  • $... \to \pi_{n+1}(B) \xrightarrow{\partial} \pi_n(F) \to \pi_n(E) \to \pi_n(B) \xrightarrow{\partial} \pi_{n-1}(F) \to ...$
• Connecting homomorphism $\partial$ is induced by the boundary map in the homotopy fiber
• Long exact sequence is a powerful tool for computing homotopy groups of spaces
  • Allows for inductive arguments and comparison of homotopy groups
• Naturality of the long exact sequence allows for the construction of commutative diagrams
• Splitting of the long exact sequence corresponds to the existence of a section of the fibration
• Exact sequences can be spliced together using the snake lemma

Applications in Algebraic Topology

• Fibrations and fiber sequences are used to study the topology of function spaces and mapping spaces
  • Loop space $\Omega X$ is the fiber of the evaluation map $ev: X^I \to X$
• Postnikov towers are constructed using fibrations with Eilenberg-MacLane spaces as fibers
  • Allows for the study of the homotopy theory of a space one homotopy group at a time
• Spectral sequences can be constructed from fiber sequences using the long exact sequence of homotopy groups
  • Serre spectral sequence relates the homology of the base and fiber to the homology of the total space
• Obstruction theory uses fibrations to study the existence and classification of lifts and extensions of maps
• Classifying spaces $BG$ are constructed as the base space of a universal principal $G$-bundle $EG \to BG$
  • Allows for the classification of principal $G$-bundles over a space $X$ in terms of homotopy classes of maps $X \to BG$

Connections to Other Topics

• Fibrations and fiber sequences are closely related to the theory of cofibrations and cofiber sequences
  • Cofibrations are dual to fibrations and satisfy the homotopy extension property
• Quillen model categories provide a unified framework for studying fibrations, cofibrations, and weak equivalences
  • Fibrations and cofibrations are characterized by lifting properties with respect to acyclic maps
• Grothendieck fibrations in category theory are a categorical analog of topological fibrations
  • Study the behavior of functors and natural transformations under base change
• Fibrations and fiber sequences play a key role in the theory of ∞-categories and higher category theory
  • Provide a way to study the homotopy theory of functors and natural transformations
• Fibrations and fiber sequences have applications in other areas of mathematics such as algebraic geometry, differential geometry, and mathematical physics

Problem-Solving Techniques

• Recognize when a map is a fibration by checking the homotopy lifting property or the existence of a path lifting function
• Use the long exact sequence of homotopy groups to compute homotopy groups of spaces in a fiber sequence
  • Identify known homotopy groups and use the exactness to fill in the unknown groups
• Construct fiber sequences by taking homotopy fibers of maps or by pulling back known fibrations
• Use the naturality of the long exact sequence to construct commutative diagrams and compare homotopy groups of different spaces
• Apply the five lemma or the snake lemma to exact sequences to deduce isomorphisms or injectivity/surjectivity of maps
• Use spectral sequences constructed from fiber sequences to compute homology or cohomology of spaces
• Apply obstruction theory to determine the existence and classification of lifts or extensions of maps
• Recognize when a space is the classifying space of a group and use this to study principal bundles over that space