8.2 The long exact sequence of a fibration

Fibrations link spaces through a special map, and their long exact sequence reveals how these spaces relate. This sequence connects homotopy groups, showing how the fiber, total space, and base space interact topologically.

The long exact sequence is a powerful tool for computing homotopy groups and understanding fibrations. It helps uncover hidden relationships between spaces and can be used to detect interesting topological phenomena in various mathematical structures.

Long Exact Sequence of a Fibration

Definition and Properties

• Given a fibration $F \to E \xrightarrow{p} B$, there exists a long exact sequence of homotopy groups: $\cdots \to \pi_n(F) \to \pi_n(E) \to \pi_n(B) \to \pi_{n-1}(F) \to \cdots \to \pi_0(E) \to \pi_0(B)$
• The maps in the long exact sequence are induced by:
  • The inclusion of the fiber $F$ into the total space $E$
  • The projection map $p: E \to B$
• Exactness means that at each stage, the image of the incoming map equals the kernel of the outgoing map
• The sequence is infinite in both directions, connecting homotopy groups of all dimensions
• The long exact sequence is natural with respect to morphisms of fibrations

Relationship to Fibration Topology

• The long exact sequence encodes the relationship between the topology of the fiber, total space, and base space of a fibration
• If the base space is contractible, the long exact sequence reveals that the total space has the same homotopy groups as the fiber
• When the fiber is discrete (has trivial homotopy groups above dimension 0), the long exact sequence shows that the total space is a covering space of the base space
• The connecting homomorphism in the long exact sequence measures the obstruction to lifting homotopy classes from the base space to the total space
• The long exact sequence can be used to detect the existence of interesting topological phenomena, such as:
  • Non-trivial bundles
  • Twisted products

Computing Homotopy Groups

Using the Long Exact Sequence

• The long exact sequence can be used to compute unknown homotopy groups of the total space, fiber, or base space when the other two are known
• If two out of three consecutive terms in the sequence are known to be zero, the third term must be isomorphic to the image or kernel of the adjacent maps
• The long exact sequence can be combined with other tools to compute homotopy groups, such as:
  • The Hurewicz theorem
  • The universal coefficients theorem
• In some cases, the long exact sequence may split into shorter exact sequences, simplifying computations

Examples and Applications

• Computing the homotopy groups of the Hopf fibration $S^1 \to S^3 \to S^2$
  • Using the long exact sequence and the known homotopy groups of $S^1$ and $S^2$
• Determining the homotopy groups of the unit tangent bundle of a sphere $S^{n-1} \to UTB(S^n) \to S^n$
  • Applying the long exact sequence and the homotopy groups of spheres
• Calculating the homotopy groups of loop spaces, such as $\Omega S^n$, using the path-loop fibration
  • $\Omega S^n \to PS^n \to S^n$, where $PS^n$ is the path space of $S^n$

Existence and Exactness of the Sequence

Proof Outline

• The proof relies on the homotopy lifting property of fibrations, which allows the construction of lifts of homotopies from the base space to the total space
• The maps in the long exact sequence are defined using the boundary maps in the long exact sequence of homotopy groups for a pair $(E, F)$
• Exactness at $\pi_n(E)$ is proved by showing that the image of the map from $\pi_n(F)$ equals the kernel of the map to $\pi_n(B)$, using the homotopy lifting property
• Exactness at $\pi_n(B)$ is proved by showing that the image of the map from $\pi_n(E)$ equals the kernel of the connecting homomorphism to $\pi_{n-1}(F)$, using:
  • The homotopy lifting property
  • The definition of the boundary map
• Naturality of the long exact sequence is proved by showing that morphisms of fibrations induce commutative diagrams connecting the corresponding long exact sequences

Key Steps and Techniques

• Constructing lifts of homotopies using the homotopy lifting property
• Defining the maps in the long exact sequence using the boundary maps in the long exact sequence of homotopy groups for a pair
• Proving exactness by showing the equality of images and kernels of adjacent maps
• Utilizing the properties of the boundary map and the homotopy lifting property to establish exactness
• Demonstrating the naturality of the long exact sequence with respect to morphisms of fibrations