5.1 Higher homotopy groups

Higher homotopy groups extend the fundamental group concept to higher dimensions. They measure "holes" in spaces using maps from n-dimensional spheres. This powerful tool helps distinguish and classify topological spaces beyond what the fundamental group alone can do.

Unlike the fundamental group, higher homotopy groups are always abelian. They play a crucial role in obstruction theory and space classification. Computing these groups often involves advanced techniques like spectral sequences and Postnikov towers.

Higher homotopy groups

Definition and role in algebraic topology

• The n-th homotopy group of a topological space X, denoted πₙ(X), consists of homotopy equivalence classes of based maps from the n-dimensional sphere Sⁿ to X
  • The group operation in πₙ(X) arises from the concatenation of based maps, making πₙ(X) an abelian group for n ≥ 2
  • Higher homotopy groups measure the number of "holes" or "voids" of different dimensions in a topological space (e.g., a 2-dimensional void in a 3-dimensional space)
  • They provide a more refined way to distinguish topological spaces than the fundamental group alone
• Higher homotopy groups play a central role in obstruction theory and the classification of topological spaces
  • Obstruction theory studies the existence and properties of continuous maps between topological spaces using higher homotopy groups
  • Classification of topological spaces involves determining when two spaces are homotopy equivalent or homeomorphic based on their higher homotopy groups

Properties and examples

• The n-th homotopy group of a point is trivial for all n ≥ 1, i.e., πₙ(point) = 0
  • This reflects the fact that a point has no higher-dimensional holes or voids
• The n-th homotopy group of the n-sphere is the integers for n ≥ 1, i.e., πₙ(Sⁿ) ≅ ℤ, and πₘ(Sⁿ) = 0 for m ≠ n
  • For example, π₂(S²) ≅ ℤ, while π₁(S²) = 0 and π₃(S²) = 0
  • The generator of πₙ(Sⁿ) corresponds to the identity map from Sⁿ to itself
• The n-th homotopy group of a contractible space, such as the Euclidean space ℝⁿ, is trivial for all n ≥ 1
  • Contractible spaces can be continuously deformed to a point, so they have no higher-dimensional holes or voids

Computing higher homotopy groups

Product spaces and wedge sums

• The higher homotopy groups of a product space can be computed using the product formula: πₙ(X × Y) ≅ πₙ(X) × πₙ(Y) for n ≥ 1
  • For example, πₙ(S¹ × S²) ≅ πₙ(S¹) × πₙ(S²) for all n ≥ 1
  • This formula reflects the fact that the holes or voids in a product space arise from the holes or voids in its constituent spaces
• The higher homotopy groups of a wedge sum of spaces can be computed using the wedge sum formula: πₙ(⋁ᵢXᵢ) ≅ ⨁ᵢπₙ(Xᵢ) for n ≥ 2
  • For example, π₂(S¹ ∨ S²) ≅ π₂(S¹) ⊕ π₂(S²) ≅ 0 ⊕ ℤ ≅ ℤ
  • The wedge sum formula holds only for n ≥ 2 because the fundamental group of a wedge sum is generally not the direct sum of the fundamental groups of its constituent spaces

Techniques and tools

• Spectral sequences, such as the Serre spectral sequence and the Atiyah-Hirzebruch spectral sequence, can be used to compute higher homotopy groups of fibrations and cell complexes
  • These spectral sequences relate the homology groups of the base space and the fiber to the homotopy groups of the total space
• The Postnikov tower of a space X is a sequence of spaces Xₙ that approximate X up to the n-th homotopy group
  • The homotopy groups of the Postnikov tower stages can be used to reconstruct the higher homotopy groups of the original space X
• The Whitehead tower of a connected space X is a sequence of spaces Xⁿ that capture the higher homotopy groups of X
  • The homotopy fibers of the maps in the Whitehead tower are Eilenberg-MacLane spaces, whose homotopy groups are related to the higher homotopy groups of X

Higher homotopy groups vs the fundamental group

Similarities and differences

• The fundamental group, π₁(X), measures the number of 1-dimensional "loops" in a topological space, while higher homotopy groups, πₙ(X) for n ≥ 2, measure higher-dimensional holes or voids
  • Both the fundamental group and higher homotopy groups are topological invariants that distinguish spaces up to homotopy equivalence
• Higher homotopy groups, πₙ(X) for n ≥ 2, are always abelian groups, while the fundamental group may be non-abelian
  • The Eckmann-Hilton argument shows that the higher homotopy groups are commutative, i.e., the group operation is independent of the order of composition
  • The non-commutativity of the fundamental group reflects the fact that loops in a space can be composed in different orders, yielding different results

Interactions and connections

• The action of the fundamental group on higher homotopy groups is given by the π₁-action, which describes how loops in π₁(X) act on elements of πₙ(X)
  • The π₁-action is a group homomorphism π₁(X) → Aut(πₙ(X)), where Aut(πₙ(X)) is the group of automorphisms of πₙ(X)
  • The π₁-action measures the extent to which the fundamental group "twists" or "permutes" the higher homotopy groups
• The Whitehead products provide a connection between the fundamental group and higher homotopy groups, measuring the non-commutativity of the π₁-action
  • The Whitehead product is a bilinear map [−,−]: πₘ(X) × πₙ(X) → πₘ₊ₙ₋₁(X) that generalizes the commutator in the fundamental group
  • Non-trivial Whitehead products indicate the presence of non-trivial interactions between the fundamental group and higher homotopy groups

Higher homotopy groups under maps

Induced homomorphisms and functoriality

• A continuous map f: X → Y induces group homomorphisms f_*: πₙ(X) → πₙ(Y) for each n ≥ 1, called the induced homomorphisms on homotopy groups
  • The induced homomorphisms are defined by composing a representative map Sⁿ → X with f to obtain a map Sⁿ → Y, and then taking the homotopy class of this composition
• The induced homomorphisms are functorial, meaning they respect the composition of continuous maps: (g ∘ f)* = g* ∘ f_*
  • Functoriality reflects the idea that the induced homomorphisms are compatible with the categorical structure of topological spaces and continuous maps
  • This property allows for the study of higher homotopy groups in the context of category theory and functorial constructions

Homotopy invariance and applications

• Homotopy equivalences induce isomorphisms on all homotopy groups, i.e., if f: X → Y is a homotopy equivalence, then f_*: πₙ(X) → πₙ(Y) is an isomorphism for all n ≥ 1
  • This property reflects the fact that homotopy equivalent spaces have the same higher homotopy groups
  • Homotopy invariance is a key tool in algebraic topology for comparing and classifying spaces using higher homotopy groups
• The behavior of induced homomorphisms under fibrations and cofibrations can be studied using the long exact sequence of homotopy groups
  • For a fibration F → E → B, there is a long exact sequence ... → πₙ(F) → πₙ(E) → πₙ(B) → πₙ₋₁(F) → ...
  • For a cofibration A → X → X/A, there is a long exact sequence ... → πₙ(A) → πₙ(X) → πₙ(X/A) → πₙ₋₁(A) → ...
  • These long exact sequences relate the higher homotopy groups of the spaces involved in a fibration or cofibration, providing a powerful tool for computation and analysis
• The Hurewicz theorem relates the higher homotopy groups to the homology groups of a space, providing a connection between homotopy theory and homology theory
  • The Hurewicz theorem states that for a simply connected space X, the first non-trivial homotopy group πₙ(X) is isomorphic to the n-th homology group Hₙ(X)
  • This theorem allows for the computation of higher homotopy groups using homology groups, which are often easier to calculate