🔢Algebraic Topology Unit 5 – Higher Homotopy Groups

Definition and Intuition

• Higher homotopy groups generalize the fundamental group to higher dimensions, capturing more intricate topological information about a space
• Intuitively, the nth homotopy group $\pi_n(X)$ describes the different ways an n-dimensional sphere $S^n$ can be mapped into the space $X$
• Two mappings are considered equivalent if they can be continuously deformed into each other within the space $X$, forming homotopy equivalence classes
• The group operation for higher homotopy groups involves concatenating the mappings and rescaling to maintain the domain as the n-dimensional sphere
  • This operation is well-defined on homotopy equivalence classes and satisfies the group axioms
• Higher homotopy groups provide a sequence of algebraic invariants $\pi_1(X), \pi_2(X), \pi_3(X), \ldots$ that progressively capture more subtle topological features of the space $X$
  • For example, $\pi_2(X)$ detects the presence of 2-dimensional holes or voids in the space

Fundamental Group Recap

• The fundamental group $\pi_1(X)$ is the first homotopy group and describes the different loops in a topological space $X$ based at a chosen point $x_0$
• Elements of the fundamental group are homotopy equivalence classes of loops, where two loops are considered equivalent if they can be continuously deformed into each other while keeping the basepoint fixed
• The group operation for the fundamental group is the concatenation of loops, with the constant loop serving as the identity element
• The fundamental group is an important algebraic invariant that captures the 1-dimensional connectivity of a space
  • For example, the fundamental group of a circle $S^1$ is isomorphic to the integers $\mathbb{Z}$, reflecting the winding number of loops around the circle
• Computing the fundamental group often involves techniques such as Van Kampen's theorem, which allows for the calculation of the fundamental group of a space by decomposing it into simpler subspaces and combining their fundamental groups
• The triviality of the fundamental group ($\pi_1(X) = 0$) indicates that the space is simply connected, meaning any loop can be contracted to a point

Higher Dimensional Spheres

• Higher dimensional spheres $S^n$ play a crucial role in the construction and study of higher homotopy groups
• The n-dimensional sphere $S^n$ is the set of points in $(n+1)$-dimensional Euclidean space that are equidistant from a fixed center point
  • For example, $S^1$ is the circle, $S^2$ is the ordinary sphere, and $S^3$ is the 3-sphere or glome
• Higher dimensional spheres exhibit interesting topological properties and serve as building blocks for more complex spaces
• The homotopy groups of spheres $\pi_k(S^n)$ are of particular interest and have been extensively studied
  • For $k < n$, $\pi_k(S^n) = 0$, meaning there are no non-trivial mappings from lower-dimensional spheres into higher-dimensional spheres
  • For $k = n$, $\pi_n(S^n) \cong \mathbb{Z}$, generated by the identity mapping
• The Hopf fibration is a famous example of a non-trivial mapping from $S^3$ to $S^2$, which generates the third homotopy group of $S^2$
• Understanding the homotopy groups of spheres is crucial for computing the homotopy groups of more general spaces using tools like the homotopy long exact sequence

Construction of Higher Homotopy Groups

• The nth homotopy group $\pi_n(X)$ is constructed by considering continuous mappings $f: S^n \to X$ from the n-dimensional sphere $S^n$ to the topological space $X$
• Two mappings $f, g: S^n \to X$ are considered homotopic if there exists a continuous family of mappings $H: S^n \times [0, 1] \to X$ such that $H(x, 0) = f(x)$ and $H(x, 1) = g(x)$ for all $x \in S^n$
  • Homotopy defines an equivalence relation on the set of mappings, partitioning them into homotopy equivalence classes
• The set of homotopy equivalence classes of mappings from $S^n$ to $X$ forms the elements of the nth homotopy group $\pi_n(X)$
• The group operation in $\pi_n(X)$ is defined by concatenating two mappings $f$ and $g$ along the equator of $S^n$ and rescaling the resulting mapping to have domain $S^n$
  • This operation, denoted by $[f] \cdot [g]$, is well-defined on homotopy equivalence classes and satisfies the group axioms
• The constant mapping from $S^n$ to a fixed point in $X$ serves as the identity element in $\pi_n(X)$
• The inverse of a mapping $f$ is obtained by composing $f$ with the antipodal map on $S^n$, which sends each point to its opposite point on the sphere
• Higher homotopy groups are abelian for $n \geq 2$, meaning the group operation is commutative

Properties and Calculations

• Higher homotopy groups are important algebraic invariants of topological spaces that capture higher-dimensional connectivity and hole structure
• Homotopy groups are homotopy invariants, meaning if two spaces $X$ and $Y$ are homotopy equivalent, then $\pi_n(X) \cong \pi_n(Y)$ for all $n$
• The homotopy groups of a product space $X \times Y$ satisfy the product formula $\pi_n(X \times Y) \cong \pi_n(X) \times \pi_n(Y)$
• The homotopy groups of a wedge sum of spaces $\bigvee_{\alpha} X_{\alpha}$ are related to the homotopy groups of the individual spaces $X_{\alpha}$ by the wedge sum formula
  • This formula involves a direct sum of the homotopy groups of the individual spaces and additional terms capturing the interaction between the spaces
• The Freudenthal suspension theorem relates the homotopy groups of a space $X$ to the homotopy groups of its suspension $\Sigma X$
  • It states that the suspension homomorphism $\pi_n(X) \to \pi_{n+1}(\Sigma X)$ is an isomorphism for $n < 2 \cdot \text{conn}(X)$, where $\text{conn}(X)$ is the connectivity of $X$
• The Hurewicz theorem establishes a connection between homotopy groups and homology groups, stating that for a simply connected space $X$, the first non-trivial homotopy group $\pi_n(X)$ is isomorphic to the nth homology group $H_n(X)$
• Computing higher homotopy groups is generally a challenging task, and many homotopy groups are still unknown or only partially understood

Homotopy Long Exact Sequence

• The homotopy long exact sequence is a powerful tool for computing homotopy groups of spaces by relating them to the homotopy groups of simpler subspaces
• Given a continuous map $f: X \to Y$ between topological spaces, there is a long exact sequence of homotopy groups: $\ldots \to \pi_{n+1}(Y) \to \pi_n(F) \to \pi_n(X) \to \pi_n(Y) \to \pi_{n-1}(F) \to \ldots$ where $F$ is the homotopy fiber of $f$
• The homotopy fiber $F$ is defined as the space of pairs $(x, \gamma)$, where $x \in X$ and $\gamma$ is a path in $Y$ from $f(x)$ to a fixed basepoint in $Y$
• The maps in the long exact sequence are induced by the inclusion of the fiber into the total space and the projection onto the base space
• 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 calculation of unknown homotopy groups in terms of known ones
• The homotopy long exact sequence is particularly useful when applied to fibrations, which are maps satisfying the homotopy lifting property
  • For a fibration $f: E \to B$ with fiber $F$, the long exact sequence takes the form: $\ldots \to \pi_{n+1}(B) \to \pi_n(F) \to \pi_n(E) \to \pi_n(B) \to \pi_{n-1}(F) \to \ldots$
• The homotopy long exact sequence can be used iteratively to compute homotopy groups of increasingly complex spaces by breaking them down into simpler pieces

Applications in Topology

• Higher homotopy groups find numerous applications in various areas of topology and related fields
• In the study of manifolds, higher homotopy groups provide obstructions to the existence of certain structures or mappings
  • For example, the vanishing of certain homotopy groups is a necessary condition for a manifold to admit a smooth structure or a Riemannian metric with positive curvature
• In the theory of fiber bundles, higher homotopy groups of the base space and the fiber can be used to classify and construct bundles
  • The long exact sequence of homotopy groups associated with a fiber bundle provides a tool for computing the homotopy groups of the total space
• Higher homotopy groups play a role in the classification of topological spaces up to homotopy equivalence
  • Spaces with isomorphic homotopy groups for all dimensions are called homotopy equivalent, and higher homotopy groups provide a way to distinguish between non-equivalent spaces
• In the study of knots and links, higher homotopy groups of the complement of a knot or link can provide invariants that distinguish different knot types
  • These invariants, such as the knot groups and the Alexander polynomials, are related to the fundamental group and higher homotopy groups of the knot complement
• Higher homotopy groups also find applications in algebraic topology, where they are used to define and study generalized cohomology theories
  • Theories such as K-theory and cobordism theory are built upon the foundations of higher homotopy groups and provide powerful tools for investigating topological spaces and their invariants

Challenges and Open Problems

• Despite significant progress in understanding higher homotopy groups, many challenges and open problems remain in this area of algebraic topology
• Computing the homotopy groups of spheres $\pi_n(S^k)$ is a central problem that has attracted much attention
  • While some homotopy groups of spheres are known, such as $\pi_n(S^n) \cong \mathbb{Z}$ and $\pi_n(S^1) \cong 0$ for $n \geq 2$, many others remain unknown or only partially understood
  • The Hopf invariant one problem, which concerns the existence of certain mappings between spheres, was a long-standing open problem until its resolution in the mid-20th century
• The homotopy groups of the spheres form a rich algebraic structure known as the stable homotopy groups of spheres
  • Understanding the structure and properties of these stable homotopy groups is an ongoing area of research
  • The Adams spectral sequence and the Adams-Novikov spectral sequence are powerful tools used to compute stable homotopy groups, but their calculations can be highly complex
• Generalizations of homotopy groups, such as the homotopy groups of spectra and the homotopy groups of simplicial sets, provide new avenues for research and have connections to other areas of mathematics
• The relationship between homotopy groups and other invariants, such as homology groups and cohomology groups, is an active area of investigation
  • The Hurewicz theorem provides a link between homotopy groups and homology groups, but the precise nature of this relationship in general settings is not fully understood
• The computational complexity of determining homotopy groups is another challenge
  • Algorithmic methods for computing homotopy groups, such as the effective Hurewicz theorem and the effective Postnikov tower, have been developed, but their efficiency and applicability are limited
• The study of homotopy groups in the context of algebraic geometry, particularly in the theory of motivic homotopy theory, has opened up new directions for research and has led to the development of new computational tools and techniques