Higher Homotopy Groups Introduction

Higher homotopy groups are the natural generalization of the fundamental group, moving from loops to higher-dimensional spheres to probe the structure of a topological space. While the fundamental group $\pi_1(X)$ can be non-abelian, these higher invariants are always abelian, providing a powerful, computable algebraic toolkit. Their study reveals profound connections between geometry and algebra, forming the backbone of classical homotopy theory and its applications.

From Loops to Spheres: Defining $\pi_n(X)$

The fundamental group $\pi_1(X, x_0)$ classifies loops in a space $X$ based on homotopy equivalence. A higher homotopy group $\pi_n(X, x_0)$ for $n \geq 2$ generalizes this idea by classifying continuous maps from the $n$-sphere $S^n$ into $X$, all sending a chosen basepoint on the sphere to the basepoint $x_0 \in X$.

Formally, an element of $\pi_n(X, x_0)$ is an equivalence class $[f]$ of a based map $f: (S^n, s_0) \to (X, x_0)$. Two such maps $f$ and $g$ represent the same class if there exists a homotopy $H: S^n \times [0,1] \to X$ such that $H(s,0)=f(s)$, $H(s,1)=g(s)$, and $H(s_0, t) = x_0$ for all $t$ (this is a based homotopy). You can visualize an element of $\pi_2(X)$ as a "film" or a continuous family of loops covering a sphere, all pinched at the basepoint.

The group operation is defined via a "pinch map." For $n \geq 1$, we collapse the equator of $S^n$ to a point, which results in a wedge sum $S^n \vee S^n$. Given two maps $f, g: S^n \to X$, we define $f + g$ as the map that applies $f$ to one hemisphere and $g$ to the other. This operation gives $\pi_n(X, x_0)$ a group structure, with the constant map as the identity.

The Abelian Nature of $\pi_n$ for $n \geq 2$

A critical departure from the fundamental group occurs when $n \geq 2$: the groups $\pi_n(X)$ are abelian. This simplifies their algebraic analysis immensely. The reason is geometric: in higher dimensions, you can physically "slide" two $n$-spheres past each other.

Consider two maps $f$ and $g$ representing elements of $\pi_n(X)$ for $n \geq 2$. Their sum $f+g$ is represented by placing the "image" of $f$ on one hemisphere and $g$ on the other. Because the hemispheres share only the basepoint at their boundary, and the dimension is high enough, you can continuously rotate the entire configuration to swap the positions of $f$ and $g$. This rotation provides a homotopy between $f+g$ and $g+f$, proving commutativity. This fails for $\pi_1$ because a loop is one-dimensional; in a one-dimensional context, you cannot slide one loop through another without cutting—the fundamental group captures this potential for entanglement.

Computations for Spheres: $\pi_k(S^n)$

Computing homotopy groups of spheres is a central, often difficult, problem in algebraic topology. Some key results form the foundation:

• $\pi_n(S^n) \cong \mathbb{Z}$ for $n \geq 1$. The integer associated with a map $f: S^n \to S^n$ is its degree, which counts, with orientation, how many times the sphere wraps around itself.
• $\pi_k(S^n) = 0$ for $k < n$. Intuitively, any map from a lower-dimensional sphere can be contracted to a point because the target sphere has "extra room."
• $\pi_k(S^1) = 0$ for $k \geq 2$. This follows because the universal cover of $S^1$ is the contractible space $\mathbb{R}$.
• The first surprising result is $\pi_3(S^2) \cong \mathbb{Z}$, generated by the Hopf fibration $S^3 \to S^2$. This shows that higher homotopy groups can be non-trivial even when the lower-dimensional ones vanish.

For $k > n$, the groups $\pi_k(S^n)$ are generally non-zero and finite (except for $\pi_{4m-1}(S^{2m})$ which contains a $\mathbb{Z}$ summand). Their computation is an active area of research and involves sophisticated tools like spectral sequences.

The Long Exact Sequence of a Fibration

A powerful computational tool is the long exact sequence of a fibration. A fibration $p: E \to B$ is a surjective map with the "homotopy lifting property" for all spaces (think of a fiber bundle, like a Mobius strip over a circle). If $F = p^{-1}(b_0)$ is the fiber over the basepoint, then the homotopy groups of $E$, $B$, and $F$ are linked in a long exact sequence:

$$\cdots \to \pi_{n}(F) \xrightarrow{i_*} \pi_{n}(E) \xrightarrow{p_*} \pi_{n}(B) \xrightarrow{\partial} \pi_{n-1}(F) \to \cdots \to \pi_0(E) \to \pi_0(B)$$

Here, $i_*$ and $p_*$ are maps induced by the inclusion $i: F \hookrightarrow E$ and the projection $p$. The crucial boundary map $\partial: \pi_n(B) \to \pi_{n-1}(F)$ is constructed by lifting a map representing an element of $\pi_n(B)$ to $E$ and then taking its restriction to the "boundary," which lies in the fiber $F$.

This sequence is exact, meaning the image of one map equals the kernel of the next. It allows you to compute unknown homotopy groups if you know the groups for two of the three spaces. For example, the Hopf fibration $S^1 \to S^3 \xrightarrow{p} S^2$ has fiber $S^1$. A portion of its long exact sequence is:

$$\pi_2(S^3) \to \pi_2(S^2) \xrightarrow{\partial} \pi_1(S^1) \to \pi_1(S^3)$$

Since $\pi_2(S^3)=0$ and $\pi_1(S^3)=0$, exactness forces $\partial$ to be an isomorphism, proving $\pi_2(S^2) \cong \pi_1(S^1) \cong \mathbb{Z}$.

Common Pitfalls

1. Confusing $\pi_n$ with homology groups $H_n$. While both are abelian groups for $n \geq 2$ and sometimes coincide, they are fundamentally different. Homotopy groups are often vastly more complex and difficult to compute. For instance, $\pi_3(S^2)$ is $\mathbb{Z}$, but $H_3(S^2)=0$. Homotopy groups are sensitive to the mapping structure, not just the shape of cycles.

1. Assuming $\pi_n(X)$ is trivial if $\pi_1(X)$ is trivial. Simply connectedness ($\pi_1=0$) does not imply higher homotopy groups vanish. The spheres $S^n$ for $n \geq 2$ are the classic counterexamples: $\pi_1(S^n)=0$, but $\pi_n(S^n)=\mathbb{Z}$.

1. Misapplying the long exact sequence. The sequence only exists for fibrations. Applying it to an arbitrary short exact sequence of spaces or maps will lead to errors. You must verify the homotopy lifting property or know your map is a fiber bundle (like a covering map or vector bundle projection).

1. Overlooking the role of the basepoint. Homotopy groups $\pi_n(X, x_0)$ are defined for a pointed space. If $X$ is path-connected, the groups are isomorphic for different basepoints, but the isomorphism depends on a choice of path and is not canonical unless the group is abelian.

Summary

• Higher homotopy groups $\pi_n(X)$ generalize the fundamental group by studying equivalence classes of based maps from the $n$-sphere $S^n$ into a space $X$.
• For all $n \geq 2$, the groups $\pi_n(X)$ are abelian, a key simplification over the potentially non-abelian fundamental group $\pi_1(X)$.
• Computing $\pi_k(S^n)$ is a deep problem; foundational results include $\pi_n(S^n) \cong \mathbb{Z}$ and the non-trivial $\pi_3(S^2) \cong \mathbb{Z}$ via the Hopf fibration.
• The long exact sequence of a fibration is an essential computational tool, linking the homotopy groups of the total space, base space, and fiber in an exact sequence: $\cdots \to \pi_{n}(F) \to \pi_{n}(E) \to \pi_{n}(B) \to \pi_{n-1}(F) \to \cdots$.