Kerodon

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3.2 Homotopy Groups

Our goal in this section is to address the following:

Question 3.2.0.1. Let $f: X \rightarrow Y$ be a morphism of Kan complexes. Under what conditions does $f$ admit a homotopy inverse $g: Y \rightarrow X$?

Let us begin with a partial answer to Question 3.2.0.1. For every Kan complex $X$, let $\pi _{\leq 1}(X)$ denote the fundamental groupoid of $X$ (Definition 1.3.6.14). For each vertex $x \in X$, we let $\pi _{1}(X,x)$ denote the automorphism group $\operatorname{Aut}_{ \pi _{\leq 1}(X) }( x ) = \operatorname{Hom}_{\pi _{\leq 1}(X)}(x,x)$; we will refer to $\pi _{1}(X,x)$ as the fundamental group of $X$ (with respect to the base point $x$). Every morphism of Kan complexes $f: X \rightarrow Y$ induces a functor $\pi _{\leq 1}(f): \pi _{\leq 1}(X) \rightarrow \pi _{\leq 1}(Y)$. Moreover, if $f$ is a homotopy equivalence, then $\pi _{\leq 1}(f)$ is an equivalence of categories (Remark 3.1.5.4). In other words, every homotopy equivalence $f: X \rightarrow Y$ satisfies the following pair of conditions:

$(W_0)$

The map $\pi _0(f): \pi _0(X) \rightarrow \pi _0(Y)$ is an isomorphism of sets: that is, $f$ induces a bijection from the set of connected components of $X$ to the set of connected components of $Y$.

$(W_1)$

For every choice of vertex $x \in X$ having image $y = f(x) \in Y$, the induced map of fundamental groups $\pi _{1}(X,x) \rightarrow \pi _1(Y,y)$ is an isomorphism.

However, these observations do not supply a complete answer to Question 3.2.0.1: conditions $(W_0)$ and $(W_1)$ are necessary for $f$ to be a homotopy equivalence, but they are not sufficient. In this section, we will remedy the situation by introducing a hierarchy of additional invariants. To each Kan complex $X$ and each vertex $x \in X$, we will associate a sequence of sets $\{ \pi _{n}(X,x) \} _{n \geq 0}$, which enjoy the following features:

  • For every nonnegative integer $n$, $\pi _{n}(X,x)$ is defined as the set of homotopy classes of pointed maps from the quotient $\Delta ^{n} / \operatorname{\partial \Delta }^{n}$ to $X$ (Construction 3.2.2.4). Here it is important to work in the homotopy theory of pointed simplicial sets, which we review in §3.2.1.

  • When $n=0$, we can identify $\pi _{n}(X,x)$ with the set $\pi _0(X)$ of connected components of $X$: in particular, it does not depend on the choice of base point $x$ (Example 3.2.2.6).

  • For $n > 0$, the set $\pi _{n}(X,x)$ comes equipped with a natural group structure (Theorem 3.2.2.10), which we will construct in §3.2.3. For this reason, we will refer to $\pi _{n}(X,x)$ as the $n$th homotopy group of $X$ (with respect to the base point $x$). Moreover, the group $\pi _{n}(X,x)$ is abelian for $n \geq 2$.

  • When $n=1$, we can identify $\pi _{1}(X,x)$ with the fundamental group of $X$ as defined earlier: that is, with the automorphism group of $x$ as an object of the homotopy category $\pi _{\leq 1}(X)$ (Example 3.2.2.12).

  • Let $f: X \rightarrow S$ be a Kan fibration between Kan complexes, let $x \in X$ be a vertex having image $s = f(x) \in S$, and let $X_{s} = \{ s\} \times _{S} X$ denote the fiber of $f$ over the vertex $s$. Then there is a long exact sequence of homotopy groups

    \[ \cdots \rightarrow \pi _{n+1}(S,s) \xrightarrow {\partial } \pi _{n}(X_ s, x) \rightarrow \pi _{n}( X, x) \rightarrow \pi _ n(S,s) \xrightarrow {\partial } \pi _{n-1}(X_ s, x) \rightarrow \cdots \]

    We construct this sequence in §3.2.4, and prove its exactness in §3.2.5 (Theorem 3.2.5.1).

  • Let $f: X \rightarrow Y$ be a morphism of Kan complexes. In §3.2.6, we show that $f$ is a homotopy equivalence if and only if it induces a bijection $\pi _0(f): \pi _0(X) \rightarrow \pi _0(Y)$ and an isomorphism of homotopy groups $\pi _{n}(X,x) \rightarrow \pi _{n}(Y, f(x) )$, for every choice of base point $x \in X$ and every positive integer $n$ (Theorem 3.2.6.1). This is a simplicial counterpart of a classical result of Whitehead ([MR30759]). In §3.2.7, we apply this result to deduce some closure properties for the class of homotopy equivalences (Propositions 3.2.7.1 and 3.2.7.3).

Structure

  • Subsection 3.2.1: Pointed Kan Complexes
  • Subsection 3.2.2: The Homotopy Groups of a Kan Complex
  • Subsection 3.2.3: The Group Structure on $\pi _{n}(X,x)$
  • Subsection 3.2.4: The Connecting Homomorphism
  • Subsection 3.2.5: The Long Exact Sequence of a Fibration
  • Subsection 3.2.6: Whitehead's Theorem
  • Subsection 3.2.7: Closure Properties of Homotopy Equivalences