\( \DeclareMathOperator{\GL}{GL} \DeclareMathOperator{\Map}{Map} \DeclareMathOperator{\Hom}{Hom} \DeclareMathOperator{\Tor}{Tor} \DeclareMathOperator{\Ext}{Ext} \DeclareMathOperator{\im}{im} \DeclareMathOperator{\id}{id} \DeclareMathOperator{\coker}{coker} \DeclareMathOperator{\CC}{\mathbf{C}} \DeclareMathOperator{\QQ}{\mathbf{Q}} \DeclareMathOperator{\RR}{\mathbf{R}} \DeclareMathOperator{\ZZ}{\mathbf{Z}} \DeclareMathOperator{\RP}{\mathbf{RP}} \DeclareMathOperator{\C}{C_*} \DeclareMathOperator{\U}{\mathcal{U}} \)

Section1.2Homotopy

Subsection1.2.1Homotopies

Definition1.2.1Homotopic maps
Maps \(f_0,f_1\colon X \to Y\) are said to be homotopic if there is a continuous map \(F\colon X\times I \to Y\) such that \[ F(x,0) = f_0(x)\text{ and }F(x,1) = f_1(x)\ \forall x\in X. \]

We let \(\Map(X,Y) = \{f\colon X \to Y \text{ continuous}\}\). Then letting \(f_t(x) = F(x,t)\) in the above definition we see that \(f_t\) is a path from \(f_0\) to \(f_1\) in \(\Map(X,Y)\).

Example1.2.2
  1. \(X = Y = \RR^n\), \(f_0(\overline{x}) = \overline{0}\) and \(f_1(\overline{x}) = \overline{x}\) are homotopic via \(f_t(\overline{x}) = t\overline{x}\).
  2. \(S^1 = \{z\in \CC : |z| = 1\}\) then
  3. \(S^n = \{ \overline{x} \in \RR^n : |\overline{x}| = 1\}\)
Definition1.2.6Contractible space
\(X\) is contractible if \(1_X\) is homotopic to a constant map.

Given a space \(X\) how can we tell if \(X\) is contractible? If \(X\) is contractible then it must be path connected for one.

Similarly if \([S^1, X]\) has more than one element then \(X\) is not contractible.

Definition1.2.8Simply connected
We say \(X\) is simply connected if \([S^1, X]\) has only one element. We say two spaces \(X\) and \(Y\) are homotopy equivalent if there exists \(f\colon X \to Y\) and \(g\colon Y \to X\) such that \(g\circ f \sim 1_X\) and \(f\circ g \sim 1_Y\).
Example1.2.9

\(X\) is contractible if and only if \(X \sim \{p\}\).

Exercise 1.2.10
If \(X_0 \sim X_1\) and \(Y_0 \sim Y_1\) then \([X_0, Y_0]\) and \([X_1, Y_1]\) are in bijection.

Given \(X\) and \(Y\) how can we determine if \(X\sim Y\)? How do we determine \([X,Y]\)? For example is \(S^n \sim S^m\).

Subsection1.2.2Homotopy groups

Definition1.2.11Map of pairs
A map of pairs\(f\colon (X, A) \to (Y, B)\) is a map \(f\colon X \to Y\) with sets \(A\subset X\) and \(B\subset Y\) such that \(f(A)\subset B\). If we have maps of pairs \(f_0, f_1\colon (X,A) \to (Y,B)\) then we write \(f_0\sim f_1\) if there exists \(F\colon(X\times I, A\times I) \to (Y,B)\) such that \(F(x,0) = f_0(x)\) and \(F(x,1) = f_1(x)\).
Definition1.2.12Homotopy groups
If \(*\in X\) then the \(n\)th homotopy group is \[\pi_n(X, *) = [(D^n, S^{n-1}) \to (X, \{*\})].\]

We now note several properties of this definition:

  1. \(\pi_0(X, *) =\) set of path components of \(X\).
  2. \(\pi_1(X, *)\) is a group.
    3. \(\pi_n(X, *)\) is an abelian group.
  3. \(\pi_n\) is a functor \[ \left\{ \begin{subarray}{l}\text{pointed spaces} \\ \text{pointed maps} \end{subarray}\right\} \to \left\{\begin{subarray}{l} \text{groups}\\ \text{group homomorphisms}\end{subarray}\right\}. \] So given \[f\colon(X, p)\to(Y,q)\] we get \[f_*\colon \pi_n(X,p)\to\pi_n(y,q)\] defined by \[f_*(\gamma) = f\circ \gamma.\]

Example1.2.13Homotopy groups of \(S^2\)
\(n\) 1 2 3 4 5 6 7
\(\pi_n(S^2)\) 0 \(\ZZ\) \(\ZZ\) \(\ZZ/2\) \(\ZZ/2\) \(\ZZ/12\) \(\ZZ/15\)