Let \(X\) be a topological space. Let \(f,g:I\to X\) be two paths from \(p\) to \(q\). Show that \(f\sim g\), that is, \(f\) is homotopic to \(g\) if and only if \(f\mathbin{\cdot}g^{-1}\sim c_p\) where \(g^{-1}\) is the inverse path to \(g\), \(c_p\) denotes the constant path based at \(p\) and the “\(\mathbin{\cdot}\)” denotes the product of (compatible) paths.
Show that \(X\) is simply connected if and only if any two paths in \(X\) with the same initial and terminal points are path homotopic.
Solution
Throughout, \(\simeq_p\) denotes path homotopy relative to the endpoints.
1. Path homotopy is compatible with concatenation, and for every path \(\alpha\) from \(a\) to \(b\) one has\[ \alpha\mathbin{\cdot}c_b\simeq_p\alpha, \qquad c_a\mathbin{\cdot}\alpha\simeq_p\alpha, \qquad \alpha\mathbin{\cdot}\alpha^{-1}\simeq_p c_a, \qquad \alpha^{-1}\mathbin{\cdot}\alpha\simeq_p c_b, \]and concatenation is associative up to path homotopy.
If \(H\) is a path homotopy from \(\alpha_0\) to \(\alpha_1\) and \(K\) is a path homotopy from \(\beta_0\) to \(\beta_1\), with the terminal point of each \(\alpha_i\) equal to the initial point of each \(\beta_i\), then \[ L(s,t)= \begin{cases} H(2s,t),&0\le s\le \frac12,\\ K(2s-1,t),&\frac12\le s\le1 \end{cases} \] is a path homotopy \[ \alpha_0\mathbin{\cdot}\beta_0 \simeq_p \alpha_1\mathbin{\cdot}\beta_1. \] Continuity follows from the pasting lemma because the two formulas agree at \(s=1/2\).
For cancellation, the loop \(\alpha\mathbin{\cdot}\alpha^{-1}\) contracts through \[ C(s,t)= \begin{cases} \alpha\bigl(2s(1-t)\bigr),&0\le s\le\frac12,\\ \alpha\bigl(2(1-s)(1-t)\bigr),&\frac12\le s\le1. \end{cases} \] At \(t=0\) this is \(\alpha\mathbin{\cdot}\alpha^{-1}\); at \(t=1\) it is constantly \(a\); and both endpoints remain \(a\) for all \(t\). Applying the same construction to \(\alpha^{-1}\) gives \[ \alpha^{-1}\mathbin{\cdot}\alpha\simeq_p c_b. \]
The constant-path identities and associativity follow from endpoint-preserving reparametrization. Indeed, if \(r_0,r_1:I\to I\) are continuous nondecreasing maps with \[ r_0(0)=r_1(0)=0, \qquad r_0(1)=r_1(1)=1, \] then for any path \(\gamma\) the formula \[ R(s,t)=\gamma\bigl((1-t)r_0(s)+t r_1(s)\bigr) \] is a path homotopy between \(\gamma\circ r_0\) and \(\gamma\circ r_1\). The paths \(\alpha\mathbin{\cdot}c_b\) and \(c_a\mathbin{\cdot}\alpha\) are reparametrizations of \(\alpha\), while the two parenthesizations of a triple concatenation are reparametrizations of the same path obtained by traversing the three factors successively.
2. If \(f\simeq_p g\), then\[ f\mathbin{\cdot}g^{-1}\simeq_p c_p. \]
By compatibility of path homotopy with concatenation from 1, \[ f\mathbin{\cdot}g^{-1} \simeq_p g\mathbin{\cdot}g^{-1}. \] The cancellation identity in 1 gives \[ g\mathbin{\cdot}g^{-1}\simeq_p c_p. \] By transitivity, \[ f\mathbin{\cdot}g^{-1}\simeq_p c_p. \]
3. If\[ f\mathbin{\cdot}g^{-1}\simeq_p c_p, \]then \(f\simeq_p g\).
Concatenate the assumed homotopy on the right with \(g\). By 1, \[ (f\mathbin{\cdot}g^{-1})\mathbin{\cdot}g \simeq_p c_p\mathbin{\cdot}g. \] Using associativity, cancellation, and the constant-path identities from 1, \[ \begin{aligned} (f\mathbin{\cdot}g^{-1})\mathbin{\cdot}g &\simeq_p f\mathbin{\cdot}(g^{-1}\mathbin{\cdot}g)\\ &\simeq_p f\mathbin{\cdot}c_q\\ &\simeq_p f, \end{aligned} \] while \[ c_p\mathbin{\cdot}g\simeq_p g. \] Hence \(f\simeq_p g\). This proves part (i).
4. Under the standard definition of simply connected, if \(X\) is simply connected, then any two paths in \(X\) with the same initial and terminal points are path homotopic.
Let \(f,g:I\to X\) both run from \(p\) to \(q\). Then \[ f\mathbin{\cdot}g^{-1} \] is a loop based at \(p\). Since \(X\) is simply connected, every based loop is path homotopic to the constant loop, so \[ f\mathbin{\cdot}g^{-1}\simeq_p c_p. \] By 3, \[ f\simeq_p g. \]
5. The converse in part (ii) is false as literally stated if “simply connected” includes path-connectedness.
Let \[ X=\{0,1\} \] with the discrete topology. Every continuous map \[ \alpha:I\to X \] is constant because \(I\) is connected and the image of a connected space is connected, whereas the only connected subsets of a discrete space are singletons. Consequently, whenever two paths in \(X\) have the same initial and terminal points, they are the same constant path and hence are path homotopic.
However, \(X\) is not path-connected: there is no path from \(0\) to \(1\). Thus \(X\) is not simply connected under the standard definition. So the path-homotopy condition alone does not imply simple connectedness.
6. The intended converse becomes correct after adding the hypothesis that \(X\) is path-connected.
Assume \(X\) is path-connected and that any two paths with the same initial and terminal points are path homotopic. Fix \(p\in X\) and let \[ \ell:I\to X \] be any loop based at \(p\). The loop \(\ell\) and the constant loop \(c_p\) have the same initial and terminal points, so by hypothesis \[ \ell\simeq_p c_p. \] Hence every loop based at \(p\) is null-homotopic, and therefore \[ \pi_1(X,p)=0. \] Together with the assumed path-connectedness, this says that \(X\) is simply connected.
7. Thus part (ii) is correct either with path-connectedness included as a hypothesis in the converse, or under the nonstandard convention that “simply connected” means only that every loop is null-homotopic componentwise.
6. If one drops path-connectedness from the definition of simply connected, the proof in 6 applies within each path component without any additional global hypothesis.