Let \(p: E \to B\) be a covering map. Let \(\alpha\) and \(\beta\) be paths in \(B\) with \(\alpha(1) = \beta(0)\); let \(\tilde{\alpha}\) and \(\tilde{\beta}\) be liftings of them such that \(\tilde{\alpha}(1) = \tilde{\beta}(0)\). Show that \(\tilde{\alpha} * \tilde{\beta}\) is a lifting of \(\alpha * \beta\).
Solution
1. Definition of path concatenation and continuity:Proof:
1.1. By definition of path product, \(\alpha * \beta: [0, 1] \to B\) is defined by:\[(\alpha * \beta)(t) = \begin{cases} \alpha(2t) & \text{if } 0 \le t \le \frac{1}{2}, \\ \beta(2t - 1) & \text{if } \frac{1}{2} \le t \le 1. \end{cases}\]Since \(\alpha(1) = \beta(0)\), \(\alpha * \beta\) is continuous by the Pasting Lemma.
1.2. Similarly, because \(\tilde{\alpha}(1) = \tilde{\beta}(0)\), the concatenated path \(\tilde{\alpha} * \tilde{\beta}: [0, 1] \to E\) is defined by:\[(\tilde{\alpha} * \tilde{\beta})(t) = \begin{cases} \tilde{\alpha}(2t) & \text{if } 0 \le t \le \frac{1}{2}, \\ \tilde{\beta}(2t - 1) & \text{if } \frac{1}{2} \le t \le 1. \end{cases}\]Since \(\tilde{\alpha}(2 \cdot \frac{1}{2}) = \tilde{\alpha}(1) = \tilde{\beta}(0) = \tilde{\beta}(2 \cdot \frac{1}{2} - 1)\), the Pasting Lemma guarantees that \(\tilde{\alpha} * \tilde{\beta}\) is a continuous map from \([0, 1]\) to \(E\).
2. Verification of the covering projection equality \(p \circ (\tilde{\alpha} * \tilde{\beta}) = \alpha * \beta\):Proof:
2.1. For \(t \in [0, 1/2]\):\[p((\tilde{\alpha} * \tilde{\beta})(t)) = p(\tilde{\alpha}(2t)) = \alpha(2t) = (\alpha * \beta)(t),\]because \(\tilde{\alpha}\) is a lifting of \(\alpha\) (\(p \circ \tilde{\alpha} = \alpha\)).
2.2. For \(t \in [1/2, 1]\):\[p((\tilde{\alpha} * \tilde{\beta})(t)) = p(\tilde{\beta}(2t - 1)) = \beta(2t - 1) = (\alpha * \beta)(t),\]because \(\tilde{\beta}\) is a lifting of \(\beta\) (\(p \circ \tilde{\beta} = \beta\)).
2.3. Thus \(p((\tilde{\alpha} * \tilde{\beta})(t)) = (\alpha * \beta)(t)\) holds for all \(t \in [0, 1]\).
3. Q.E.D.By steps 1 and 2, \(\tilde{\alpha} * \tilde{\beta}\) is a continuous path into \(E\) satisfying \(p \circ (\tilde{\alpha} * \tilde{\beta}) = \alpha * \beta\).