Let \(A\) be a subspace of a topological space \(X\text{.}\) The space \(A\) is a retract of \(X\) if there is a continuous function \(r: X \to A\) satisfying \(r(a) = a\) for all \(a \in A\text{.}\) The function \(r\) is called a retraction from \(X\) to \(A\text{.}\)
Definition4.2.2.
Let \(Y\) be a subspace of a topological space \(X\text{.}\) A deformation retraction of \(X\) onto \(Y\) is a family of maps \(f_t: X \to X\) for all \(t \in I\) such that:
\(f_0 = 1_X\text{,}\)
\(f_1(X) = Y\text{,}\)
\(f_t|_Y = 1_Y\) for all \(t \in I\text{,}\) and
the function \(H: X \times I \to X\) defined by \(H(p,t) := ft(p)\text{,}\) for all \(p \in X\) and \(t \in I\text{,}\) is continuous.
The function \(H\) is also called a deformation retraction from \(X\) to \(Y\text{.}\) The space \(Y\) is called a deformation retract of \(X\text{,}\) written \(X ⇝ Y\text{.}\)
Lemma4.2.3.
If \(\{f_t\}_{t \in I}\) is a deformation retraction from a topological space \(X\) onto a subspace \(Y\text{,}\) then \(f_t: X \to X\) is continuous for all \(t \in I\text{.}\) Moreover, \(f_1\) is the composition of a retraction from \(X\) to \(Y\) with the inclusion map \(i: Y \into X\text{.}\)
Definition4.2.4.
Let \(X\) and \(Y\) be topological spaces and let \(f:X \to Y\) be a continuous map. Let \((X \times I) ∐ Y\) have the disjoint union topology (where \(X \times I\) has the product topology), and let \(\sim\) be the minimal equivalence relation on \((X \times I) ∐ Y\) such that \((p,1) \sim f(p)\) for all \(p \in X\text{.}\) The mapping cylinder of \(f\) is the quotient space \(X_f := ((X \times I) ∐ Y)/\sim\text{.}\)
Theorem4.2.5.
If \(X_f\) is the mapping cylinder associated to a continuous function \(f:X \to Y\text{,}\) then the functions \(g:X \to X_f\) defined by \(g(p) := [(p,0)]\) for all \(p \in X\) and \(h:Y \to X_f\) defined by \(h(y) := [y]\) for all \(y \in Y\) are embeddings.
Theorem4.2.6.
If \(X\) and \(Y\) are topological spaces and \(f:X \to Y\) is a continuous map, then \(Y\) is homeomorphic to a deformation retract of the mapping cylinder \(X_f\) associated to \(f\text{.}\)
Definition4.2.7.
Let \(X\) be a topological space and let \(f:X \to X\) be a continuous map. Let \(X \times I\) have the product topology, and let \(\sim\) be the minimal equivalence relation on \(X \times I\) such that \((p,1) \sim (f(p),0)\) for all \(p \in X\text{.}\) The mapping torus of \(f\) is the quotient space \(T_f := (X \times I)/\sim\text{.}\)
Remark4.2.8.
If \(X\) is a topological space and \(f:X \to X\) is a continuous map, then \(X\) may not be homeomorphic to a deformation retract of the mapping cylinder \(T_f\) associated to \(f\text{.}\)
