Motivation: Deformation retraction is not symmetric and so is not an equivalence relation on topological spaces; build a (smallest) equivalence relation that includes deformation retraction.
Definition4.3.1.
Let \(X\) and \(Y\) be topological spaces and let \(g,h: X \to Y\) continuous functions. A homotopy from \(g\) to \(h\) is a family of maps \(f_t: X \to Y\) for all \(t \in I\) such that
\(f_0 = g\text{,}\)
\(f_1 = h\text{,}\) and
the function \(H: X \times I \to Y\) defined by \(H(p,t) := f_t(p)\text{,}\) for all \(p \in X\) and \(t \in I\text{,}\) is continuous.
The function \(H\) is also called a homotopy from \(g\) to \(h\text{.}\) The function \(g\) is said to be homotopic to \(h\text{,}\) written \(g \simeq h\text{.}\)
Lemma4.3.2.
If \(g,h: X \to Y\) are continuous functions and \(\{f_t\}_{t \in I}\) is a homotopy from \(g\) to \(h\text{,}\) then \(f_t: X \to X\) is continuous for all \(t \in I\text{.}\)
Lemma4.3.3.
Let \(X\) and \(Y\) be topological spaces. Homotopy is an equivalence relation on the set of functions \(X \to Y\text{.}\)
Proposition4.3.4.
If \(Y\) is a subspace of a topological space \(X\) and \(\{f_t\}_{t \in I}\) is a deformation retraction from \(X\) onto \(Y\text{,}\) then \(\{f_t\}_{t \in I}\) is a homotopy from \(f_0 = 1_X\) to \(f_1\text{,}\) and \(f_1\) is the composition of a retraction from \(X\) to \(Y\) with the inclusion map \(i: Y \into X\text{.}\)
Example4.3.5.
Examples
Definition4.3.6.
If \(Z\) is a topological space and \(f,g: Z \to (\R^n,\cT_{\text{Eucl}})\) are continuous functions, then the straight line homotopy from \(f\) to \(g\) is the function \(H: Z \times I \to \R^n\) defined by \(H(p,t) := (1-t)f(p) + tg(p)\) for all \(p \in X\) and \(t \in I\text{.}\)
Proposition4.3.7.
If \(H\) is straight line homotopy from \(f\) to \(g\text{,}\) then \(H\) is a homotopy from \(f\) to \(g\text{.}\)
Definition4.3.8.
Let \(X\) and \(Y\) be topological spaces, let \(Z\) be a subspace of \(X\text{,}\) and let \(g,h: X \to Y\) continuous functions that satisfy \(g|_Z = h|_Z\text{.}\) A homotopy from \(g\) to \(h\) relative to \(Z\) is a family of maps \(f_t: X \to Y\) for all \(t \in I\) such that
\(f_0 = g\text{,}\)
\(f_1 = h\text{,}\)
the function \(H: X \times I \to Y\) defined by \(H(p,t) := ft(p)\text{,}\) for all \(p \in X\) and \(t \in I\text{,}\) is continuous, and
\(f_t|_Z = g|_Z\) for all \(t \in I\text{.}\)
The function \(H\) is also called a homotopy from \(g\) to \(h\) relative to \(Z\text{.}\) The function \(g\) is said to be homotopic to \(h\) relative to \(Z\text{,}\) written \(g \simeq_Z h\text{.}\)
Definition4.3.9.
Topological spaces \(X\) and \(Y\) are homotopy equivalent, written \(X \simeq Y\text{,}\) if there are continuous functions \(f:X \to Y\) and \(g:Y \to X\) such that \(f \circ g \simeq 1_Y\) and \(g \circ f \simeq 1_X\text{.}\) The functions \(f\) and \(g\) are called homotopy equivalences, and each function is a homotopy inverse of the other.
Theorem4.3.10.
Homotopy equivalence is an equivalence relation on topological spaces.
Lemma4.3.11.
Let \(X\) and \(Y\) be topological spaces.
If \(X \cong Y\) then \(X \simeq Y\text{.}\)
If \(f,g: X \to Y\) and \(f = g\) then \(f \simeq g\text{.}\) If moreover \(Z\) is any subspace of \(X\) then \(f \simeq_Z g\) also.
Definition4.3.12.
A property \(P\) of topological spaces is a homotopy invariant if whenever \(X \simeq Y\) and \(X\) has property \(P\text{,}\) then \(Y\) has property \(P\text{.}\)
Remark4.3.13.
Use homotopy invariants to prove two spaces are NOT homotopy equivalent.
Theorem4.3.14.
Path-connectedness is a homotopy invariant.
Proposition4.3.15.
Compactness is not a homotopy invariant.
Theorem4.3.16.
If \(Y\) is a deformation retract of \(X\text{,}\) then \(X \simeq Y\text{.}\)
Theorem4.3.17.
Let \(X\) and \(Y\) be topological spaces. Then \(X \simeq Y\) if and only if there is a space \(Z\) such that \(X\) and \(Y\) are both deformation retracts of \(Z\text{.}\)
Corollary4.3.18.
Let \(\approx\) be the smallest equivalence relation on spaces such that whenever \(Y\) is a deformation retract of \(X\) then \(Y \approx X\text{.}\) Then \(X \simeq Z\) iff \(X \approx Z\text{.}\)
Definition4.3.19.
A space \(X\) is contractible if \(X \simeq P\text{,}\) where \(P =\{*\}\) is the topological space with one point.
Definition4.3.20.
Let \(X\) and \(Y\) be topological spaces. A map \(f:X \to Y\) is null homotopic if \(f\) is homotopic to a constant function.
Theorem4.3.21.
A space \(X\) is contractible iff the identity map \(1_X\) on \(X\) is nullhomotopic.
