This note is a reading record, not a replacement for Hatcher’s exposition. The main source is Allen Hatcher, Algebraic Topology, Chapter 0, pp. 1–4 (PDF pp. 10–13). The Stanford MATH 215A notes are used only as a parallel explanation of the same first ideas. I will keep one question in view: when may a complicated space be replaced by a simpler one without changing its homotopy-level information?
1. A homotopy is a map
Let $X$ and $Y$ be topological spaces, and write $I=[0,1]$ for the unit interval. A map is assumed continuous unless stated otherwise. Given maps $f,g:X\to Y$, a homotopy from $f$ to $g$ is a continuous map
$$ H:X\times I\longrightarrow Y $$such that
$$ H(x,0)=f(x),\qquad H(x,1)=g(x)\qquad (x\in X). $$For each $t\in I$, the map $H_t:X\to Y$ defined by $H_t(x)=H(x,t)$ is one stage of the deformation. The parameter $t$ records time; it is not a second coordinate of the point $x$. We write $f\simeq g$ when such an $H$ exists.
The important point is that continuity belongs to the joint map $H(x,t)$. A collection of pictures, or a separate continuous path for each $x$, is not enough until these paths vary continuously with $x$.
A first calculation. For $f,g:\mathbb R\to\mathbb R$ given by $f(x)=x$ and $g(x)=x^2$, set $H(x,t)=(1-t)x+tx^2$. This is a polynomial in $(x,t)$, hence continuous, and it satisfies the two endpoint conditions. Therefore $f\simeq g$. The formula is more informative than the picture: it tells us exactly what every point does at every time.
The relation $\simeq$ is an equivalence relation on maps with the same domain and codomain. Reflexivity uses the constant homotopy $H(x,t)=f(x)$. If $H$ runs from $f$ to $g$, then $H(x,1-t)$ runs from $g$ to $f$. Finally, if $H$ runs from $f$ to $g$ and $K$ runs from $g$ to $h$, concatenate them by
$$ L(x,t)= \begin{cases} H(x,2t),&0\leq t\leq\frac12,\\ K(x,2t-1),&\frac12\leq t\leq1. \end{cases} $$The two formulas agree at $t=\frac12$ because $H(x,1)=g(x)=K(x,0)$, so the pasting lemma makes $L$ continuous. This elementary fact is what allows us to speak of a map’s homotopy class.
2. Retractions and the fixed part of a deformation
Let $A\subseteq X$ be a subspace. A retraction of $X$ onto $A$ is a map $r:X\to A$ satisfying
$$ r(a)=a\qquad (a\in A). $$The map $r$ is allowed to move points outside $A$, but it must leave every point of $A$ fixed. Let $i:A\hookrightarrow X$ denote the inclusion map. We may also view the retraction as the map $i\circ r:X\to X$ whose image is contained in $A$.
A deformation retraction of $X$ onto $A$ is a homotopy $H:X\times I\to X$ satisfying
$$ H(x,0)=x,\qquad H(x,1)=i(r(x)),\qquad H(a,t)=a. $$The first condition starts at the identity map of $X$; the second ends at a retraction; the third says that the target subspace is fixed throughout the deformation. With this notation,
$$ i\circ r\simeq\operatorname{id}_X\quad\text{and}\quad r\circ i=\operatorname{id}_A. $$The last equality is strict, while the first is only a homotopy. This distinction is easy to lose in an informal drawing and is one reason to write the three conditions separately.
3. Our complete example: an annulus retracts to its core circle
Consider the closed annulus
$$ A=\{x\in\mathbb R^2:1\leq\lVert x\rVert\leq 2\} $$and its core circle
$$ C=S^1=\{x\in\mathbb R^2:\lVert x\rVert=1\}. $$Here $\lVert x\rVert$ is the Euclidean norm. Define
$$ H:A\times I\longrightarrow A, \qquad H(x,t)=\left((1-t)+\frac{t}{\lVert x\rVert}\right)x. $$Why does this formula work?
- Since $1\leq\lVert x\rVert\leq2$, the denominator never vanishes, so $H$ is continuous.
- At $t=0$, $H(x,0)=x$.
- At $t=1$, $H(x,1)=x/\lVert x\rVert\in C$.
- If $x\in C$, then $\lVert x\rVert=1$, so $H(x,t)=x$ for every $t$.
- The norm of $H(x,t)$ is $(1-t)\lVert x\rVert+t$, which lies between $1$ and $2$; hence the image stays inside $A$.
Thus $H$ is a deformation retraction of the annulus onto its core circle. The geometric phrase “push every point radially inward” becomes a theorem only after these five checks.
4. What the source figure is showing
Source excerpt: Allen Hatcher, Algebraic Topology, Chapter 0, p. 2 (PDF p. 11).
The figure comes from Hatcher’s discussion of deformation retractions. It shows a Möbius band being collapsed onto its core circle through intermediate stages. The picture supplies the geometric idea; it does not replace the continuity condition in the definition. In the annulus calculation above, the formula plays the role that the sequence of drawings plays in the source.
5. Homotopy equivalence
Maps $f:X\to Y$ and $g:Y\to X$ are homotopy inverses if
$$ g\circ f\simeq\operatorname{id}_X, \qquad f\circ g\simeq\operatorname{id}_Y. $$In this case $X$ and $Y$ are homotopy equivalent, written $X\simeq Y$. A deformation retraction is therefore a particularly concrete homotopy equivalence: the inclusion $i:A\hookrightarrow X$ and the retraction $r:X\to A$ are homotopy inverses.
This relation is weaker than homeomorphism. A homeomorphism preserves the whole topological structure, while a homotopy equivalence preserves only the structure visible to homotopy-invariant constructions. The point of algebraic topology is that fundamental groups, homology groups, and cohomology groups will be examples of such constructions.
Reading rule. Whenever a source says that one space “shrinks to” another, ask three questions: What are the two spaces? What is the actual map $H(x,t)$? Which points are required to remain fixed? If one of these is missing, the sentence is intuition rather than a completed proof.
6. Exercises for this reading
- Verify directly that the annulus formula above is continuous and fixes $C$ pointwise.
- For a point $a\in X$, show that the constant map $X\to\{a\}$ is a retraction onto the subspace $\{a\}$, but that the existence of this retraction alone does not give a deformation retraction.
- Write an explicit homotopy between two constant maps $\mathbb R^m\to\mathbb R^n$.
The next note will read quotient spaces and attaching maps before turning to Hatcher’s CW-complex construction. This will be our first place where a geometric picture and a precise topological construction must be kept together.
Sources
- Allen Hatcher, Algebraic Topology, Chapter 0, pp. 1–4: author’s PDF.
- Andrew Lin, notes for Zhenkun Li’s Stanford MATH 215A, pp. 1–3: course notes.