(0.00) In this section, we will define a notion of homotopy
equivalence of spaces, which you should think of as saying two
spaces can be deformed into one another; for example, the symbols
\(\theta\) and 8 are homotopy equivalent (just by contracting the
contractible bar midway up the \(\theta\)). The fundamental group is a
homotopy invariant, and we will eventually be able to compute it
simply by looking at a space, and noticing that it's homotopy
equivalent to a simpler space whose fundamental group we already know.
Homotopy of maps
(1.23) We first need the notion of homotopy of maps.
Let \(X,Y\) be spaces. Two continuous maps \(F,G\colon X\to Y\) are
homotopic if there is a continuous homotopy \(H\colon
X\times[0,1]\to Y\) such that \(H(x,0)=F(x)\) and \(H(x,1)=G(x)\)
for all \(x\in X\). We write \(F\simeq G\).
(2.57) When \(X=S^1\), this coincides with the notion of free
homotopy of loops. In general, we will think of this as a
one-parameter family of maps \(F_s\colon X\to Y\) (where
\(F_s(x)=H(x,s)\)) interpolating between \(F_0=F\) and \(F_1=G\).
(4.13) The analogue of a based homotopy is the following.
Let \(X,Y\) be spaces and let \(A\subset X\) be a subset. Two
continuous maps \(F,G\colon X\to Y\) satisfying \(F|_A=G|_A\) are
homotopic rel \(A\) if there is a homotopy \(H\colon
X\times[0,1]\to Y\) such that \(H(x,0)=F(x)\) and \(H(x,1)=G(x)\)
for all \(x\in X\) and \(H(x,s)\) is independent of \(s\) for all
\(x\in A\).
This gives based homotopy (or homotopy with fixed endpoints) if
\(A=\{0,1\}\subset[0,1]=X\).
Contractible spaces
(5.40) Let \(X=Y=\mathbf{R}^n\), let
\(F\colon\mathbf{R}^n\to\mathbf{R}^n\) be the identity map
\(F(x)=x\), and let \(G\colon\mathbf{R}^n\to\mathbf{R}^n\) be the
constant map \(G(x)=0\). Then \(F\simeq G\) via the homotopy \(H(x,s)=sx\).
(7.19) A space \(X\) is contractible if the identity map
\(id_X\) is homotopic to a constant map. A homotopy from the
identity to a constant map is called a nullhomotopy. The previous
example shows that \(\mathbf{R}^n\) is contractible.
Homotopy equivalence
(8.55) A map \(F\colon X\to Y\) is a homotopy equivalence if
there exists a map \(G\colon Y\to X\) such that \(F\circ G\simeq
id_Y\) and \(G\circ F\simeq id_X\). We say that \(G\) is a homotopy
inverse for \(F\) (note that \(F\) does not need to be invertible).
(11.08) If there exists a homotopy equivalence \(F\colon X\to Y\),
we say that \(X\) and \(Y\) are homotopy equivalent and we write
\(X\simeq Y\).
(11.44) A contractible space \(X\) is homotopy equivalent to a
point \(\{p\}\).
Let \(F\colon X\to \{p\}\) be the constant map \(F(x)=p\) and let
\(G\colon \{p\}\to X\) be the inclusion of a point \(G(p)=x_0\). We
have
\[F\circ G(p)=F(x_0)=p,\quad\Rightarrow\quad F\circ G=id_{\{p\}},\]
and
\[G\circ F(x)=G(p)=x_0,\quad\Rightarrow\quad G\circ
F=\mbox{const}.\]
Since \(X\) is contractible, \(G\circ F\) is homotopic to the
identity on \(X\).
Examples
(14.56) In the next section, we will see that homotopy equivalent
spaces have isomorphic fundamental groups. For the rest of this
section, we will look at examples of homotopy equivalent spaces.
(15.24) Let \(Y\) be the following (grey) subset of the plane:
and let \(X\) be the blue figure 8 (drawn as a subset of
\(Y\)). Then \(X\simeq Y\). Let \(G\colon Y\to X\) be the map which
projects along the red arrows in the figure and let \(F\colon X\to
Y\) be the inclusion map. Then \(G\circ F\) is the identity and
\(F\circ G\) is homotopic to the identity: a homotopy from the
identity to \(F\circ G\) can be obtained by flowing along the red
arrows (i.e. \(H(x,s)\) is the point you get from \(x\) by moving a
fraction \(s\) along the red arrow through \(x\) towards the figure
8).
(18.21) Let \(B\) be the unit ball in \(\mathbf{R}^3\) and let
\(C\) be a small tubular neighbourhood of the \(x\)-axis. Let
\(X=B\setminus C\). Then \(X\) is homotopy equivalent to the circle
(you can see it by imagining the circle sitting in the \(yz\)-plane
at radius \(1/2\) (so inside \(X\)) and ``squishing'' \(X\) down
onto this core circle by following arrows in much the same way as
the previous example.
(19.47) Some examples you might like to think about:
\(\mathbf{R}^3\setminus K\) where \(K\) is a neighbourhood of the
unit circle in the \((z=0)\)-plane.
\(\mathbf{R}^3\setminus (K_{-1}\cup K_1)\), where \(K_j\) is a
neighbourhood of the unit circle in the plane \(z=j\), \(j=\pm 1\).
Pre-class questions
How many homotopy equivalence classes of symbols are there in the
following list? ABCDEFGHIJKLMNOPQRSTUVWXYZ0123456789
Show that homotopy equivalence of spaces is an equivalence
relation.
