(0.00) If \(X\) is a CW complex and \(A\) is a closed subcomplex
then the pair \((X,A)\) has the HEP.
(0.30) A closed subcomplex is a union of closed cells of \(X\)
such that \(X\) is obtained by adding cells to \(A\). The pair
\((X,A)\) (where \(X\) is a CW complex and \(A\) is a closed
subcomplex) is sometimes called a CW pair.
(1.18) If \(e\) is a \(k\)-dimensional disc then there is a
continuous map \(e\times[0,1]\to(\partial e\times[0,1])\cup
e\times\{0\}\).
Below is a picture of the map in the case \(k=1\). If we embed
\(e\times[0,1]\) into \(\mathbf{R}^{k+1}\) as the subset
\[\{(x,y)\in\mathbf{R}^k\times\mathbf{R}\ :\ |x|\leq 1,\
y\in[0,1]\}\] then the linear projection from the point
\((0,\ldots,0,2)\) to the indicated subset (\(d\), in red) of the
boundary \(\partial(e\times[0,1])\) is the required map.
(4.16) If \(X\) is a space obtained from \(A\) by attaching a
single \(k\)-cell \(e\) then the pair \((X,A)\) has the HEP.
(4.58) Recall that the HEP states that if we are given a function
\(F\colon X\to Y\) and a homotopy \(h_t\colon A\to A\) of
\(h_0=F|_A\) then there is a homotopy \(H_t\colon X\to X\) of
\(H_0=F\) such that \((H_t)|_A=h_t\). Recall that \(H_t(x)\) means
\(H(x,t)\) for some continuous map \(H\colon X\times[0,1]\to Y\).
(6.38) In our case, \(X=A\cup_{\varphi}e\) for some attaching map
\(\varphi\colon \partial e\to A\); we have \(H\) already defined on
\(A\times[0,1]\) (where it is equal to \(h\)) and on
\(e\times\{0\}\) (where it is equal to \(F\)) and we want to define
the homotopy \(H\) on \(X\times[0,1]\).
(7.50) The only points where we do not already know \(H\) are the
points of \(e\times[0,1]\subset X\times[0,1]\). If we let \(G\colon
e\times[0,1]\to(\partial e\times[0,1])\cup e\times\{0\}\) be the map
from the previous lemma then we can define \(H|_{e\times[0,1]}\) to
be the composition \(H\circ G\): \(G\) takes points of
\(e\times[0,1]\) into the region \((\partial e\times[0,1])\cup
e\times\{0\}\), which is attached to \((A\times[0,1])\cup
e\times\{0\}\), where we already know how to define \(H\), so this
circular-looking definition actually makes sense.
(10.30) The theorem now follows by induction: \(X\) is obtained from
\(A\) by attaching cells, and each time you attach a cell you can
extend a homotopy from \(A\) over the new cell.
1-dimensional CW complexes
(11.03) Any connected 1-dimensional CW complex \(X\) is homotopy
equivalent to a wedge of (possibly infinitely many) circles (in
other words, a single 0-cell with a bunch of 1-cells attached).
(12.01) We need to find a contractible subcomplex \(A\) containing
all the 0-cells: by the previous theorem, \((X,A)\) has the HEP. By
the theorem we proved last time, this means that \(X\simeq X/A\),
and \(X/A\) is a CW complex with a single 0-cell (the image of \(A\)
under the quotient map) and a bunch of 1-cells (the 1-cells not
contained in \(A\)).
(13.17) To ``find'' a contractible subcomplex, we partially order
the set of contractible subcomplexes by inclusion. By Zorn's
lemma, there is a contractible subcomplex which is maximal with
respect to this partial order (i.e. one which is not contained in
any bigger contractible subcomplex). There may be many choices of
maximal contractible subcomplex.
(14.56) A maximal contractible subcomplex \(A\) necessarily passes
through all the 0-cells: otherwise we could add an edge connecting
\(A\) to a 0-cell it does not contain to obtain a bigger
contractible subcomplex.
(16.26) I'm taking Zorn's lemma as an axiom here: it's equivalent to
another axiom of set theory known as the Axiom of Choice. This is the
assumption that you can make infinitely many choices simultaneously
(more precisely, that an infinite product of nonempty sets is
nonempty). You could think of our proof as saying: start with a single
edge and start adding more edges to get a maximal contractible
subcomplex; you have to make infinitely many choices along the way,
and there's no algorithm for doing this if you don't know anything
more about your CW complex. Therefore we just assume we can make these
choices. This is a non-constructive proof (we don't give an algorithm
for making the choices) and some mathematicians, logicians and
philosophers find this kind of argument troubling.
Pre-class questions
Look back at the argument that any 1-dimensional connected CW
complex is homotopy equivalent to a wedge of circles. Where did we
use the fact that it is connected?
