5.03 Fundamental group of a mapping torus
Below the video you will find accompanying notes and some pre-class questions.
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Notes
Mapping tori
(0.00) Given a space \(X\) and a continuous map \(\phi\colon X\to
X\), we can form a new space \(MT(\phi)\) called the /mapping torus/
of \(\phi\) defined as follows: \[MT(\phi)=(X\times[0,1])/\sim,\]
where \((\phi(x),0)\sim(x,1)\). [NOTE: I got this the wrong way
around in the video!] In other words, you take your space \(X\),
multiply it with an interval and glue the two ends \(X\times\{0\}\)
and \(X\times \{1\}\) using the map \(\phi\).
(1.27) If \(\phi=id_X\) then you just get \(MT(id_X)=X\times
S^1\). For example, when \(X=S^1\) we have \(MT(id_X)\) is a
2-torus, hence the name mapping torus.
(2.36) Let \(X=S^1\) and \(\phi\colon S^1\to S^1\) be a reflection
(which switches clockwise to anticlockwise). Then \(MT(\phi)\) is
the Klein bottle.
Fundamental group of a mapping torus
(3.46) Let \(X\) be a CW complex and let \(\phi\colon X\to X\) be
a cellular map (i.e. it takes the \(n\)-skeleton to the
\(n\)-skeleton for each \(n\): the cellular approximation theorem
guarantees that any map of CW complexes is homotopic to a cellular
map). Then there is a CW structure on \(MT(\phi)\) where:
(6.25) For example, the 0-skeleton of \(MT(\phi)\) is just the
0-skeleton \(X^0\) of \(X\) (placed at \(X\times\{0\}\)). A 0-cell
\(\{x\}\) in \(X\) gives an interval \(\{x\}\times[0,1]\) in
\(MT(\phi)\) and the attaching map sends \((x,0\) to \(x\in X^0\) and
\((x,1)\) to \(\phi(x)\in X^0\).
- each \(k\)-cell \(e\) of \(X\) gives us a \(k\)-cell \(e\times\{0\}\) in \(X\times\{0\}\).
- each \(k\)-cell \(e\) of \(X\) also gives us a \((k+1)\)-cell
\(e\times[0,1]\) in \((X\times[0,1])/\sim\).
(8.10) Each 1-cell \(e\) in \(X\) gives us a 2-cell \(e\times[0,1]\) in \(MT(\phi)\). We can think of this 2-cell as a square and read its boundary off in the usual way. We see that if \(e\) attaches to the 0-cells \(x\) and \(y\) at either end then the boundary of the 2-cell \(e\times[0,1]\) attaches along:
- the path \(e\times\{0\}\) followed by,
- the path \(y\times[0,1]\) followed by,
- the path \(e\times\{1\}\) backwards followed by,
- the path \(y\times[0,1]\) backwards.
(9.29) Assume that \(X\) has only one 0-cell \(x\) (for
simplicity) and \(\phi\colon X\to X\) is a cellular map. Then
\[\pi_1(MT(\phi),(x,0))=\langle\mbox{generators of }\pi_1(X,x),\
c\ |\ \mbox{relations in }\pi_1(X,x),\ \mbox{a new relation for each
}1\mbox{-cell in }X\rangle.\] Here, \(c\) is the new generator
coming from the 1-cell \(\{x\}\times[0,1]\).
This follows from the previous theorem (which gave a CW structure on
the mapping torus) and the previous video (which gave us a way to
compute the fundamental group of any CW complex).
(13.33) Each 1-cell \(e\) in \(X\) gives the new relation \(c^{-1}\phi(e)^{-1}ce=1\), or \(cec^{-1}=\phi(e)\).
Examples
(14.50) For the Klein bottle, we have \(X=S^1\) and \(\phi\) is a
reflection. Take the cell structure on \(S^1\) with one 0-cell \(x\)
and one 1-cell \(e\) (if the 0-cell is a fixed point of the
reflection then the reflection is a cellular map). Since the
reflection switches the orientation of the circle, we have
\(\phi(e)=e^{-1}\). Our theorem gives us \[\pi_1(MT(\phi),x)=\langle
e,c\ |\ cec^{-1}=\phi(e)\rangle=\langle e,c\ |\ ce=e^{-1}c\rangle.\]
Computing fundamental groups of mapping tori will come in handy for
finding fundamental groups of knot complements.
Pre-class questions
- Let \(F\colon T^2\to T^2\) be the map \(F(x,y)=(y,x)\). What is the
fundamental group of the mapping torus \(MT(F)\)?
Navigation
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- Index of all lectures.