5.03 Fundamental group of a mapping torus


Below the video you will find accompanying notes and some pre-class questions.


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:

  • 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 \((X\times[0,1])/\sim\).

(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 \(\phi(x)\in X^0\).

(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.

(9.18) We needed \(\phi\) to be cellular in order for the cells to attach to the skeleton of the correct dimension.

(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 \(\{x\}\times[0,1]\).

(13.33) Each 1-cell \(e\) in \(X\) gives the new relation \(c^{-1}\phi(e)^{-1}ce=1\), or \(cec^{-1}=\phi(e)\).

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).


(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

  1. 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 \(MT(F)\)?


CC-BY-SA, Jonny Evans 2017