# 5.03 Fundamental group of a mapping torus

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

# 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:
• 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).

## 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

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 $$MT(F)$$?