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:

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

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 $MT(F)$ ?

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Index of all lectures.