Homotopy

A homotopy is a very general kind of mathematical eqivalence between functions. In the case of winding numbers, we are interested in equivalent curves, so we want to think of a curve as a function <applet code="webeq.Main" width=121 height=36 align=middle> <param name=eq value="\gamma:[0,1] \mapsto R^2"> <param name=parser value="webtex"> <param name=allow_selection value=true> <param name=linewrap value=true> </applet> <IMG WIDTH="102" HEIGHT="33" ALIGN="MIDDLE" BORDER="0" SRC="img7.gif" ALT="$ \gamma:[0,1] \mapsto R^2$"> .

If <applet code="webeq.Main" width=13 height=24 align=middle> <param name=eq value="\gamma"> <param name=parser value="webtex"> <param name=allow_selection value=true> <param name=linewrap value=true> </applet> <IMG WIDTH="13" HEIGHT="28" ALIGN="MIDDLE" BORDER="0" SRC="img8.gif" ALT="$ \gamma$"> and <applet code="webeq.Main" width=14 height=24 align=middle> <param name=eq value="\sigma"> <param name=parser value="webtex"> <param name=allow_selection value=true> <param name=linewrap value=true> </applet> <IMG WIDTH="14" HEIGHT="14" ALIGN="BOTTOM" BORDER="0" SRC="img9.gif" ALT="$ \sigma$"> are two curves, we say they are homotopic if there exists a continuous function <applet code="webeq.Main" width=190 height=36 align=middle> <param name=eq value="H: [0,1] \times [0,1] \rightarrow R^2"> <param name=parser value="webtex"> <param name=allow_selection value=true> <param name=linewrap value=true> </applet> <IMG WIDTH="159" HEIGHT="33" ALIGN="MIDDLE" BORDER="0" SRC="img10.gif" ALT="$ H: [0,1] \times [0,1] \rightarrow R^2$"> with the following properties:

1.

<applet code="webeq.Main" width=108 height=28 align=middle> <param name=eq value="H(0,t) = \gamma(t)"> <param name=parser value="webtex"> <param name=allow_selection value=true> <param name=linewrap value=true> </applet> <IMG WIDTH="100" HEIGHT="31" ALIGN="MIDDLE" BORDER="0" SRC="img11.gif" ALT="$ H(0,t) = \gamma(t)$">

2.

<applet code="webeq.Main" width=109 height=28 align=middle> <param name=eq value="H(1,t)=\sigma(t)"> <param name=parser value="webtex"> <param name=allow_selection value=true> <param name=linewrap value=true> </applet> <IMG WIDTH="101" HEIGHT="31" ALIGN="MIDDLE" BORDER="0" SRC="img12.gif" ALT="$ H(1,t)=\sigma(t)$">

3.

<applet code="webeq.Main" width=132 height=28 align=middle> <param name=eq value="H(s,0) = H(s,1) \mbox{ for all } s \in [0,1]"> <param name=parser value="webtex"> <param name=allow_selection value=true> <param name=linewrap value=true> </applet> <IMG WIDTH="124" HEIGHT="31" ALIGN="MIDDLE" BORDER="0" SRC="img13.gif" ALT="$ H(s,0) = H(s,1)$">&nbsp; &nbsp; for all <IMG WIDTH="63" HEIGHT="31" ALIGN="MIDDLE" BORDER="0" SRC="img14.gif" ALT="$ s \in [0,1]$"> .

Note for fixed $s_0$, <applet code="webeq.Main" width=127 height=28 align=middle> <param name=eq value="\gamma_{s_0}(t) = H(s_0,t)"> <param name=parser value="webtex"> <param name=allow_selection value=true> <param name=linewrap value=true> </applet> <IMG WIDTH="119" HEIGHT="31" ALIGN="MIDDLE" BORDER="0" SRC="img16.gif" ALT="$ \gamma_{s_0}(t) = H(s_0,t)$"> is a curve in the plane. Because of the third condition, its ends must meet up. The other two conditions say the when $s=0$, the curve you get is <applet code="webeq.Main" width=13 height=24 align=middle> <param name=eq value="\gamma"> <param name=parser value="webtex"> <param name=allow_selection value=true> <param name=linewrap value=true> </applet> <IMG WIDTH="13" HEIGHT="28" ALIGN="MIDDLE" BORDER="0" SRC="img8.gif" ALT="$ \gamma$"> and when $s=1$, you get <applet code="webeq.Main" width=14 height=24 align=middle> <param name=eq value="\sigma"> <param name=parser value="webtex"> <param name=allow_selection value=true> <param name=linewrap value=true> </applet> <IMG WIDTH="14" HEIGHT="14" ALIGN="BOTTOM" BORDER="0" SRC="img9.gif" ALT="$ \sigma$"> .

If you think of $s$ as a time parameter, we begin deforming <applet code="webeq.Main" width=13 height=24 align=middle> <param name=eq value="\gamma"> <param name=parser value="webtex"> <param name=allow_selection value=true> <param name=linewrap value=true> </applet> <IMG WIDTH="13" HEIGHT="28" ALIGN="MIDDLE" BORDER="0" SRC="img8.gif" ALT="$ \gamma$"> at $s=0$, and when we reach time $s=1$, the curve has evolved into <applet code="webeq.Main" width=14 height=24 align=middle> <param name=eq value="\sigma"> <param name=parser value="webtex"> <param name=allow_selection value=true> <param name=linewrap value=true> </applet> <IMG WIDTH="14" HEIGHT="14" ALIGN="BOTTOM" BORDER="0" SRC="img9.gif" ALT="$ \sigma$"> . The curve <applet code="webeq.Main" width=127 height=28 align=middle> <param name=eq value="\gamma_{s_0}(t) = H(s_0,t)"> <param name=parser value="webtex"> <param name=allow_selection value=true> <param name=linewrap value=true> </applet> <IMG WIDTH="119" HEIGHT="31" ALIGN="MIDDLE" BORDER="0" SRC="img16.gif" ALT="$ \gamma_{s_0}(t) = H(s_0,t)$"> is the curve it has evolved into at time $s_0$.