Homology

Author

Eric W. Weisstein
February 15, 2008

This notebook downloaded from http://mathworld.wolfram.com/notebooks/Topology/Homology.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/Homology.html.

©2008 Wolfram Research, Inc. except for portions noted otherwise

Three cycles

Graphics[{{GrayLevel[.8],Rectangle[{-1,-1},{2,1}]},
{Blue,Rectangle[{-1/2,-1/2},{3/2,1/2}]},
{GrayLevel[.8],Disk[{0,0},1/3],Disk[{1,0},1/3]},
{PointSize[.02],Point[{0,0}],Point[{1,0}]}}]

The boundary of a region is homologous to zero

Graphics[{{GrayLevel[.8],Rectangle[{-1,-1},{2,1}]},
{RGBColor[1,1,0],Line[{{-1/2,-1/2},{3/2,-1/2},{3/2,1/2},{-1/2,1/2},{-1/2,-1/2}}],
Arrow[{{1/4,1/2},{3/4,1/2}}],
Arrow[{{3/4,-1/2},{1/4,-1/2}}]},
{RGBColor[1,0,0],Circle[{0,0},1/3],
Arrow[{{0,1/3},{.1,1/3}}],
Arrow[{{0,-1/3},{-.1,-1/3}}],
Circle[{1,0},1/3],
Arrow[{{1,1/3},{1.1,1/3}}],
Arrow[{{1,-1/3},{.9,-1/3}}]
},
{PointSize[.02],Point[{0,0}],Point[{1,0}],
Text["α",{0,1/5}],
Text["β",{1,1/5}],
Text["γ",{1/2,.6},{0,-1}]
}}]