= Euler characteristic of a torus
{c}
{title2=$\chi(T^2)=1-2+1=0$}
Identify opposite sides of a <square> by translations to obtain a <torus>. All four corners become one <vertex>; the horizontal sides give one <topological cell> $a$ and the vertical sides one <topological cell> $b$. The open <square> supplies one two-dimensional <topological cell>. Thus $(c_0,c_1,c_2)=(1,2,1)$ and $\chi=0$. The <cellular boundary> of each one-dimensional <topological cell> is zero because its endpoints agree. Traversing the face boundary gives the <group commutator> $aba^{-1}b^{-1}$, whose signed counts of both $a$ and $b$ vanish. Hence its <cellular boundary> is also zero, and $(b_0,b_1,b_2)=(1,2,1)$.<A Concise Course in Algebraic Topology by J. P. May>[A Concise Course in Algebraic Topology, Chapter 13, §5, p. 103, Example (i), the torus cellular chains and integral homology]{ref}
\Image[/euler-characteristic-sphere-torus-cells.png]
{title=The tetrahedral sphere and the square presentation of a torus have different alternating cell counts}
The boundary of a <tetrahedron> is a <two-sphere>, with four <vertices>, six <edges> and four triangular <topological cells>: $\chi=4-6+4=2$. Its rational <Betti numbers> are $(1,0,1)$ by <homology of a sphere>. The <torus>'s two independent one-dimensional <homology classes> cancel its degree-zero and degree-two contributions. The <torus> and <annulus> both have <Euler characteristic> zero, yet different <homology groups>; one integer cannot recover all of <homology>.
Back to article page
