Attaching a Two-Cell to a Circle

[Euge]

Let ##X## be the space obtained from ##S^1## by attaching a two-cell by the map ##S^1 \to S^1##, ##z\mapsto z^n##. For ##k## an arbitrary field, compute the first homology ##H_1(X; k)##.

 

[Infrared]

Spoiler

X is a CW complex with 1 cell each in dimension ##0,1,2.## Call these cells respectively ##a,b,c.## Since the attaching map for ##c## onto the ##1##-skeleton has degree ##n##, we find ##\partial c=nb.## Also, ##\partial b=a-a=0.## So the cellular chain complex with coefficients in ##k## is: ##k\to k\to k,## where the left map is multiplication by ##n## and the second map is zero. In a field, multiplication by a nonzero element is surjective, so the image of the first map is ##k## unless ##n## is divisible by the characteristic of ##k##. Since the kernel of the second map is also ##k##, we find ##H_1(X,k)=k/k=0.##

In the case that ##n## is a multiple of the characteristic of ##k##, then both maps are zero and ##H_1(X,k)=k/0=k.##