POTW Attaching a Two-Cell to a Circle

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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 $H_1(X; k)$.

 

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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 $H_1(X,k)=k/0=k.$