Reduce homology of $H_1(\mathbb{R}^n,x)$

Solution 1:

Yes this is correct, and you can actually compute $\tilde H_0(x) = 0$ directly from the definition of singular homology because there's only one $0-$simplex and one $1-$simplex in this space so all the relevant groups in the augmented chain complex $C_*(x)$ are finitely generated. It's a good exercise to get you more familiar with the definitions.

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