Compute the homology of R^3 - S^1.(adsbygoogle = window.adsbygoogle || []).push({});

Actually a friend of mine asked me this question and I came up with the following way to solve this but I'm not sure if it's correct.

My analysis:

H_0 = Z (the integers) because it's path connected.

H_1 = Z (the friend said so but I don't believe him)

H_2 = ??

R^3 - {point} = S^2 (= means homeomorphic to or homotopic to)

R^3 - {line} = S^2 - {2 points} = R^2 - {1 point}

So R^3 - S^1 = R^3 - {a line together with a point at infinity} = R^2 - {2 points} = figure eight

Is this a valid reasoning? If so, then H_0 = Z, H_1 = Z direct sum Z, H_2 = 0.

Thanks.

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# Computing the homology of R^3 - S^1

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