I am currently working through an old qualifying exam problem:

Calculate the fundamental group and homology groups of the space $X$ obtained from the union $S^2\cup T$ where $S^2$ is the 2-sphere and $T$ is the torus by identifying 2 disjoint circles on the sphere to 2 disjoint circles on the torus. (Picture half of the torus being inside the sphere and half outside)

As for the fundamental group, I know that I can't use Seifert-Van Kampen on $A=S^2$ and $B=T$ because $A\cap B$ is 2 disjoint circles and hence is not path connected.

Without Seifert-Van Kampen, I am not sure how to approach the problem. I would appreciate any hints or outlines on how to proceed.

  • $\begingroup$ Let $A$ be the two circles on the sphere, their interiors, and an arc joining them. Then $A$ is contractible, so $S^2\cup T$ is homotopy equivalent to $(S^2\cup T)/A$, which is $S^2 \vee (S^2/S^0) \vee (S^2/S^0)$. (The two "halves" of the torus each become $S^2$ with two points identified.) $\endgroup$ – kccu May 28 '16 at 17:06
  • $\begingroup$ Ah I see! But in Hatcher (p. 11) we see that $S^2/S^0$ is homotopy equivalent to $S^2\bigvee S^1$, so I am I correct in saying that the whole space is homotopy equivalent to $S^2\bigvee S^2\bigvee S^2\bigvee S^1\bigvee S^1$? Then $\pi_1=Z\ast Z$ and $H_1=Z^2$, $H_2=Z^3$? $\endgroup$ – 1234 May 28 '16 at 19:18
  • $\begingroup$ That looks correct to me. $\endgroup$ – kccu May 28 '16 at 21:03

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