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I was working in some problems and I found this answer. Homology of Torus with two discs glued on the inside As the answer states, a torus with two disks attached inside is homotopically equivalent to $S^{1}\lor S^{2} \lor S^{2} $, my question is, a torus with just one disk attached inside would be homotopically equivalent to $S^{1}\lor S^{2}$ ?. Thanks in advance.

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    $\begingroup$ If the disk is attached in the same way as in the link, then yes, and the argument is analogous to the answer of the linked question. $\endgroup$
    – Kevin.S
    Jul 24 at 19:15

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