Let $M^3$ and $N^3$ be two compact connected $3$-dimensional handlebodies. It is easy to see that they have the same homotopy type of $1$-dimensional CW complexes. Is it true that their connected sum $M \# N$ has this same property (not boundary connected sum)?
2026-03-28 00:49:22.1774658962
Connected sum of handlebodies has the homotopy type of a $1$-dimensional CW-complex
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It does not. That sphere in $M \# N$, which results from gluing $M - \text{(open ball)}$ and $N - \text{(open ball)}$, has nontrivial homology class in $H_2(M \# N)$.