Let $M$ and $N$ be two connected manifolds of the same dimension $n$. What is the fundamental group of their connected sum $M \# N$ in terms of $\pi_1(M)$ and $\pi_1(N)$ ?
Fundamental group of the connected sum of manifolds
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algebraic-topology
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1This suggests that "it depends" : http://topospaces.subwiki.org/wiki/Fundamental_group_of_connected_sum_is_free_product_of_fundamental_groups_in_dimension_at_least_three – 2012-09-20