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  "body": "\n## Definition\n\nLet $n \\ge 1$ and let $M$ and $N$ be [[connected|0V032VWTHD]] [[topological manifolds|QQQ9B1DK4P]] of dimension $n$.\nLet $D^n$ be the [[closed unit ball|0Q0H13G8ZB]] of $\\R^n$, with interior $B^n$ and boundary sphere $S^{n-1}$.\nLet $2D^n$ be the closed ball of radius $2$.\nLet $f \\colon 2D^n \\to M$ and $g \\colon 2D^n \\to N$ be [[topological embeddings|00ZG9Y9TR4]] onto subsets of the interiors.\nThe *connected sum* $M \\# N$ is the [[quotient space|0NQ1H2X495]] of the disjoint union\n$$\\bigl(M \\setminus f(B^n)\\bigr) \\sqcup \\bigl(N \\setminus g(B^n)\\bigr)$$\nby the relation that identifies $f(x)$ with $g(x)$ for every $x \\in S^{n-1}$.\nIf $M$ and $N$ are [[oriented|702SNFQG0W]], one takes $f$ preserving and $g$ reversing orientation.\nThen $M \\# N$ gets an orientation that extends those of both pieces.\n\n## Remarks\n\nFor two tori $T^2 \\# T^2$ is the closed orientable surface of genus $2$.\nEach torus minus an open disk is a torus with one boundary circle, and gluing the two circles gives a connected surface with two handles.\n\nThe disjoint union $M \\sqcup N$ is not a connected sum: no balls are removed and no spheres are glued, and the result is not connected.\n\nThe result is a connected topological $n$-manifold.\nUp to homeomorphism, it does not depend on the choice of the embeddings when $M$ and $N$ are oriented and the orientation rule above is followed.\nWithout orientations the choice can matter: $\\mathbb{CP}^2 \\# \\mathbb{CP}^2$ and $\\mathbb{CP}^2 \\# \\overline{\\mathbb{CP}^2}$ are not homotopy equivalent, since their intersection forms differ.\nThe sphere is a unit: $M \\# S^n$ is homeomorphic to $M$.\nThe real projective plane gives $\\mathbb{RP}^2 \\# \\mathbb{RP}^2$ homeomorphic to the [[Klein bottle|6J4C2P3ZHC]].\n",
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