Connected Sum

The connected sum M_1#M_2 of n-manifolds M_1 and M_2 is formed by deleting the interiors of n-balls B_i^n in M_i^n and attaching the resulting punctured manifolds M_i-B^._i to each other by a homeomorphism h:partialB_2->partialB_1, so

 M_1#M_2=(M_1-B^._1) union _h(M_2-B^._2).

B_i is required to be interior to M_i and partialB_i bicollared in M_i to ensure that the connected sum is a manifold.

Topologically, if M_1 and M_2 are pathwise-connected, then the connected sum is independent of the choice of locations on M_1 and M_2 where the connection is glued.


The illustrations above show the connected sums of two tori (top figure) and of two pairs of multi-handled tori.

The connected sum of two knots is called a knot sum.

See also

Knot Sum

Portions of this entry contributed by Todd Rowland

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Rolfsen, D. Knots and Links. Wilmington, DE: Publish or Perish Press, p. 39, 1976.

Referenced on Wolfram|Alpha

Connected Sum

Cite this as:

Rowland, Todd and Weisstein, Eric W. "Connected Sum." From MathWorld--A Wolfram Web Resource.

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