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Manifolds and cell complexes Definition

Connected sum

Builds on Topological manifoldConnected spaceTopological embeddingClosed ballQuotient topologyOrientability

Definition

Let n≥1n \ge 1 and let MM and NN be connected topological manifolds of dimension nn. Let DnD^n be the closed unit ball of ℝn\R^n, with interior BnB^n and boundary sphere Sn−1S^{n-1}. Let 2Dn2D^n be the closed ball of radius 22. Let f ⁣:2Dn→Mf \colon 2D^n \to M and g ⁣:2Dn→Ng \colon 2D^n \to N be topological embeddings onto subsets of the interiors. The connected sum M#NM \# N is the quotient space of the disjoint union (M∖f(Bn))⊔(N∖g(Bn))\bigl(M \setminus f(B^n)\bigr) \sqcup \bigl(N \setminus g(B^n)\bigr) by the relation that identifies f(x)f(x) with g(x)g(x) for every x∈Sn−1x \in S^{n-1}. If MM and NN are oriented, one takes ff preserving and gg reversing orientation. Then M#NM \# N gets an orientation that extends those of both pieces.

Remarks

For two tori T2#T2T^2 \# T^2 is the closed orientable surface of genus 22. Each torus minus an open disk is a torus with one boundary circle, and gluing the two circles gives a connected surface with two handles.

The disjoint union M⊔NM \sqcup N is not a connected sum: no balls are removed and no spheres are glued, and the result is not connected.

The result is a connected topological nn-manifold. Up to homeomorphism, it does not depend on the choice of the embeddings when MM and NN are oriented and the orientation rule above is followed. Without orientations the choice can matter: ℂℙ2#ℂℙ2\mathbb{CP}^2 \# \mathbb{CP}^2 and ℂℙ2#ℂℙ2‾\mathbb{CP}^2 \# \overline{\mathbb{CP}^2} are not homotopy equivalent, since their intersection forms differ. The sphere is a unit: M#SnM \# S^n is homeomorphic to MM. The real projective plane gives ℝℙ2#ℝℙ2\mathbb{RP}^2 \# \mathbb{RP}^2 homeomorphic to the Klein bottle.

Reading path

To understand this page from the ground up, read these 7 pages in order. The foundations come first.

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Closed ball, concept, no evidence levelClosed ballconceptConnected space, concept, no evidence levelConnected spaceconceptOrientability, concept, no evidence levelOrientabilityconceptQuotient topology, concept, no evidence levelQuotient topologyconceptTopological embedding, concept, no evidence levelTopological embeddingconceptTopological manifold, concept, no evidence levelTopological manifoldconceptConnected sum, definition, no evidence levelConnected sumdefinition
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
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  1. Closed ball concept
  2. Connected space concept
  3. Orientability concept
  4. Quotient topology concept
  5. Topological embedding concept
  6. Topological manifold concept
  7. Connected sum definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Connected_sum
About this page
TagBW8X6DXTHG
URIm2://BW8X6DXTHG
Kinddefinition
WritingW0 stub
MSC57N99, 57R65
DefinesConnected sum
UsesTopological manifold, Connected space, Topological embedding, Closed ball, Quotient topology, Orientability
Changed2 November 2026

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