Connected sum of oriented manifolds ($M\#N$)

For connected oriented closed $d$-dimensional manifolds with $d\ge 2$, their connected sum removes an open $d$-dimensional ball from each and identifies the resulting boundary spheres by an orientation-reversing diffeomorphism. The remaining orientations fit together. Collapsing the separating sphere gives a pinch map to the wedge sum of the two closed manifolds, and projection to either summand has degree of a continuous mapping one. In intermediate positive degrees the cohomology is the direct sum of the summand groups. Products of classes from different summands vanish; products landing in top degree use the single common orientation class. These statements follow from excision, the Mayer–Vietoris sequence and the degree-one projections.

Remove small disks around a nondegenerate maximum of a smooth function on one closed surface and a nondegenerate minimum on another. Rescale their values so the boundary values agree with the two ends of a neck. Their regular level collars allow a strictly monotone function on the neck, with nonzero derivative, matching smoothly to both sides. The resulting connected sum has all the old critical points except the deleted maximum and minimum. This permits one degenerate torus saddle to be retained while adding ordinary handles for higher genus.

For a connected sum of identical oriented manifolds of dimension at least three, one can permute the attachment balls in a punctured sphere and match the attached copies by their chosen parametrizations. This gives orientation-preserving diffeomorphisms realizing the permutations on the middle cohomology summands. For sums of complex projective planes, these permutations preserve the diagonal positive intersection form.

// Destination: geometry-and-topology.bigb

Tube the projective lines in the two summands of ${\mathbb{CP}}^2\#{\mathbb{CP}}^2$ through the connecting neck. Their connected sum is an embedded two-sphere in the class $h_1+h_2$, with self-intersection two. Its oriented normal disk bundle has Euler number two, so its boundary is the unit tangent bundle of $S^2$, namely $SO(3)\cong {\mathbb{RP}}^3$. The boundary separates the interior of the tubular neighborhood from its nonempty exterior.

// Destination: geometry-and-topology.bigb

For closed connected oriented $d$-manifolds with $d\ge 2$,Deleting an open ball changes the Euler characteristic by $-1+\chi (S^{d-1})$, and gluing subtracts $\chi (S^{d-1})$. Since $\chi (S^{d-1})+\chi (S^d)=2$, the formula follows. Thus a connected sum of oriented manifolds subtracts two from the sum of Euler characteristics in even dimension and subtracts zero in odd dimension.

For the standard complex orientations, the two summand classes have zero product and equal squares, the oriented degree-four class. The quadratic relations imply all cubic monomials vanish. The rational Betti numbers are $1,2,1$ in degrees zero, two and four.

For odd $r\ge 1$ and $W_g={\#}_{i=1}^g(S^r\times S^r)$, the integral cohomology has one copy of $\mathbb{Z}$ in degrees zero and $2r$, ${\mathbb{Z}}^{2g}$ in degree $r$, and zero elsewhere. Choose the top orientation class $\omega$ and degree-$r$ classes ${\alpha }_i,{\beta }_i$ from the two factors of summand $i$. Their nonzero positive-degree basis products areAll ${\alpha }_i{\alpha }_j$ and ${\beta }_i{\beta }_j$ vanish, and $\omega$ annihilates positive-degree classes. The Künneth theorem computes each sphere product; the Mayer–Vietoris sequence computes the connected sum groups, and its degree-one pinch maps determine the products. Thus its middle-degree Poincare duality pairing is a symplectic vector space after changing coefficients to a field of characteristic different from two. The case $r=1$ is the usual cohomology ring of a closed oriented surface.