Return to Answer

The answer is: yes, the formula holds.

For any $4n$-dimensional manifold and $2n$-dimensional immersed submanifolds $A$ and $B$. (Everything must be oriented if you want consider an integer formula, intersections with signs. If you drop the condition on orientability, then you still can count the points of intersections mod 2.) Now the formula holds, because intersection is dual to the cohomological multiplication, and  $A\#B$ is equal to $A+B$.

The pairity of the dimensions is necessary, because cohomological multiplication (as well as the intersection number) is not commutative, but anticommutative.

Actually one can generalize the statement further (the dimensions of the immersed submanfolds might be greater than the half of the dimension of the ambient manifold). In this case the intersection number one can be replaced by the homology class represented by the transversal intersection of the immersed submanifolds.

The answer is: yes, the formula holds.

For any $4n$-dimensional manifold and $2n$-dimensional immersed submanifolds $A$ and $B$. (Everything must be oriented if you want consider an integer formula, intersections with signs. If you drop the condition on orientability, then you still can count the points of intersections mod 2.) Now the formula holds, because intersection is dual to the cohomological multiplication, and  $A\#B$ is equal to $A+B$.

The parity of the dimensions is necessary, because cohomological multiplication (as well as the intersection number) is not commutative, but anticommutative.

Actually one can generalize the statement further (the dimensions of the immersed submanifolds might be greater than the half of the dimension of the ambient manifold). In this case the intersection number one can be replaced by the homology class represented by the transversal intersection of the immersed submanifolds.

The answer is: yes, the formula holds.

For any 4n-dimensional manifold and 2n-dimensional immersed submanifolds $A$ and $B$. (Everything must be oriented if you want consider an integer formula, intersections with signs. If you drop the condition on orientability, then you still can count the points of intersections mod 2.) Now the formula holds, because intersection is dual to the cohomological multiplication, and $A#B$ is equal to $A+B.$

The pairity of the dimensions is necessary, because cohomological multiplication (as well as the intersection number) is not commutative, but anticommutative.

Actually one can generalize the statement further (the dimensions of the immersed submanfolds might be greater than the half of the dimension of the ambient manifold). In this case the intersection number one can be replaced by the homology class represented by the transversal intersection of the immersed submanifolds.

The answer is: yes, the formula holds.

For any $4n$-dimensional manifold and $2n$-dimensional immersed submanifolds $A$ and $B$. (Everything must be oriented if you want consider an integer formula, intersections with signs. If you drop the condition on orientability, then you still can count the points of intersections mod 2.) Now the formula holds, because intersection is dual to the cohomological multiplication, and $A\#B$ is equal to $A+B$.

The pairity of the dimensions is necessary, because cohomological multiplication (as well as the intersection number) is not commutative, but anticommutative.

Actually one can generalize the statement further (the dimensions of the immersed submanfolds might be greater than the half of the dimension of the ambient manifold). In this case the intersection number one can be replaced by the homology class represented by the transversal intersection of the immersed submanifolds.

The answer is: yes, the formula holds.

For any 2n-dimensional manifold and n-dimensional immersed submanifolds $A$ and $B$. (Everything must be oriented if you want consider an integer formula, intersections with signs. If you drop the condition on orientability, then you still can count the points of intersections mod 2.) Now the formula holds, because intersection is dual to the cohomological multiplication, and $A#B$ is equal to $A+B.$

Actually you can generalize the statement further (the dimensions of the immersed submanfolds might be greater than the half of the dimension of the ambient manifold). In this case the intersection number one can be replaced by the homology class represented by the transversal intersection of the immersed submanifolds.

The answer is: yes, the formula holds.

For any 4n-dimensional manifold and 2n-dimensional immersed submanifolds $A$ and $B$. (Everything must be oriented if you want consider an integer formula, intersections with signs. If you drop the condition on orientability, then you still can count the points of intersections mod 2.) Now the formula holds, because intersection is dual to the cohomological multiplication, and $A#B$ is equal to $A+B.$

The pairity of the dimensions is necessary, because cohomological multiplication (as well as the intersection number) is not commutative, but anticommutative.

Actually one can generalize the statement further (the dimensions of the immersed submanfolds might be greater than the half of the dimension of the ambient manifold). In this case the intersection number one can be replaced by the homology class represented by the transversal intersection of the immersed submanifolds.