## New answers tagged 4-manifolds

10

If an $n$-dimensional smooth manifold $X'$ is obtained from $X$ by doing some surgeries, then there is an $(n+1)$-dimensional smooth cobordism $W$ from $X$ to $X'$; vice-versa, any handle decomposition of such a cobordism induces a sequence of surgeries.
Hence, the equivalence relation of "being obtained from one another by surgeries" is equivalent to the ...

2

Mathematical physics is the study of physical questions from the point of view of full mathematical rigor. Physical questions are phrased as well-defined mathematical problems, to be attacked with methods from differential geometry, functional analysis, Lie groups, topology, etc..
The mathematically rigorous construction of interacting 4-dimensional ...

1

Thank you.
The following gives an answer in the case that all the curves are assume to have genus $g=1$, and $b\geq 2$.
One of the curves, say $C=C_1$ has $C^2>0$. As $p_g=q=0$ we have $\chi(O_X)=1-q+p_g=1$. By Riemann-Roch, we have $\chi(C)=1+ \frac12 (C^2-K\cdot C)$. As $C^2+K\cdot C=2g(C)-2=0$ (by adjunction), we have $\chi(C)=1+C^2$. Also as ...

0

The answer is clearly affirmative for $b=1$, which unfortunately corresponds only to 101 surfaces, including ${\mathbb P}^2$. I doubt that there is a single other case in which it works.
Indeed I do not know a single example with $b>1$ for which your statement holds. It clearly fails for $({\mathbb P}^1)^2$, as mentioned by ulrich, and a similar argument ...

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