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Let $M^4$ be a closed orientable smooth manifold and $$ I: H_2(M,\mathbb Z) \times H_2(M,\mathbb Z) \to \mathbb Z $$ its intersection form. If $b^+ \ge 1$, where $b^+$ denotes the number of positive ...

What can we tell about non-transverse intersection points of (smooth) submanifolds? Especially, in the case of complementary dimensions, how can we define and calculate the ''multiplicity'' of an ...

Sorry for the confusion from earlier. I tried to fix the thread. The old version can be found below. For $6$-dimensional subspaces $V$ of the space $\mathbb{R}^{3\times 3}$ of real three-times-three ...

This is an exercise from Gromov's Partial differential relations. (page 5) Let $V$ and $V'$ be two closed complex submanifolds in $\mathbb{C}^N$ of complimentory dimension. Prove that $V$ and $V'$ ...

In Dubrovin/Fomenko/Novikov Modern geometry--Methods and applications, Part II, the (Poincare-)Hopf theorem is treated in section 15.2 (see theorem 15.2.7 on page 129), while the Lefschetz theorem on ...

I'm guessing that the answer to this question is well-known, but I'm struggling to find anything to help me. Let $X,Y$ be compact manifolds of dimension $n,m$ respectively. Let $f:X \to Y$ be a ...