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Let X be a real orientable compact differentiable manifold. Is the (co)homology of X generated by the fundamental classes of oriented subvarieties? And if not, what is known about the subgroup ...

Now I want to calculate the intersection form of a logarithmic transformation which is defined as follows. Let $X$ be an oriented, closed, simply-connected 4-manifold and $T^2\subset X$ be an ...

Let $f: X\to Y$ be a smooth map between smooth manifolds, both connected. Let $Y=\cup_{i=1}^k Y_i$ be a finite union of disjoint locally closed submanifold $Y_i$ such that $f^{-1}(Y_i)\to Y_i$ is ...