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Bezout's theorem says that the intersection number (defined algebraically as the sum of the intersection multiplicities) of two algebraic curves $C$ and $D$ in $\mathbb{P}^2$ is equal to $\deg(C)\cdot …

When is a complete toric variety a Poincare duality space? Is there an "if and only if" condition? And is this condition local? Given an analytically-locally-toric compactification of a smooth variet …

Let $M$ be a compact oriented manifold of dimension $n$. It is well-known that there is a perfect intersection pairing $$H_k(M;\mathbb{Z})_{torsion\,\,free}\otimes H_{n-k}(M;\mathbb{Z})_{torsion\,\,fr …

The torus has two functions $f$ and $g$ which are (1) relatively prime, (2) each have two square roots, and (3) whose product has $4$ square roots. For instance take two functions which vanish on disj …

Let $X$ be a topological manifold of dimension at least five. The Kirby-Siebenmann invariant $ks(X)\in H^4(X,\mathbb{Z}_2)$ is an obstruction to the existence of a PL structure on $X$. If it vanishes, …