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The classic MO thread Ways to prove the fundamental theorem of algebra contains $60$ proofs of FTA, and I'm sure there are many more in the literature. It would be nice to have some way to organize th …

The study of $n$-manifolds has some well-known periodicities in $n$ with period a power of $2$: $n \bmod 2$ is important. Poincaré duality implies that odd-dimensional compact oriented manifolds ha …

I wrote a blog post that turned into quite a nice exercise in characteristic classes. The goal was to compute the cohomology of a smooth hypersurface of degree $d$ in $\mathbb{CP}^3$, as a ring. This …

The papers I was thinking of are actually by Propp: Euler measure as generalized cardinality, and Exponentiation and Euler measure (although I think these papers are outdated and more is known these …

The Lie bracket on rational homotopy has a relatively simple conceptual explanation: if $X$ is simply connected, then the rational homology $H_{\bullet}(\Omega X, \mathbb{Q})$ of its loop space is a c …

This is an extended comment on KConrad's discussion of symmetry groups. We can think of $k$-forms on a vector space $V$ (homogeneous polynomials of degree $k$) abstractly as elements of the symmetric …

Commutative Poisson algebras $A$ can be thought of as commutative algebras equipped with a first-order deformation into a noncommutative algebra given by the Poisson bracket. A simple example is the s …

We have $$\log \sum_{k \ge 0} T_k t^k = \sum_{i=1}^n \log \frac{x_i t}{1 - e^{-x_i t}}$$ so if we write $$\log \frac{x_i t}{1 - e^{-x_i t}} = \log \sum_{k \ge 0} B_k^{+} x_i^k \frac{t^k}{k!} = \sum_{k …

The action of $\mathbb{Z}_n$ on $\mathbb{C}^n$ can be diagonalized: it's conjugate to the action sending a vector $(z_0, z_1, \dots z_{n-1}) \in \mathbb{C}^n$ to $$(z_0, \zeta_n z_1, \zeta_n^2 z_2, \d …

Acyclic groups must in particular have trivial abelianization, so all of their quotients must be perfect. This is the only obstruction; A.J. Berrick shows in The acyclic group dichotomy (which I just …

One of the old classic MO questions is a big-list of proofs of the fundamental theorem of algebra. Here is a second big-list question about this big list: Which of the FTA proofs can, even in prin …

Is there a notion of cohomology ring of X with coefficients in A? Yes, and nothing new is needed. The underlying additive group of $A$ is abelian so you take cohomology with coefficients in that …

As a real vector bundle, $E^{\ast} \otimes E$ decomposes as the direct sum of two copies of the bundle $\mathfrak{su}(E)$ of trace-free skew-adjoint endomorphisms and two copies of the trivial bundle. …

Let $V$ be a real vector bundle on a space $X$, perhaps the tangent bundle of a smooth compact manifold. I'm interested in understanding the obstructions to $V$ admitting a stable complex structure, a …

1) some hypotheses are needed for them to work perfectly, like semi-locally simply-connectedness for the existence of coverings Tim Porter already said this but I'll say it again with a slightly …