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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 …

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 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 …

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 …

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. …

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 …

There are many things to say here. Here's one. Suppose you want to classify all spaces up to (weak) homotopy equivalence, or equivalently all CW complexes up to homotopy equivalence. The zeroth step i …

Any finitely presented group occurs as the fundamental group of a smooth compact manifold, so the question reduces to whether we can find a finitely presented group $\pi$ acting on a smooth compact ma …

The rationalization of this ring can be understood in a very nice way, as follows. Suppose for simplicity that $X$ is simply connected. Then we can define its rational homotopy groups $$\pi_n(X, \math …

Yes, but it's more complicated. First let me describe the special case where $\pi_1$ is trivial. If $G$ is such a 3-group, then $X = BG$ is a pointed connected simply connected space. It fits into a f …

No. The Stiefel-Whitney and Pontryagin numbers of a closed oriented manifold are cobordism invariants, but the Betti numbers are not. More explicitly, all closed oriented $3$-manifolds are frameable …