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I'm guessing that you heard this from someone whose reasoning goes "Every finite presentation of a group can be made to give the $\pi_1$ of a smooth 4-manifold. If we could put any 4-manifold into the …

I'm guessing that you mean that $X$ is a scheme defined over $\mathbb R$, with real points $X_{\mathbb R}$, and complex points $X_{\mathbb C}$ that you also denoted $X$. (I won't.) I'll assume that $ …

Mathai and Quillen have a theorem that computes the Euler characteristic as an integral of a form defined using a section of and a connection on the tangent bundle. If one scales the section by a fact …

As Mariano's comment indicates, you need more conditions. The usual one is that $G$ is a torus. Using $G$ compact, you can average a metric to get a $G$-invariant metric. Then the exponential map giv …

Such an endomorphism of $M$ gives an automorphism of the cohomology ring that acts by $-1$ on top cohomology. The cohomology ring of your example $M = {\mathbb C \mathbb P}^{2n}$ doesn't have such aut …

I agree that it's very confusing that curl is again a vector field, and that there's this wacky determinant. I recommend doing the 2-d version, and "discovering" that curl is more naturally thought of …

To expand on Oscar's answer: the principal $SO(4)$ bundle over $G_2/SO(4)$ gives us $$ \qquad \qquad \qquad \ldots \to \pi_2(G_2) \to \pi_2(G_2/SO(4)) $$ $$ \to \pi_1(SO(4)) \to \pi_1(G_2) \to \pi_1(G …

There's a generalization of Morse-Bott called Morse-Bott-Kirwan that you can read about in Kirwan's book. Basically this condition guarantees that the unstable sets are manifolds, but not the stable …

Use the Haar measure on $G$ (compact!) to average a metric, obtaining a $G\times G$-invariant metric, and thus an identification $\mathfrak g \cong \mathfrak g^*$. Also, the geodesic spray $\mathfrak …

In your question, you may as well take $M=M_j=$ a ($G$-invariant) tubular neighborhood of a single orbit $G/S$. Near the point $S/S$, the space looks like $\mathfrak g/\mathfrak s \times N$, where $N$ …

Check out Almost commuting elements in compact Lie groups by Borel, Freedman, and Morgan. "We describe the components of the moduli space of conjugacy classes of commuting pairs and triples of elemen …

Usual story: vector fields on $M$, with their Lie bracket, form a Lie algebra. We can consider "actions" of some other Lie algebra ${\mathfrak g}$ on $M$ by looking at Lie homomorphisms ${\mathfrak g} …

Flag manifolds $G/B$ are nice: the Kähler cone is the positive Weyl chamber, with edges coming from the Poincaré duals of the Schubert divisors.

The connected double cover of $S^1$ (boundary of the Möbius strip) is an $S^0$ bundle that is not the double of the unique $0$-disc bundle over $S^1$.

Let $K$ be a maximal compact subgroup of $G$. Then $K$ acts transitively on each $G/P$, and up to $K$-isomorphism, the $K$-spaces obtained exactly match those occurring as coadjoint orbits of $K$ (act …