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This is going to basically be a cross-post of a MSE question: https://math.stackexchange.com/questions/147146/mayer-vietoris-implies-excision. I suspect that the answer to this question will turn out …

Let $G$ be a compact Lie group, let $T$ be a maximal torus, and let $W$ be the Weyl group. My main question is as follows: How does one prove that $H^\ast(BG,\mathbb{Q})$ is isomorphic to the $W$- …

Since you expressed interest in the hyperbolic case: it follows from the Mostow rigidity theorem and the Hurewicz theorem that an isomorphism $\phi \colon H_1(M) \to H_1(M)$ is induced by an isometry …

For any connected and oriented $n$-manifold $M$, the sequence: $$\Omega_c^{n-1}(M) \to \Omega_c^n(M) \to \mathbb{R} \to 0$$ is exact, where the first map is $d$ and the second map is $\int_M$. A deta …

My first remark about this question is a little bit pithy - the standard cohomological index formula works only for even dimensional manifolds while Chern-Weil theory is of course more general. For …

I think it is fairly straightforward in the polygonal case, and I'd wager that the general case follows from the polygonal case. Let $c(t)$ be a polygonal curve whose maximal winding number is $q$, a …

It is well-known that topological K-theory is blessed with the Bott periodicity theorem, which specifies an isomorphism between $K^2(X)$ and $K^0(X)$ (where $K^n$ is defined from $K^0$ by taking suspe …

Every symplectic form on a manifold $M^n$ determines a De Rham cohomology class in $H^2(M)$ (often a nontrivial class), and this in turn determines a class in $H_{n-2}(M)$. What in general can be sai …

I'm looking for a book, article, or lecture notes that does basic cohomology theory from a relative point of view (including the Thom isomorphism, the excision theorem, Lefschetz duality, the Gysin se …

Let $M$ be a compact manifold and let $f$ be a Morse function with exactly one critical point at each critical level. Then one can recover a CW-complex with the homotopy type of $M$ from just the cri …

Let $X$ be a finite simplicial complex and let $B$ denote the set of barycenters of the simplices of $X$. McCord constructed a $T_0$ topology on $B$ with the property that the inclusion $B \to X$ is …

Of course it isn't really that hard - nowhere near as hard as $\pi_k(S^n)$ for $k>n$, for instance. The hardness that I'm referring to is based on the observation that apparently nobody knows how to d …

Probably the argument that you're looking for is based on the "Dirac / Dual Dirac" method. The idea is to exploit the product structure in KK-theory as much as possible - this makes the proof functor …

As Fabian pointed out in the comments, you have to be more careful about how you trivialize $SO(D^2)$. I'm going to use the standard coordinates $(x,y)$ on $\mathbb{R}^2$ (note that these are not glo …

I used to think that the entire theory was intellectual masturbation, but two examples in particular completely changed my mind. The first is the Pontryagin-Thom construction, which exhibits an isomo …