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In my experience the answer is not really. My favorite example of a Hausdorff manifold which is not paracompact (which incidentally is also my all-time favorite counter example to most point-set topo …

I believe that the answer is yes, this needs Yau's theorem. Rather, I think that the statement of your question can be phrased in such a way as to be equivalent to Yau's theorem. My understanding of …

For certain $F$ and $\epsilon_i$ the answer is no. But it is probably yes for generic choices. Here is an example with n=2 and $M$ the Klein bottle. We start with F being a standard projection of the …

Consider ordinary singular cohomology with varying coefficients. You can look at the short exact sequence of abelian groups: $$0 \to \mathbb{Z}/2 \to \mathbb{Z}/4 \to \mathbb{Z}/2 \to 0$$ This give …

The Ricci curvature is a local quantity, so I am only going to focus on the case that the local holonomy group is SO(n). Philosophically, the local holonomy group and the curvature of a connection att …

It sounds like what you are after is a theory of genuine equivariant K-theory (Which exists!) If you think of a cohomology theory as a sequence of functors to abelian groups together with some proper …

Here is a partial answer. If the order of $\Gamma$ is odd, then this is a trivial application of transfer maps. You have described your manifold as a quotient $\pi:S^7 \to M = S^7/\Gamma$, and hence $ …

Let $(M, \omega)$ be a holomorphic symplectic manifold of (complex) dimension $2n$. Let $x$ be a point in $M$. My understanding from the discussion and answers to this MO question is that there exists …

This is a famous open-problem. It is still unknown.

The torsion is a notoriously slippery concept. Personally I think the best way to understand it is to generalize past the place people first learn about torsion, which is usually in the context of Rie …

I glanced at the paper briefly. Let me try to explain what I understand. Let us suppose that M is compact and without boundary. Let $f: M \to \mathbb{R}$ be a Morse function. Let us further suppose fo …

Let S be a surface of genus g with some parked points (n of them). Assume $n \geq 2$ and fix two of the marked points. Consider the set of embedded arcs going between these two special points. The gro …

What are the derivations of the algebra of continuous functions on a topological manifold? A supermanifold is a locally ringed space (X,O) whose underlying space is a smooth manifold X, and whose …

The part I'm still hesitant about is that the manifold you call $C$ is $n$-dimensional. Codimension arguments in the infinite dimensional setting are always a little sticky. So, I'm just going to trea …

There are a number of technical issues with making what you describe precise, for example: what precisely is a supermanifold with boundary? how can you glue/compose bordisms? etc. I am going to ignore …