I have started reading "Introduction to differentiable manifolds" by Serge Lang. In this book, Lang takes a different approach, by immediately introducing manifolds on arbitrary Ba …

What is a motivation to study topological conjugacy of a real-valued functions on a manifold? (The importance of notion of a topologically conjugate homeomorphisms is clear for me) …

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If $n$ is odd then $S^{n-1}$ doesn't admit a nowhere-vanishing vector field, and if $n$ is even then there does exist one (Hairy Ball Theorem). We can then ask, on $S^{n-1}$, what …

Due to Cerf, there's exist a certain homotopy between two Morse functions. It is said that this homotopy is "generic". What is a precise definition of the property to be "generic" …

Why do currents, functionals on compactly supported differentiable n-forms, bear the name they do? I've assumed that it has something to do with an electrical current being formal …

I posted this question on math stack exchange and didn't receive an answer. If it is too elementary for this forum I will be happy to delete it. Let $M^m$ be a smooth manifold wi …

Assume $M^n$ and $N^n$ are null bordant, i.e. each can be realized as boundary of an $n+1$ dimensional manifold. Suppose $M^n \times \mathbb R$ is homeomorphic to $N^n\times \mathb …

Hello, I have two n*n correlation matrices with values ranging between -1,1. (2 correlation matrix because I have the same n terms under 2 different conditions) I then transformed …

I was wondering if the sum $TS^{2n}\oplus TS^{2n}$ is a trivial bundle? The same is true for spheres of odd dimension (one can find a nowhere zero section of the second bundle, add …

I know the construction of a 3D topological quantum field theory (TQFT) from a modular tensor category. I heard that we can even (mathematically) construct 4D TQFT from a modular …

Suppose you are on a manifold. Suppose you have a trivial bundle and a trivial subbundle of it. If you divide this trivial bundle with its trivial subbundle, do you get a trivial b …

Short question: Let $M$ and $N$ be smooth manifold, with appropriate smooth function algebras $C^\infty(M,\mathbb{R})$ and $C^\infty(N,\mathbb{R})$. Can we express the smooth fu …

Let $M$ be a compact manifold (without boundary) and let $f:M\to \mathbb{R}$ be a fixed Morse-function. Question: Is the space $GVect(M,f)$ of all gradient-like vector-fields f …

I'm afraid this might be an exercise in differential topology (in which case a reference to a book where it is would be very much appreciated); apologies in advance. Given an analy …