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 …

Suppose that you have decomposed a manifold $M$ into cells (I care most, if it matters, about compact oriented smooth manifolds; but if my question can be solved in the PL category …

Is there a known definition of vector fields on a simplicial manifold? For me, it seems natural that the definition should be something along the lines: Let $M_{\bullet}$ be a si …

I would guess that the following is true, and that somebody has worked it out, but I don't recall ever seeing it. Can anyone point me to any literature on it? Let $G$ be a finite …

Are there simple necessary and sufficient conditions for an (oriented) even-dimensional compact smooth manifold to fiber over an (oriented) odd-dimensional manifold (with orient …

If $M$ is a connected smooth manifold, then it is easy to show that there is a sequence of connected compact smooth submanifolds with boundary $M_1\subseteq M_2\subseteq\cdots$ suc …

Clarification: My question concerns the homotopy type of the space of $C^k$ diffeomorphisms with the compact-open $C^k$ topology, where $0< k \leq\infty$. I have stated my quest …

Recall that a Weil algebra is a finite-dimensional real unital algebra that admits exactly one homomorphism to R. Such algebras form the basis of the Weil approach to differential …

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 …

Suppose that $S$ and $T$ are two smooth manifolds and '$ \Re$' be the reals with the normal manifold structure. And here I use '$=$' to mean diffeomorphism. Is the statement below …

Let $\Sigma$ be a compact Riemann surface with complex structure $j$. Let $(M,J)$ be an almost complex manifold. A map $u: \Sigma \rightarrow M$ is called $J$-holomorphic if $$ du …

I apologize in advance if this is somewhat elementary, but: Let $(M,g)$ be a compact Riemannian manifold. Is there a "characterization" of which symmetric bilinear tensors $B\i …

My question is: Does the forgetful functor F:(Mfd) $\to$ (Top) preserve pullbacks? Detailed explanation is following. A pullback is defined as a manifold/topological space satisf …

If one has a smooth simply connected manifold $M^n$ which we know to bound a an $n+1$ manifold $N$ what can be said about a handle decomposition for one in terms of a handle decomp …

Usually, the most general condition for fiber product of manifolds (or vector bundles) to exist is that we require the images cleanly intersects. See e.g. http://mathoverflow.net/q …