11 votes

Can we define Whitney stratification algebraically?

There is a purely algebraic characterisation of Condition (B) due to Le and Teissier, see Proposition 1.3.8 of the paper Lê Dũng Tráng; Teissier, Bernard, Limites d’espaces tangents en géométrie ...
  • 14.9k
11 votes
Accepted

Is it possible to define contact manifolds as manifolds with a G-structure?

A contact structure on $M^{2n+1}$ defines a $G$-structure (actually, it defines more than one, but there is a 'minimal' $G$-structure that is preserved by all contact transformations, and that is the ...
10 votes
Accepted

Can a smooth manifold be realised as the image of a smooth function?

My comment turned answer: Any smooth $m$-manifold $M$ admits a complete Riemannian metric (for example, as this answer says, any manifold embeds into some Euclidean space as a closed subset by Whitney ...
  • 3,577
6 votes
Accepted

Regularity of lipschitz and derivable function

I claim that a function with these properties need not be $C^1$. We start with the function $f: t \in (-1,1)\setminus \{ 0 \} \mapsto \operatorname{sin}(1/t)$, and we also set $f(0) = 0$. The ...
  • 2,949
4 votes

Regularity of lipschitz and derivable function

Unless I'm overlooking something, you're simply asking whether a function $f:[0,1] \rightarrow \mathbb R$ with a bounded derivative must be continuously differentiable. This can fail quite ...
4 votes
Accepted

Index formula for elliptic operators acting on Sobolev sections vanishing on the boundary (say $D: H_0^k(\Omega) \to H_0^{k-1}(\Omega)$)

Local boundary conditions such as the Dirichlet condition you mention were considered in Atiyah-Bott, The index problem for manifolds with boundary. 1964 Differential Analysis, Bombay Colloq., 1964 pp....
3 votes

When are bundles of odd and even differential forms isomorphic?

I will explain that as long as $n>2$ the real vector bundles $\Omega^{even}$ and $\Omega^{odd}$ over $M$ are isomorphic. If $n>2$ then $dim(\Omega^{even}) = dim(\Omega^{odd}) = 2^{n-1} > n$ ...
1 vote

Why do we define the pants complex and the pants decomposition?

Cut systems are not the same as pants decompositions. The former are defined in section one of the paper and consist of $g$ curves. The later are defined in the appendix (and there called "...
  • 20.4k
1 vote

Patching up two trivial fibre bundles induces homology equivalence

I think the answer is yes, even if the bundles are not trivial. Assume that $\dim X_1 < \dim X_2$. $X_1$ is a closed subset of $X$ and therefore a properly embedded submanifold. Let $Y_i=f^{-1}(X_i)...

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