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Given smooth manifold $M$, let $Bl_\Delta(M\times M)$ be the (say oriented; you can ask this question for the unoriented case too) real blow up of $M\times M$ along the diagonal and let $\pi:Bl_\Delta(M\times M)\to M\times M$ denote the blow down map. Let $G(M)$ be the space of homeomorphisms $f:M\to M$ satisfying: there exists a homeomorphism $\tilde{f}:Bl_\Delta(M\times M)\to Bl_\Delta(M\times M)$ lifting $(f,f):M\times M\to M\times M$, i.e., $\pi\circ\tilde{f}=(f,f)\circ\pi$.

Clearly $\{C^1\text{-homeomorphisms of }M\}\subset G(M)\subset \text{Homeo}(M)$ and it is easy to give examples showing $G(M)\neq\text{Homeo}M$. My question is: is $G(M)$ the same as the space of $C^1$-homeomorphisms of $M$? I have trouble coming up with counterexamples. My only counterexample is when $M$ is 1-dimensional, in which case it is easy. But are there counterexamples if we assume $\dim(M)\ge2$?

Locally, a (continuous, injective) map $f:\mathbb{R}^n\to\mathbb{R}^n$ satisfies the lifting condition above implies that: there exists a homeomorphism $\phi\in\text{Homeo}(S^{n-1})$, such that for any sequance of pairs of points $\{(x_i,y_i)\in\mathbb{R}^n\times\mathbb{R}^n\}_{i=1}^\infty$, $$\lim_{i\to\infty}x_i,y_i=0,\ \lim_{i\to\infty}\frac{y_i-x_i}{|y_i-x_i|}=v\in S^{n-1}\implies\lim_{i\to\infty}\frac{f(y_i)-f(x_i)}{|f(y_i)-f(x_i)|}=\phi(v).$$ Are there such $f$'s that are not $C^1$?

Given smooth manifold $M$, let $Bl_\Delta(M\times M)$ be the (say oriented; you can ask this question for the unoriented case too) real blow up of $M\times M$ along the diagonal and let $\pi:Bl_\Delta(M\times M)\to M\times M$ denote the blow down map. Let $G(M)$ be the space of homeomorphisms $f:M\to M$ satisfying: there exists a homeomorphism $\tilde{f}:Bl_\Delta(M\times M)\to Bl_\Delta(M\times M)$ lifting $(f,f):M\times M\to M\times M$, i.e., $\pi\circ\tilde{f}=(f,f)\circ\pi$.

Clearly $\{C^1\text{-homeomorphisms of }M\}\subset G(M)\subset \text{Homeo}(M)$ and it is easy to give examples showing $G(M)\neq\text{Homeo}M$. My question is: is $G(M)$ the same as the space of $C^1$-homeomorphisms of $M$? I have trouble coming up with counterexamples. My only counterexample is when $M$ is 1-dimensional, in which case it is easy. But are there counterexamples if we assume $\dim(M)\ge2$?

Locally, a (continuous, injective) map $f:\mathbb{R}^n\to\mathbb{R}^n$ satisfies the lifting condition above implies that: there exists a homeomorphism $\phi\in\text{Homeo}(S^{n-1})$, such that for any sequance of pairs of points $\{(x_i,y_i)\in\mathbb{R}^n\times\mathbb{R}^n\}_{i=1}^\infty$, $$\lim_{i\to\infty}x_i,y_i=0,\ \lim_{i\to\infty}\frac{y_i-x_i}{|y_i-x_i|}=v\in S^{n-1}\implies\lim_{i\to\infty}\frac{f(y_i)-f(x_i)}{|f(y_i)-f(x_i)|}=\phi(v).$$ Are there such $f$'s that are not $C^1$ at 0?

Given smooth manifold $M$, let $Bl_\Delta(M\times M)$ be the (say oriented; you can ask this question for the unoriented case too) real blow up of $M\times M$ along the diagonal and let $\pi:Bl_\Delta(M\times M)\to M\times M$ denote the blow down map. Let $G(M)$ be the space of homeomorphisms $f:M\to M$ satisfying: there exists a homeomorphism $\tilde{f}:Bl_\Delta(M\times M)\to Bl_\Delta(M\times M)$ lifting $(f,f):M\times M\to M\times M$, i.e., $\pi\circ\tilde{f}=(f,f)\circ\pi$.

Clearly $\{C^1\text{-homeomorphisms of }M\}\subset G(M)\subset \text{Homeo}(M)$ and it is easy to give examples showing $G(M)\neq\text{Homeo}M$. My question is: is $G(M)$ the same as the space of $C^1$-homeomorphisms of $M$? I have trouble coming up with counterexamples. My only counterexample is when $M$ is 1-dimensional, in which case it is easy. But are there counterexamples if we assume $\dim(M)\ge2$?

Locally, a (continuous, injective) map $f:\mathbb{R}^n\to\mathbb{R}^n$ satisfies the lifting condition above implies that: there exists a homeomorphism $\phi\in\text{Homeo}(S^{n-1})$, such that for any sequance of pairs of points $\{(x_i,y_i)\in\mathbb{R}^n\times\mathbb{R}^n\}_{i=1}^\infty$, $$\lim_{i\to\infty}x_i,y_i=0,\ \lim_{i\to\infty}\frac{y_i-x_i}{|y_i-x_i|}=v\in S^{n-1}\implies\lim_{i\to\infty}\frac{f(y_i)-f(x_i)}{|f(y_i)-f(x_i)|}=\phi_x(v).$$ Are there such $f$'s that are not $C^1$?

Given smooth manifold $M$, let $Bl_\Delta(M\times M)$ be the (say oriented; you can ask this question for the unoriented case too) real blow up of $M\times M$ along the diagonal and let $\pi:Bl_\Delta(M\times M)\to M\times M$ denote the blow down map. Let $G(M)$ be the space of homeomorphisms $f:M\to M$ satisfying: there exists a homeomorphism $\tilde{f}:Bl_\Delta(M\times M)\to Bl_\Delta(M\times M)$ lifting $(f,f):M\times M\to M\times M$, i.e., $\pi\circ\tilde{f}=(f,f)\circ\pi$.

Clearly $\{C^1\text{-homeomorphisms of }M\}\subset G(M)\subset \text{Homeo}(M)$ and it is easy to give examples showing $G(M)\neq\text{Homeo}M$. My question is: is $G(M)$ the same as the space of $C^1$-homeomorphisms of $M$? I have trouble coming up with counterexamples. My only counterexample is when $M$ is 1-dimensional, in which case it is easy. But are there counterexamples if we assume $\dim(M)\ge2$?

Locally, a (continuous, injective) map $f:\mathbb{R}^n\to\mathbb{R}^n$ satisfies the lifting condition above implies that: there exists a homeomorphism $\phi\in\text{Homeo}(S^{n-1})$, such that for any sequance of pairs of points $\{(x_i,y_i)\in\mathbb{R}^n\times\mathbb{R}^n\}_{i=1}^\infty$, $$\lim_{i\to\infty}x_i,y_i=0,\ \lim_{i\to\infty}\frac{y_i-x_i}{|y_i-x_i|}=v\in S^{n-1}\implies\lim_{i\to\infty}\frac{f(y_i)-f(x_i)}{|f(y_i)-f(x_i)|}=\phi(v).$$ Are there such $f$'s that are not $C^1$?

Given smooth manifold $M$, let $Bl_\Delta(M\times M)$ be the (say oriented; you can ask this question for the unoriented case too) real blow up of $M\times M$ along the diagonal and let $\pi:Bl_\Delta(M\times M)\to M\times M$ denote the blow down map. Let $G(M)$ be the space of homeomorphisms $f:M\to M$ satisfying: there exists a homeomorphism $\tilde{f}:Bl_\Delta(M\times M)\to Bl_\Delta(M\times M)$ lifting $(f,f):M\times M\to M\times M$, i.e., $\pi\circ\tilde{f}=(f,f)\circ\pi$.

Clearly $\{C^1\text{-homeomorphisms of }M\}\subset G(M)\subset \text{Homeo}(M)$ and it is easy to give examples showing $G(M)\neq\text{Homeo}M$. My question is: is $G(M)$ the same as the space of $C^1$-homeomorphisms of $M$? I have trouble coming up with counterexamples. My only counterexample is when $M$ is 1-dimensional, in which case it is easy. But are there counterexamples if we assume $\dim(M)\ge2$?

Locally, a (continuous, injective) map $f:\mathbb{R}^n\to\mathbb{R}^n$ satisfies the lifting condition above implies that: there exists a homeomorphism $\phi\in\text{Homeo}(S^{n-1})$, such that for any sequance of pairs pf points $\{(x_i,y_i)\in\mathbb{R}^n\times\mathbb{R}^n\}_{i=1}^\infty$, $$\lim_{i\to\infty}x_i,y_i=0,\ \lim_{i\to\infty}\frac{y_i-x_i}{|y_i-x_i|}=v\in S^{n-1}\implies\lim_{i\to\infty}\frac{f(y_i)-f(x_i)}{|f(y_i)-f(x_i)|}=\phi_x(v).$$ Are there such $f$'s that are not $C^1$?

Given smooth manifold $M$, let $Bl_\Delta(M\times M)$ be the (say oriented; you can ask this question for the unoriented case too) real blow up of $M\times M$ along the diagonal and let $\pi:Bl_\Delta(M\times M)\to M\times M$ denote the blow down map. Let $G(M)$ be the space of homeomorphisms $f:M\to M$ satisfying: there exists a homeomorphism $\tilde{f}:Bl_\Delta(M\times M)\to Bl_\Delta(M\times M)$ lifting $(f,f):M\times M\to M\times M$, i.e., $\pi\circ\tilde{f}=(f,f)\circ\pi$.

Clearly $\{C^1\text{-homeomorphisms of }M\}\subset G(M)\subset \text{Homeo}(M)$ and it is easy to give examples showing $G(M)\neq\text{Homeo}M$. My question is: is $G(M)$ the same as the space of $C^1$-homeomorphisms of $M$? I have trouble coming up with counterexamples. My only counterexample is when $M$ is 1-dimensional, in which case it is easy. But are there counterexamples if we assume $\dim(M)\ge2$?

Locally, a (continuous, injective) map $f:\mathbb{R}^n\to\mathbb{R}^n$ satisfies the lifting condition above implies that: there exists a homeomorphism $\phi\in\text{Homeo}(S^{n-1})$, such that for any sequance of pairs of points $\{(x_i,y_i)\in\mathbb{R}^n\times\mathbb{R}^n\}_{i=1}^\infty$, $$\lim_{i\to\infty}x_i,y_i=0,\ \lim_{i\to\infty}\frac{y_i-x_i}{|y_i-x_i|}=v\in S^{n-1}\implies\lim_{i\to\infty}\frac{f(y_i)-f(x_i)}{|f(y_i)-f(x_i)|}=\phi_x(v).$$ Are there such $f$'s that are not $C^1$?