Return to Answer

added 22 characters in body

Source Link

edited Oct 23, 2023 at 16:55

Summarizing what has been said before and adding a bit:

If the normalization map $\tilde X\to X$ is bijective then it is a homeomorphism in the Hausdorff topology. So if $\tilde X$ is smooth and $X$ not then $X$ will be a topological manifold without being smooth. This happens for example for the cusp $y^2=x^3$.
If $X$ is normal then the problem is ony interesting for $\dim X>1$. HereFor $\dim X=2$, the main result is the affirmative answer by Mumford, linked to by Jason Starr, that a point is non-singular if and only if it has arbitrary small simply connected punctured neighborhoods.
In higher dimension there is Brieskorn's paper where he shows that the variety given by $z_1^3+z_2^2+\ldots+z_n^2=0$ in $\mathbb C^n$ with $n\ge4$ even is normal, singular and a topological manifold. Later on he shows in a famous paper that the Milnor link of the hypersurface $$z_1^2+z_2^2+z_3^2+z_4^3+z_5^{6k-1}=0,$$ runs for $k=1,\ldots,28$ through all $28$ smooth structures of $S^7$. SoIn particular, also these hypersurfaces provide counterexamples.

Summarizing what has been said before and adding a bit:

If the normalization map $\tilde X\to X$ is bijective then it is a homeomorphism in the Hausdorff topology. So if $\tilde X$ is smooth and $X$ not then $X$ will be a topological manifold without being smooth. This happens for example for the cusp $y^2=x^3$.
If $X$ is normal then the problem is ony interesting for $\dim X>1$. Here, the main result is the affirmative answer by Mumford, linked to by Jason Starr, that a point is non-singular if and only if it has arbitrary small simply connected punctured neighborhoods.
In higher dimension there is Brieskorn's paper where he shows that the variety given by $z_1^3+z_2^2+\ldots+z_n^2=0$ in $\mathbb C^n$ with $n\ge4$ even is normal, singular and a topological manifold. Later on he shows in a famous paper that the Milnor link of the hypersurface $$z_1^2+z_2^2+z_3^2+z_4^3+z_5^{6k-1}=0,$$ runs for $k=1,\ldots,28$ through all $28$ smooth structures of $S^7$. So also these hypersurfaces provide counterexamples.

Summarizing what has been said before and adding a bit:

If the normalization map $\tilde X\to X$ is bijective then it is a homeomorphism in the Hausdorff topology. So if $\tilde X$ is smooth and $X$ not then $X$ will be a topological manifold without being smooth. This happens for example for the cusp $y^2=x^3$.
If $X$ is normal then the problem is ony interesting for $\dim X>1$. For $\dim X=2$, the main result is the affirmative answer by Mumford, linked to by Jason Starr, that a point is non-singular if and only if it has arbitrary small simply connected punctured neighborhoods.
In higher dimension there is Brieskorn's paper where he shows that the variety given by $z_1^3+z_2^2+\ldots+z_n^2=0$ in $\mathbb C^n$ with $n\ge4$ even is normal, singular and a topological manifold. Later on he shows in a famous paper that the Milnor link of the hypersurface $$z_1^2+z_2^2+z_3^2+z_4^3+z_5^{6k-1}=0,$$ runs for $k=1,\ldots,28$ through all $28$ smooth structures of $S^7$. In particular, also these hypersurfaces provide counterexamples.

Source Link

created Oct 23, 2023 at 13:32

Summarizing what has been said before and adding a bit:

If the normalization map $\tilde X\to X$ is bijective then it is a homeomorphism in the Hausdorff topology. So if $\tilde X$ is smooth and $X$ not then $X$ will be a topological manifold without being smooth. This happens for example for the cusp $y^2=x^3$.
If $X$ is normal then the problem is ony interesting for $\dim X>1$. Here, the main result is the affirmative answer by Mumford, linked to by Jason Starr, that a point is non-singular if and only if it has arbitrary small simply connected punctured neighborhoods.
In higher dimension there is Brieskorn's paper where he shows that the variety given by $z_1^3+z_2^2+\ldots+z_n^2=0$ in $\mathbb C^n$ with $n\ge4$ even is normal, singular and a topological manifold. Later on he shows in a famous paper that the Milnor link of the hypersurface $$z_1^2+z_2^2+z_3^2+z_4^3+z_5^{6k-1}=0,$$ runs for $k=1,\ldots,28$ through all $28$ smooth structures of $S^7$. So also these hypersurfaces provide counterexamples.