All Questions
11 questions
2
votes
1
answer
365
views
Correspondence between fundamental group and geometric properties of $X$
At the time of studing some algebraic topology I was wondering about the following.
Let $X$ be a topological space and $\pi_1(X)$ be its fundamental group.
If we assume some algebraic property of $\...
- 613
13
votes
3
answers
2k
views
A quotient space of complex projective space
Let $\mathbb{C}P^n$ be the $n$-dimensional complex projective space and denote $[z_0:\dots:z_n]$ its points. If we glue $[z_0:\dots:z_n]$ and $[\overline{z_0}:\dots:\overline{z_n}]$ for any $[z_0:\...
- 331
7
votes
0
answers
407
views
Understanding that a simplicial complex is a PL manifold via links
Suppose $X$ it a simplicial complex homeomorphic to a topological $n$-manifold. Suppose we know that the link of each $k$-simplex $\Delta^k$ is homeomorphic (as a topological space) to the sphere $S^{...
- 14.3k
2
votes
0
answers
315
views
smooth structure on complete intersection
A complete intersection is an algebraic variety cut out by homogenous polynomials. Geometrically, this is the intersection of hypersurfaces in complex projective space.
Below, let's confine to the ...
- 59
4
votes
3
answers
382
views
Extending a continuous map over projective space
Let $X = P^{n-1}(\Bbb C)$ ($(n-1)$-dimensional projective space) with $n \geq 3$, and let $K \subset X$ denote a compact subset. I have a bijective, continuous map $\phi:K \to K$ which satisfies the ...
- 356
4
votes
1
answer
304
views
Local product structure of determinantal variety
The variety $X_n$ of singular $n\times n$ real matrices is stratified by smooth strata $X_{n,k}$ where $k$ is the rank. Choose a rank $k$ matrix $A\in X_{n,k}$. Is there a local diffeomorphism sending ...
- 16.6k
19
votes
1
answer
842
views
Vector field on a K3 surface with 24 zeroes
In https://mathoverflow.net/a/44885/4177, Tilman points out that one can use a $K3$ surface minus the zeroes of a generic vector field to build a nullcobordism for $24[SU(2)]$. Given that a) this is a ...
- 35.5k
10
votes
2
answers
940
views
Morgan Shalen compactification of $\mathbb C^2$
I'm reading the Otal's survey on the compactification of Morgan Shalen.
(available here)
He claims in an example (page 8) that the compactification of $\mathbb C^2$ is $S^4$, which sounds completely ...
- 829
8
votes
1
answer
998
views
Relation between Milnor ring and middle dimensional homology of hypersurface
I have suspected that the following is well-known:
If $P$ is a homogeneous polynomial of degree $d$ in $n$ variables (for example, Fermat quintic $x_1^5 + \cdots + x_5^5$). The Milnor ring is ${\...
- 803
15
votes
3
answers
1k
views
moduli spaces are kahler?
I often heard from experts that "moduli spaces are Kahler". This sounds as a meta-theorem asserting that every time one defines reasonable moduli spaces, then there is a standard strategy to see (...
- 829
7
votes
1
answer
826
views
Weight filtration for smooth analytic manifolds
In his ICM 2002 talk (Topology of singular algebraic varieties, available also on arXiv) B. Totaro says on p. 3 (of the arXiv version): "Using the method of Guillen and Navarro Aznar I was able to ...
- 23.5k