Daniele Zuddas
  • Member for 9 years, 9 months
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CW structures on unitary sphere of a banach/Hilbert space
12 votes

The answer is no in infinite dimension (and yes in the finite dimensional case, of course). Because the sphere in a Banach space is Baire. But an infinite dimensional CW-complex is not, being a ...

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Morse theory in TOP and PL categories?
11 votes

For TOP Morse function a reference is the classical book of Kirby and Siebenmann "Foundational essays on topological manifolds, smoothings, and triangulations" (1977). The key point is to consider the ...

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Open immersions of open manifolds
8 votes

This is the special case announced in the comment. We assume $M$ to be an oriented 4-manifold with the homotopy type of a 2-complex. We prove that $M$ immerses in $\mathbb{C}P^2$. In the case of ...

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The complex structure on $S^{2n}$
8 votes

The answer is not. Calabi and Eckmann ("A class of compact, complex manifolds which are not algebraic", Ann. of Math. 58 (1953) 494-500) proved that $\Bbb R^{2n}$, $n > 1$, has a complex structure ...

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How does pseudoconvexity restrict the topology?
Accepted answer
8 votes

A theorem of Eliashberg implies that an open subset of $\Bbb C^n$, $n \neq 2$, is isotopic to a Stein domain (hence to a domain of holomorphy) if and only if it admits a handlebody structure with all ...

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Can cobordisms of 3 or 4 manifolds be visualized by moves on kirby diagrams?
Accepted answer
7 votes

I think that a complete set of moves for cobordisms are the following: Kirby moves (that preserve the 4-manifold), handle trading (dotted circles become 0-framed 2-handles), addition/deletion of pairs ...

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Classification of complex structures on $\mathbb{R}^{2n}$
7 votes

There are even infinitely many inequivalent complex structures without non-constant holomorphic functions on $\Bbb R^4$ download.

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The boundary of a domain whose interior is diffeomorphic to the ball
7 votes

First, your assumption imply that $\bar D$ is a compact smooth manifold with boundary a topological sphere (because is a simply connected homology sphere). So, $\bar D$ is a topological $n$-ball, by ...

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The quotient of $\mathbb{R}^{n}$ by a closed subset
7 votes

For $A$ compact the answer is that given by Joseph Van Name. If $A$ is not compact then the answer is negative. For example take the standard $\Bbb R \subset \Bbb R^2$. Then the quotient is not II-...

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Topological Classification of Four-Manifolds
6 votes

As Paul Siegel pointed out, the fundamental group of a smooth closed orientable 4-manifold can be an arbitrary finitely presented group, and for this reason a general classification is not possible, ...

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Results true in a dimension and false for higher dimensions
6 votes

The Schoenflies theorem: if $\Sigma^n \subset \Bbb R^{n+1}$ is homeomorphic to $S^n$, then there is a homeomorphism $h \colon \Bbb R^{n+1} \to \Bbb R^{n+1}$ such that $h(\Sigma^n) = S^n$ (the round $n$...

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Kähler structure on open complex manifold
5 votes

In this G&T paper there are examples of complex structures on $\Bbb R^4$ withouth compatible Kähler metrics. The point is that in this $(\Bbb R^4, J)$ there are holomorphic elliptic compact curves,...

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How to specify a compact topological 4-manifold with a finite amount of data
4 votes

In my understanding, I guess that the following strategy could be be attempted. Let $M$ be a closed connected orientable 4-manifold (while nonorientable 4-manifolds are doubly covered by orientable ...

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Two commuting mappings in the disk
4 votes

May be a possible way to solve the problem is to set up it in the context of closed relations instead of functions, by putting $R = g^{-1} \circ f : B^n \to B^n$, and assuming for instance $f(B^n) \...

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Source on the proof that codimension 2 is sufficient for knottings?
Accepted answer
4 votes

This question is badly posed. If you have a not simply connected manifold (of any dimension $> 3$) then two connected closed curves are isotopic iff they are homotopic, hence you have as many ...

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Continuous relations?
3 votes

This is a remarkable application in topology. It has been given by Mike Freedman in his work about the classification of simply connected closed topological 4-manifolds, which had as a main ...

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Existence of orientation preserving, finite order self homeomorphism on a genus 2 surface without fixed point
Accepted answer
3 votes

The answer is yes, as it has been remarked in the previous answers. Here is a very explicit construction. Let $T$ be the sphere $S^2 \subset \Bbb R^3$ with three disks removed. Take these three disks ...

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Video lectures of mathematics courses available online for free
3 votes

The videos of Mike Freedman lectures on the topology of 4-manifolds, broadcasted from UC Santa Barbara: Freedman's Lectures Also other videos on 4-manifolds and related topics given at MPIM during ...

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Smooth extension of maps on Manifolds
2 votes

This is a well-known basic fact, so it might be not suitable for MO, but let me sketch an answer. You are looking for a local extension, and everything can be arranged to happen in a local chart. So, ...

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Algorithm for computing the Arf invariant of a knot
2 votes

This is too long for a comment, it's just an idea for this computation. Start with a Seifert surface $F \subset \Bbb R^3$ for the knot $K$. Up to isotopy, we can assume that $F$ projects regularly to ...

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Circle bundles over $CP^1$ and self-intersection number of $CP^1$ embeddings
2 votes

Another approach (for any base surface $S$): take the disk bundle over $S$ with Euler number $e$ (any integer). The total space is a smooth 4-manifold with boundary with the required property (the 0-...

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Visualize Fourth Homotopy Group of $S^2$
2 votes

The most explicit and geometric generator for $\pi_4(S^2)$ is probably the achiral genus-1 Lefschetz fibration $f : S^4 \to S^2$ with two Lefschetz singularies (one positive and one negative, with ...

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complex manifold with corner
Accepted answer
2 votes

There are some related results about compact Stein 4-manifolds with boundary as Lefschetz fibrations over the disk (whose fiber has non-empty boundary). Corners in this case arise naturally on the ...

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Mapping class between coverings of Riemann surfaces
2 votes

This is a simple exercise and the answer is no (cf. comments of Lee Mosher and Misha). Consider the two degree-4 coverings of the torus to itself with monodromies $\omega_1, \omega_2 : \pi_1(T^2) \to ...

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Finite pre-images implies (local) branch cover?
Accepted answer
1 votes

No. Under your assumptions the map can have fold singularities (and also other kinds of singularities), namely those of the form $f(x,y) = (x, y^2)$ in local coordinates. In order to have a local ...

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Dehn twist generators for mapping class group of a genus zero surface with boundary
Accepted answer
1 votes

The answer is in Wajnryb's paper from which I attach Figure 12.

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Characterization of discs
1 votes

I show that $\partial D$ is a Jordan curve in the plane, may be this will be of some help. To do this, define a map $\phi : S^1 \to \Bbb R^2$ such that $\phi(v) = (\Bbb R_+ v) \cap \partial D$. Your ...

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Homotopy of Unitary sphere in a Banach space and finite dimensional spheres
Accepted answer
1 votes

Well, it is well-known that both of $X$ and $Y$ are contractible (hence they are homotopy equivalent, this should answer your question). Notice that $Y$ is exactly the set of unit vectors of $E$. To ...

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Intersection forms of 4-manifolds with boundary
1 votes

As the question is posed, the answer is not. If you take connected sum with $\Bbb{CP}^2$ you preserve the determinant, but change the intersection form.

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gluing along a real analytic manifold
1 votes

The question should be posed in this way: given a real analytic $l$-form $\alpha$ on $M$, is there a real analytic extension $\widetilde \alpha$ defined in an open neighborhood of $M$ in $X$? The ...

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