Questions tagged [pl-manifolds]
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15 questions
17
votes
1
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644
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Can the product of an exotic torus and a circle be the standard torus?
As discussed in this question from last week, if $M$ is a closed manifold such that $M\times S^1$ is homeomorphic to the torus $T^{n+1}$, then $M$ is homeomorphic to $T^n$. Is the corresponding ...
- 19.3k
1
vote
0
answers
48
views
When is a PL function linear?
We have a simplicial decomposition of a $n$-dimensional disk $D_n$ an a PL function $\mu$ on the decomposition. For each set of vertices $v_1,\dots,v_n,v_{n+1}$, so that both $v_1,\dots,v_{n-1},v_n$ ...
- 405
7
votes
0
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405
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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
4
votes
0
answers
123
views
Is finding boundary-reducing discs for PL 3-manifolds with boundary pattern computationally efficient?
I am working with manifolds with boundary pattern, as defined by Matveev in his book Algorithmic Topology and Classification of 3-Manifolds. A manifold with boundary pattern is a pair $(M, P)$ where $...
- 41
10
votes
0
answers
455
views
Exotic analytic triangulations of $S^5$?
I would like to understand better the nature of bad triangulations of $S^5$, discussed in two Lectures of Jacob Lurie
https://www.math.ias.edu/~lurie/937notes/937Lecture2.pdf
http://www-math.mit.edu/~...
- 14.3k
7
votes
1
answer
231
views
Singularities of PL embedding of surface in a contractible 4-manifold
I am trying to understand the article "A solution to a conjecture of Zeeman" by Akbulut, but I am not an expert in PL-geometry.
As far as I understand, two statements should be true, but I ...
- 674
11
votes
0
answers
237
views
Torus trick without surgery theory
It follows from surgery theory that in dimension $\geq 5$ every closed PL manifold homotopy equivalent to a torus has a finite cover which is PL homeomorphic to a torus. This is an important ...
- 637
17
votes
1
answer
792
views
Is there a constructive proof that in four dimensions, the PL and the smooth category are equivalent?
Summary
Famously, the categories of 4-dimensional smooth manifolds and 4-dimensional piecewise linear manifolds are equivalent. Is there a constructive proof for this theorem or does it depend on the ...
- 5,565
3
votes
0
answers
114
views
Reference request concerning PL tangent Stiefel-Whitney classes
I am hoping for a reference for the fact that PL manifolds have tangent Stiefel-Whitney classes.
I understand this as follows: they have tangent microbundles, which in turn lead to spherical ...
- 55.9k
2
votes
0
answers
239
views
Are there analogs of smooth partitions of unity and good open covers for PL-manifolds?
Smooth partitions of unity and differentiable good open covers are important technical tools in the realm of smooth manifolds.
Are there analogs of these tools for piecewise linear manifolds?
A PL ...
- 37.8k
14
votes
1
answer
623
views
What is the status of the PL-pseudoisotopy stability theorem?
Suppose that $M$ is a compact PL-manifold (possibly with boundary) and let $C^{PL}(M)$ denote the (simplicial) group of PL isomorphisms of $M \times I$ relative to $M \times \{0\} \cup \partial M \...
- 8,167
8
votes
0
answers
193
views
PL surface projections - is there a theory of folds and cusps?
For smooth surfaces, the generic singularities of a map of one surface to another are folds and cusps (Whitney). It is a standard result in singularity theory that the generic isotopy of such a map is ...
- 81
6
votes
1
answer
352
views
What is known about the distribution of average edge-degrees for 3-manifold triangulations (with the number of 3-simplices less than a fixed constant)
This is my first question on mathoverflow! It relates to a project I'm undertaking with a student.
Work by Tamura (extending results by Luo and Stong) shows that for any closed 3-manifold $M$ and any ...
- 175
3
votes
2
answers
864
views
uniqueness of regular/tubular neighborhood with equivariant boundary
Let $N$ and $N'$ be regular neighborhoods of a subpolyhedron $P$ in a closed PL manifold $M$, and suppose that $t$ is a free PL involution on $M$ such that each of $\partial N$, $\partial N'$ is ...
- 5,575
3
votes
1
answer
335
views
When is the neighbourhood of a set a ball?
In euclidean n-space, it's easy to show that given a set $S$ of radius $< r$, the $a$-neighbourhood of $S$ is a ball, for any $a \geq 2r$.
Proof: Let $S$ be contained in $B_r(y)$, $y \in \mathbb{...
- 7,800