All Questions
Tagged with gt.geometric-topology reference-request
361 questions
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
10
votes
1
answer
534
views
The Tits alternative for $\operatorname{Out}(F_n)$
Not sure if this is the right place to ask this, but the paper I am reading seems to be too specialised for mathstack (if you do not agree, pleas let me know and I will take down this question)
I am ...
- 275
10
votes
1
answer
503
views
Is there an "exponential law" for differentiable maps between smooth manifolds?
Although it seems like a textbook question, I was not able to find a textbook or even a research article answering the following question:
Let $M$, $N$ and $P$ be finite-dimensional smooth manifolds ...
- 843
10
votes
1
answer
588
views
Explicit Computations of Examples in Spin Geometry
I have been trying to learn about spin geometry, Dirac operators, and index theory by reading Lawson/Michelson's "Spin Geometry" and Friedrich's "Dirac Operators in Riemannian Geometry." Both are ...
- 1,010
10
votes
1
answer
828
views
Different definitions of the linking number
Assume that
$$
\iota_1:\mathbb{S}^k\to\mathbb{R}^n,
\quad
\iota_2:\mathbb{S}^\ell\to\mathbb{R}^n,
\quad
k+\ell=n-1,
$$
are two embeddings of spheres with disjoint images $\iota_1(\mathbb{S}^k)\cap\...
- 28k
10
votes
2
answers
538
views
Is there an analogue of Mostow-Palais equivariant embedding theorem for noncompact groups
Let $M$ be a (Hausdorff) smooth compact manifold and $G$ a Lie group acting smoothly on $M$. If $G$ is compact then, by Mostow-Palais theorem, there exists an equivariant smooth embedding $M\to {\...
- 31.2k
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
10
votes
0
answers
415
views
Lipschitz homotopy groups
There is an extensive literature on Lipschitz homotopies of Lipschitz maps. But I haven't seen anything about Lipschitz homotopy groups. We have introduced this notion in an article that you can find
...
- 28k
10
votes
0
answers
303
views
When were bordered Heegaard Floer homology's DA bimodules invented?
This is less of a strictly mathematical question and more of a reference request.
In their paper "Bordered Heegaard Floer Homology (http://arxiv.org/pdf/0810.0687.pdf)," Lipshitz-Ozsvath-Thurston (...
- 1,474
9
votes
3
answers
797
views
Reference request for wild 3-manifolds
I’m looking for a text on 3-manifolds that focuses on wild/pathological objects, similar to Bing’s work in the field. I know basic algebraic topology (homotopy, homology, cohomology) and have read ...
- 2,069
9
votes
2
answers
2k
views
How to find Colin Day's PhD Thesis
A week or so ago, I was saddened to read Jim Stasheff's post on the AlgTop mailing list, announcing the passing of Colin Day, after a long bout with cancer.
I was thinking of reading Colin Day's PhD ...
- 22.1k
9
votes
3
answers
1k
views
Realizing a homology by a smooth immersion
An alternative title is: When can I homotope a continuous map to a smooth immersion?
I have a simple topology problem but it's outside my area of expertise and I worry may be rather subtle. Any ...
- 2,299
9
votes
2
answers
7k
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Reference on Geometric Topology
Geometric topology is more motivated by objects it wants to prove theorems about. Geometric topology is very much motivated by low-dimensional phenomena -- and the very notion of low-dimensional ...
- 193
9
votes
2
answers
267
views
group actions in dimension 2 and 3
I am looking for a reference to the following claims:
Any compact group (connected or not) acting on $S^2$ is differentiably conjugate to a linear action. This must be classical.
A circle $S^1$ ...
9
votes
1
answer
530
views
Does there exist a discrete gauge theory as a TQFT detecting the figure-8 knot?
My question: Does there exist a discrete gauge theory as TQFT detecting the figure-8 knot?
By detecting, I mean that computing the path integral (partition function with insertions of the knot/...
- 10.5k
9
votes
4
answers
469
views
Notion of smoothness for set-valued functions
Is there a way of talking about continuity and smoothness for set valued functions? More precisely, consider $M$ and $N$ topological/smooth manifolds, and let $f$ a function that associates to each ...
- 39.1k
9
votes
1
answer
520
views
Are any embeddings $[0,1]\to\mathbb{R}^3$ topologically equivalent?
Suppose we are given embeddings $f_1,f_2:[0,1]\to\mathbb R^3$.
Does there exist a homeomorphism $g:\mathbb R^3\to\mathbb R^3$ such that $g\circ f_1=f_2$?
This question seems to be classical eighty ...
- 1,095
9
votes
1
answer
458
views
Fuchsian groups and Eichler's result
Let $G$ be a Fuchsian group of first kind contained in $\text{PSL}_2(\mathbb{R})$. A result of Eichler says, there exists a finite set $S\subset G$ such that any $\gamma$ in $G$ can be written as a ...
- 521
9
votes
1
answer
816
views
Who invented the Morse-Bott-complex?
In the "Morse-Bott theory and equivariant cohomology" paper by D.M. Austin and P.J. Braam, the authors introduce the Morse-Bott-complex to calculate the de-Rham-cohomology of a compact manifold (using ...
- 93
9
votes
1
answer
638
views
Reference request: A knot is tame if and only if it has a tubular neighbourhood
Definitions:
A knot is an embedding $\kappa:S^1\hookrightarrow S^3$ (we do not require smooth or polygonal).
Two knots $\kappa,\,\lambda:S^1\hookrightarrow S^3$ are equivalent if one of the following ...
- 93
9
votes
1
answer
323
views
Reference for a PL flat torus embedding in $\mathbb{R}^3$
A piecewise linear flat torus embedded in $\mathbb{R}^3$ is shown at http://www.mathcurve.com/polyedres/toreplat/toreplat.shtml.
It is flat in the sense that the angle defect at the vertices is zero.
...
- 1,916
9
votes
1
answer
296
views
The works of González-Acuña and Duchon from 70s and 80s
I would like to access the following two works of González-Acuña from 1970:
González-Acuña, F. Dehn’s construction on knots. Bol. Soc. Mat. Mexicana (2) 15 (1970), 58–79.
and
González-Acuña, F. On ...
- 1,292
8
votes
7
answers
1k
views
Knot theory without planar diagrams?
I remember reading somewhere that there are just a few contributions to knot theory which do not involve knot diagrams. Hence my question:
Does anybody know about papers concerning knot theory which ...
- 785
8
votes
3
answers
4k
views
A simple and good reference on surgery theory
Can anyone help me to find a simple and good reference (a book, lecture notes or a website) for learning the surgery theory and its applications? I seek a reference together with many examples and ...
- 1,303
8
votes
2
answers
430
views
Number of Reflections in a Circle between Two Points
For my research I am interested in the transmission characteristics between a transmitter (Tx) and a receiver (Rx) situated in a circular room. In particular, it is important for me to know the number ...
- 183
8
votes
1
answer
2k
views
Relation of SW and Donaldson Invariant
My question is:
I am request for the reference that Is there any relationship between the Seiberg-Witten Invariant and Donaldson's Invariant? Or the relationship between Seiberg-Witten Moduli Space ...
- 703
8
votes
2
answers
710
views
Stratification of smooth maps from R^n to R?
I'm interested in stratifications of smooth maps $\mathbb{R}^n\to\mathbb{R}$ (or more generally of any $n$-manifold $M^n\to\mathbb{R}$). The codimension 0 stratum should be Morse functions, and the ...
- 12.8k
8
votes
1
answer
281
views
Non-compact three-manifolds with the same proper homotopy type are homeomorphic?
I am looking for some literature with some (counter) examples of the following fact (though I don't know if the fact is true or not):
Let $M, M'$ be two non-compact connected $3$-manifolds with the ...
- 1,097
8
votes
1
answer
1k
views
Example of a triangulable topological manifold which does not admit a PL structure
I know there are some examples of manifolds which don't admit a PL structure (combinatorial triangulation), and that it has been recently proven that in dimension $n\geq5$ there are manifold which are ...
- 683
8
votes
1
answer
200
views
For which planar topological spaces $Z$ does there exist a hyperbolic group $\Gamma$ with $\partial \Gamma \cong Z$?
Recall a topological space is called planar if it can be embedded in $S^2$. I'm interested in understanding hyperbolic groups with planar boundaries.
In [1], it is shown that if a one-ended hyperbolic ...
- 639
8
votes
1
answer
573
views
Majorana modes and the first Stiefel–Whitney class
The first Stiefel–Whitney class of a vector bundle is an element in the first cohomology group of the base space. Namely, the first Stiefel–Whitney class for a vector bundle $E$ over a $d$-dimensional ...
- 10.5k
8
votes
2
answers
631
views
Teichmüller space on non-orientable closed surfaces
It is known that any closed orientable surface of genus $g \geq 2$ admits a hyperbolic metric, and the Teichmüller space of such metrics has dimension $6g - 6$. I was wondering if there is a ...
- 81
8
votes
1
answer
943
views
Freedman's work on non-simply-connected 4-manifolds
In the late 1970's and in the 1980's, Michael Freedman showed a relationship between the topological surgery problem in 4-dimensions, the slice problem for links, and the classification of non-simply-...
- 22.1k
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
8
votes
2
answers
394
views
Maps to $K(\pi,1)$ spaces from manifolds with $S^1$-action
Suppose $M$ is a connected smooth manifold with a smooth $S^1$-action that fixes a point in $M$. Let $X$ be a $K(\pi,1)$-space and let $\varphi: M\to X$ be a continuous map.
Question. How to prove ...
- 14.3k
8
votes
1
answer
498
views
Loday's characterization and enumeration of faces of associahedra (Stasheff polytopes)
From "The multiple facets of the associahedra" by Loday:
Let us consider the formal power series
$$f(x) = x+a_1 x^2 +a_2 x^3 + \cdots+ a_n x^{n+1} + \cdots$$
and let
$$ g(x) = x+b_1 x^2 + ...
- 10.5k
8
votes
1
answer
428
views
Is there a combinatorial version of PL ambient isotopy in dimension $>3$?
The classical PL Reidemeister Theorem reads:
Reidemeister Theorem: Two knots in $S^3$ are PL ambient isotopic if and only if any diagram of one can be transformed into a diagram of the other by ...
- 22.1k
8
votes
1
answer
241
views
Piecewise linear vs smooth high dimensional knots
A knot to me is the image of a smooth (pl locally flat) embedding $S^n \to S^m$ under the equivalence of smooth (ambient pl) isotopies. And more generally embeddings of closed manifolds $N \to M$.
...
- 313
8
votes
1
answer
704
views
Is this knot invariant already treated somewhere in the literature?
Fix a knot type $K \subset S^3$, and consider the set $$Y_K = \{ \mbox{Diagrams of }K \} / \mbox{planar isotopy}.$$
We can turn $Y_K$ into a metric space by considering the distance induced by ...
- 197
8
votes
0
answers
222
views
references on categorification of knot invariants
I am extremely sorry if this is not the right place for this kind of question.
I have studied some knot theory, quantum invariants and would like to study more about categorification of knot ...
- 81
8
votes
0
answers
200
views
Disciplining dunce hats
I'm wondering if anyone has a copy of a preprint by Charles Giffen from 1977, with the enjoyable title, Disciplining dunce hats in 4-manifolds. I've seen it referred to in various places, including ...
- 19.4k
8
votes
0
answers
502
views
Reference request: Mapping class group action on homology of surface with boundary
This is a request for a reference to a proof of a result. The result is not very hard, but I'd rather cite than reprove.
I'm looking for a generalization of the following result (Farb and Margalit, ...
- 366
8
votes
0
answers
251
views
Exact triangle for monopole Floer homology with $\mathbb{Z}$-coefficient
Let $Y$ be oriented 3 manifold with torus boundary and let $\gamma_{j}$ (j=0,1,2) be three curves on its boundary with $\#(\gamma_{j}\cap \gamma_{j+1})=-1$. We denote by $Y_{j}$ the manifold obtained ...
- 1,069
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
7
votes
2
answers
1k
views
All mapping space between CW complexes is a CW complex?
Let $\mathrm{Map}(X,Y)$ denote the (unbased) cellular mapping space from $X$ to $Y$.
If $X$ and $Y$ are finite CW complexes, is $\mathrm{Map}(X,Y)$ a CW complex?
Can we know the cell structure of $\...
- 699
7
votes
2
answers
355
views
Convex subcomplexes of CAT(0) cubical complexes
Is the following statement true? If so, can anyone provide a reference?
Let $X$ be a CAT(0) cubical complex, and let $Y$ be a connected
subcomplex of $X$. Then the following are equivalent:
...
- 8,493
7
votes
1
answer
416
views
Free $\mathbb{Z}_2$-actions match at some point
I have in front of me a proof of this lemma:
If $f$ and $g$ are free $\mathbb{Z}_2$-actions on $S^1$, then $f(x)=g(x)$ for some $x \in S^1$.
A $\mathbb{Z}_2$-action on the unit circle $S^1$ is a ...
- 151k
7
votes
1
answer
723
views
Surgery and homology: a reference request
I need a reference (or a short proof) for the following statement:
Suppose a closed manifold $N$ is the result of a surgery (along an embedded sphere) on a closed manifold $M$. Then the difference $\...
- 4,736
7
votes
1
answer
824
views
Which Abelian Group sequences arise as the Homology of Embedded CW Complexes?
Background
Let $\mathcal{A} = \lbrace A_0, \ldots, A_M \rbrace$ be an arbitrary sequence of finitely generated Abelian groups. It is well-known that a finite CW complex $X_\mathcal{A}$ may be ...
- 15.5k
7
votes
2
answers
695
views
Variants and Generalizations of Arf (-Brown-Kervaire) invariants
(1) I encounter the Arf invariants in Kirby-Taylor, Pin structures on low-dimensional manifolds. The form that I looked at was:
$$
S(q)=|H^1(M^2,\mathbb{Z}_2)|^{-1/2} \sum_{x\in H^1(M^2,\mathbb{Z}_2)} ...
- 10.5k