All Questions
Tagged with gt.geometric-topology at.algebraic-topology
1,145 questions
10
votes
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answer
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views
Ranicki symmetric L-groups of finite fields?
Can anyone tell me what the Ranicki symmetric L-groups $L^*(F)$ are when $F$ is a finite field? (and maybe provide a reference?) Thanks!
- 5,534
10
votes
1
answer
635
views
Self-homomorphisms of surface groups
Let $X$ be a closed, orientable surface of genus at least 2, and let $\phi: \pi_1(X) \to \pi_1(X)$ be a surjective homomorphism. Is $\phi$ necessarily injective?
- 932
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4
answers
2k
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Complements of Simply Connected Subsets of the Plane
this is my first question here! Hopefully it is appropriate. Let $\mathbb{A}$ be the punctured plane, i.e. the 'standard' annulus. For compact, connected subsets of the plane (planar continua) $X \...
- 439
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votes
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positions of a methane molecule with carbon atom at the origin
Let $\text{CH}_4$ be the molecule of Methane:
The four hydrogen atoms form vertices of a regular tetrahedron with the carbon atom in the center of the regular tetrahedron.
Here we regard all atoms to ...
- 2,223
10
votes
1
answer
847
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Applications of Faber's conjecture
Faber's perfect pairing conjecture states that the tautological ring $R^*$ of the moduli space $\mathcal{M}_g$ of curves of genus $g$ behaves as if it were the rational cohomology of a closed, ...
- 4,133
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1
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637
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Cell structures of Dold manifold and Wu manifold
In Dold's 1956 paper Erzeugende der Thomschen Algebra N, Dold studied the Dold manifold $P(m,n)=(S^m\times\mathbb{CP}^n)/\tau$ where $\tau$ acts as $-1$ on $S^m$ and a complex conjugation on $\mathbb{...
- 1,329
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1
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587
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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
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1
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828
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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
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1
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378
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Discrete Pin structures
It is clear that an oriented manifold $M^n$ (with dimension $n$) admits spin structures if and only if its second Stiefel-Whitney class $[w^2]\in H^2(M,\mathbb Z_2)$ vanishes. In the construction of ...
- 10.5k
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2
answers
757
views
Embedded (framed) cobordisms
[The title initially was "Actions of gauge groups on framed cobordisms. This has been changed.]
This question is a follow-up to my answer to When is a submanifold of $\mathbf R^n$ given by global ...
- 23.5k
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Classification of oriented vector bundles of rank 5 over closed oriented 5-manifolds
I am looking for a complete classification in terms of characteristic classes and "computable" (preferably geometric) invariants. There is this work where the authors classify oriented vector bundles ...
- 2,830
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1
answer
557
views
Elements of infinite order in the topological mapping class group
Let $M$ be a closed topological manifold, and let $\operatorname{MCG}(M):=\operatorname{Homeo}(M)/\operatorname{Homeo}_0(M)$ denote the topological mapping class group of $M$ ($\operatorname{Homeo}_0(...
- 18.7k
10
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1
answer
388
views
Wild half-line in a Euclidean space
Is there an $m$-dimensional simplicial complex $S$ with the following properties:
The cone over $S$ is homeomorphic to $\mathbb{E}^{m+1}$. Here $\mathbb{E}^{m+1}$ denoes the $(m+1)$-dimensional ...
- 45k
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1
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536
views
Inducing up the group homomorphism between mapping class groups
There are many ways to embed the braid group into the mapping class group of a surface. To describe one of them, let ${C}_{2g+2}(\mathbb{D}^2)$ be the configuration of unordered $2g+2$ points in the ...
- 1,003
10
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1
answer
489
views
Visualising locally flat embeddings of surfaces in R^4
As far as I understand it follows from the work of M. Freedman that there exist locally flat embeddings of two dimensional surfaces in $\mathbb R^4$ that can not be smoothed in the class of locally ...
- 14.3k
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362
views
Topological factors of complex projective manifolds
Let $M$ be a closed orientable smooth 4-manifold. Assume $\pi_1(M)=\{0\}$ and $b_2(M)>0$.
Let $S$ be a closed orientable surface. Denote $P=M\times S$.
Can it so happen that there is no complex ...
user164740
10
votes
1
answer
378
views
Finiteness of $\pi_n(Top/O)$
For $n>4$ one may identify $\pi_n(Top/O)$ via smoothing theory as concordance class of smooth structures on $S^n$ where two structures are concordant if they bound a smooth structure on the product ...
- 5,839
10
votes
0
answers
199
views
"Homotopy homomorphisms" of homeomorphisms of Euclidean space
For a topological group $G$, an older term for a map $BG \to BG$ is a "homotopy homomorphism". If $G$ is connected, taking based loops shows that a homotopy class of such a map is the same ...
- 8,167
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votes
0
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187
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A minimal $\mathbb Z/2$-invariant Morse function on $U(n)$
Consider the group $U(2n)$ of unitary matrices. This has two standard and important homomorphic involutions. The most famous one $A \mapsto \overline A$ is complex conjugation, with fixed-set $O(2n)$, ...
- 9,580
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455
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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
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0
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176
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Embedding 2-complexes null homotopically into 2-complexes
Whitehead's conjecture states that if $L$ is an aspherical 2-complex and $K$ is a subcomplex of $L$, then $K$ is also aspherical. It is known by work of Howie and Luft that if the Whitehead ...
- 5,349
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372
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Steenrod Problem and realization of rational homology classes by manifolds
Steenrod's problem asks wheter a simplicial homology class of a topological space $x$,
$$ x\in H_n(X, \mathbb{Z})$$
can be represented by a triangulation of an $n$-dimensional, ...
- 2,630
10
votes
0
answers
484
views
Space of embeddings of an $n$-ball into an $n$-manifold
Let $M$ be a smooth $n$-manifold without boundary, and let $B$ be the open unit ball in $\mathbb{R}^n$. I am trying to understand the space $\text{Emb}(B,M)$ of smooth embeddings of $B$ into $M$. ...
- 101
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0
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458
views
is a group $G$, that admits finite $k(G, 1)$ and has no Baumslag-Solitar subgroups, necessarily hyperbolic?
This is the first question asked in Bestvina's article "Questions in Geometric Group Theory". Does anyone know if there has been any progress made on this problem? Is the question answered if $G$ is ...
- 101
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votes
4
answers
2k
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How to prove the connected sum of two closed aspherical n-manfolds (n >2) is not asperical?
The intuitive idea is that the sphere connected the two manifolds is not contractible, which implies the (n-1)th homotopy group is not zero. Another argument, which I am not totally understand, uses ...
- 1,598
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5
answers
2k
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Is the topological concept of collapsible useful?
I ask this question because in the process of reviewing for my topology comp, I began rereading Alg Topology by Hatcher. In the introduction is the famous Bing's House of Two Rooms. I thought this ...
- 91
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1k
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Homology sphere with $\mathbb{R}^3$ as the universal cover
Question. Is there a $3$-dimensional integer homology sphere whose universal cover is $\mathbb{R}^3$?
I believe the answer is in the positive and I am looking for (precise) references. If not in ...
- 28k
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4
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3k
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Associativity of topological join and join of spheres
This must be an elementary question, as I couldn't find any proofs on the Internet, but I still can't do it. And yes, Hatcher says that the join is not actually associative for general topological ...
- 1,000
9
votes
3
answers
735
views
Judging whether a finitely presented group is a 3-manifold group?
Given a finitely presented group $G$, how many necessary conditions do people know for $G$ to be isomorphic to the fundamental group of some closed connected 3-manifold? (e.g. residually finite)
9
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1
answer
5k
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Manifolds are paracompact
By Definition, smooth manifolds are assumed to be Hausdorff and to satisfy the second countability axiom.
I have heard (but never seen written) that these assumptions imply paracompactness (and thus ...
- 10.4k
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3
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3k
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Does a connected manifold with vanishing Euler characteristic admit a nowhere-vanishing vector field?
A version of the "hairy ball" theorem, due probably to Chern, says that the Euler-characteristic of a closed (i.e. compact without boundary) manifold $M$ can be computed as follows. Choose any vector ...
- 54.6k
9
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1
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916
views
explicitly embedding a simplicial $d$-complex into $\mathbb{R}^{2d+1}$, or algorithms for doing so
A classical result in topology for which I can't find a reference for is that a simplicial complex $K$ of dimension $d$ with $n$ vertices can be linearly embedded into $\mathbb{R}^{k}$ when $k=2d+1$. ...
- 103
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2
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367
views
Is compact flat manifold cusp cross-sections of a complete finite volume hyperbolic manifold?
Let $M^{n-1}$ be a closed flat manifold. Is it true that there exists a hyperbolic manifold $N^n$ with finite volume such that $M$ is a cusp cross-section of $N$?
It was proved in "On the geometric ...
- 2,623
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3
answers
1k
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Contractibility of space of embeddings of a disc
I'm pretty sure that both of the following spaces are contractible. However, I can't seem to find a proof or a reference. Can anyone provide one? Let $D^2$ be the unit disc in $\mathbb{R}^2$.
The ...
- 99
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votes
3
answers
1k
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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
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votes
2
answers
641
views
Künneth formulas/theorem for bordism groups and cobordisms?
We are familiar with Künneth theorem:
The Kunneth formula is given by $R$ as a ring, $M,M'$ as the R-modules, $X,X'$ are some chain complex. The Kunneth formula shows the cohomology of a chain ...
- 10.5k
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votes
1
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588
views
Oriented bordism in higher dimensions (e.g. $12 \leq d \leq 28$)
The classification of oriented compact smooth manifolds up to oriented cobordism is one
of the landmarks of 20th century topology. The techniques used there form the part of the foundations of ...
- 10.5k
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votes
2
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753
views
Deformation equivalent vs diffeomorphic to projective manifold
Let $M$ be a closed complex manifold that is not deformation equivalent to a complex projective manifold.
Can $M$ be orientedly diffeomorphic to a complex projective manifold? What if $M$ is moreover ...
user164740
9
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4
answers
1k
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Where should I learn about immersion theory?
I'd like to learn the basics of Hirsch-Smale immersion theory. What sources are best for this? My background is mostly topological; however, many of the sources I've found on the internet focused on ...
9
votes
2
answers
621
views
Generalization of the sphere theorem in dimension at least 4
In 1956, Papakyriakopoulos proved Dehn's lemma, loop theorem and the sphere theorem. The proofs are based on a clever technique called "tower construction". Later, Whitehead, Shaprio, ...
- 2,083
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votes
1
answer
777
views
Intuition for torsion of a chain complex and application to lens spaces
I have read a bit about the torsion of an acyclic complex. One of my concrete hopes was that I could understand why $L(7,1)$ and $L(7,2)$ are not homeomorphic - I am under the impression that ...
- 5,349
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votes
2
answers
1k
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Reference request: Spin structures on surfaces and the spin mapping class group
I am looking for references on the following: Spin structures on surfaces, and particularly the spin mapping class group.
What is known about generating the spin mapping class group? Has anybody ...
- 2,136
9
votes
1
answer
5k
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Universal covering of compact surfaces
Is there any elementary (i.e. without using analytical methods like the theory of Riemann surfaces or more elaborate results from differential geometry) way to show that the universal covering of the ...
- 11.9k
9
votes
2
answers
763
views
Knot complement diffeomorphism groups and embedding spaces
I'm interested in the following collection of questions: Let $S^n_k = \sqcup_k S^n$ be a disjoint union of $k$ distinct $n$-dimensional spheres. Write $Emb(S_k^n, S^{n+2})$ for the space of ...
- 4,133
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1
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384
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embedding of quaternionic projective spaces
Let $\mathbb{H}P^m$ be the $m$-th quaternionic projective space. What is the smallest integer $N$ such that there exists an embedding
$$
\mathbb{H}P^2\longrightarrow \mathbb{R}^N?
$$
Are there any ...
- 1,990
9
votes
4
answers
3k
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Poincaré dodecahedron space
The Poincaré homology sphere $X$ is constructed by identifying opposite faces of a dodecahedron by a (clockwise) twist of 36 degree.
Many books say its fundamental group $\pi_1(X)$ is the binary ...
- 133
9
votes
2
answers
659
views
Can the fundamental group of an intersection of a homeomorphic image of a ball with a complement of a ball in $R^3$ be perfect?
I have the following problem: Let $A, B\subset R^3$, $A$ is homeomorphic to a ball, while $B$ is a standard Euclidean ball. Can it happen that the fundamental group of $A\setminus B$ is a perfect ...
- 135
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votes
1
answer
374
views
Can an action of a compact Lie group be nontrivial if it is trivial on the boundary?
Let $G$ be a compact Lie group acting on a connected topological manifold $M$ with boundary. Suppose the action on one boundary component is trivial. Does it follow that the action on the whole of $M$ ...
- 23.5k
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votes
2
answers
1k
views
Fibrewise homotopy-equivalence of unit sphere bundles vs isomorphism of tangent bundles
Let $M$ be a smooth $m$-dimensional manifold, $TM$ its tangent bundle and $SM$ its unit sphere bundle.
Are there some simple examples where $SM$ is fibrewise homotopy-equivalent to the trivial ...
- 44.4k
9
votes
1
answer
372
views
A strong form of Mostow rigidity without geometrization?
Mostow rigidity theorem says that two closed hyperbolic manifolds with isomorphic fundamental groups are isometric.
Here is my question: suppose that $M$ and $N$ are two closed 3-manifolds such that $...
- 855