All Questions
Tagged with gt.geometric-topology at.algebraic-topology
1,145 questions
96
votes
4
answers
10k
views
Which manifolds are homeomorphic to simplicial complexes?
This question is only motivated by curiosity; I don't know a lot about manifold topology.
Suppose $M$ is a compact topological manifold of dimension $n$. I'll assume $n$ is large, say $n\geq 4$. The ...
- 27.2k
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
19
votes
3
answers
1k
views
Degrees of self-maps of aspherical manifolds
In "Infinitesimal computations in Topology", Publ IHES, page 318, Dennis Sullivan writes "Recall any self-mapping
of a Riemann surface of genus $g>1$ either has
degree $0$ or degree $\pm 1$." ...
- 20.9k
5
votes
1
answer
240
views
Self-linkage of the orthogonal group $O_n({\mathbb R})$.
In Exercise 153 of my list, it is proved that the connected components $SO_2({\mathbb R})$ and $O_2^-({\mathbb R})$ of the orthogonal group are linked as curves in the three-dimensional sphere defined ...
- 52.3k
8
votes
1
answer
436
views
A sphere bundle map
I think this may all be classical bundle-theory. But I'm trying to read some old papers on classifications of bundles and the following came up as questions I couldn't immediately answer:
Consider the ...
- 2,734
35
votes
5
answers
9k
views
Intuition behind Alexander duality
I was wondering if anyone could offer some intuition for why Alexander duality holds. Of course, the proof is easy enough to check, and it is also easy to work out many examples by hand. However, I ...
- 361
3
votes
1
answer
806
views
a CW-complex homotopic to a manifold
I'm reading a paper and here the authors say that a connected 4-manifold with zero rational top homology has a homotopy type of 3-dimensional CW-structure. I can't figure out how it can be done.
- 537
1
vote
1
answer
929
views
Complement of lines and wedges of spheres
Let $L=L_1 \cup ... \cup L_n$ be the union of $n$ distinct lines through the origin in $\mathbb{R}^{3}$. I'd like a convincing argument that $\mathbb{R}^{3} \setminus L$ is homotopy equivalent to a ...
- 1,329
5
votes
1
answer
550
views
Local homology of degenerate critical points
Given a smooth function $f:M\rightarrow \mathbb R$ on a manifold, its local homology at a critical point $x$ is the group
$$ C_\star(x) := H_\star ( M_{ < c} \cup \{ x \} , M_{ < c} ) ,$$
where ...
82
votes
12
answers
15k
views
Compelling evidence that two basepoints are better than one
This question is inspired by an answer of Tim Porter.
Ronnie Brown pioneered a framework for homotopy theory in which one may consider multiple basepoints. These ideas are accessibly presented in his ...
- 22.1k
8
votes
1
answer
1k
views
Beyond an intro to topological graph theory...
I'm looking to find out what active areas of research there are in topological graph theory, particularly those that interface strongly with other areas of math (say, group theory, algebraic topology, ...
- 1,180
27
votes
1
answer
4k
views
connectivity of the group of orientation-preserving homeomorphisms of the sphere
In the paper "Local Contractions and a Theorem of Poincare" Sternberg has mentioned the following question which was open when the paper was written:
Is the group of orientation-preserving ...
- 6,224
13
votes
2
answers
2k
views
Surface bundles over a surface
What can be used to distinguish two $\Sigma_g$-bundles over $\Sigma_h$ up to
(1) homotopy?
(2) homeomorphism?
(3) fiberwise homeomorphism?
(4) bundle isomorphism?
And can these always be computed ...
- 2,734
4
votes
3
answers
348
views
Cohomological dimension of a group acting on a cellular complex
Let $G$ be a group acting on $X$, $X$ a cellular complex and $cd(G)$ the cohomological dimension of $G$.
2 things:
(1) I'm looking for a reference (or proof!) of this:
Suppose $X$ is acyclic. Then $...
- 2,734
5
votes
1
answer
799
views
Exotic spheres detected in higher homotopy
Thinking about exotic 7-spheres, one can look at the maps $\cdots \rightarrow \Omega^2Diff(D^4, rel \space \partial) \rightarrow \Omega Diff(D^5, rel \space \partial) \rightarrow Diff(D^6, rel \space \...
- 2,734
15
votes
2
answers
2k
views
The space of framed functions
Framed functions arose in the work of K. Igusa defining cohomology invariants for smooth manifold bundles (Igusa-Klein torsion). In the late 80's, he proved a strong connectivity result about the "...
- 2,734
15
votes
2
answers
927
views
$\pi_4$ of simply-connected 4-manifold
In Baues "The homotopy category of simply conected 4-manifolds" there is some
algebraic description of $\pi_4(M^4)$ where $M^4$ is simply-connected closed 4-manifold, but this description is pure ...
- 5,063
12
votes
1
answer
2k
views
Conventions for definitions of the cap product
In singular (co)homology, if $\alpha\in C^*(X)$ and $x\in C_*(X)$, then the cap product $\alpha \cap x$ is generally defined by the following process:
Apply to $x$ the diagonal map $C_*(X)\to C_*(X\...
- 5,509
1
vote
1
answer
414
views
Equivariant maps inducing isomorphism in integral cohomology
Consider the following statement.
Suppose $X$, $Y$ are finite CW-complexes with free involution
and $\mu:X\to Y$ is an equivariant map.
If $\mu^*:H^i(Y;\mathbb{Z})\to H^i(X;\mathbb{Z})$ is an ...
- 11
61
votes
2
answers
3k
views
The topological analog of flatness?
Recall that a map $f:X\to Y$ of schemes is called flat iff for any $x\in X$ the ring $O_{X,x}$ is a flat $O_{Y,f(x)}$-module.
Briefly the question is: what is the topological analog of this?
Many ...
- 23.5k
32
votes
3
answers
2k
views
A Pachner complex for triangulated manifolds
A theorem of Pachner's states that if two triangulated PL-manifolds are PL-homeomorphic, the two triangulations are related via a finite sequence of moves, nowadays called "Pachner moves".
A ...
- 44.4k
9
votes
5
answers
2k
views
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
3
votes
0
answers
446
views
When does the normal bundle of a submanifold of Euclidean space admit a flat connection?
Given a smooth submanifold of $R^n$, I was wondering if there is a reasonably simple criterion for deciding whether its normal bundle admits a flat connection. I am not ruling out monodromy in the ...
- 313
3
votes
1
answer
265
views
Equivariant Surgery problem
I have a question about surgery.
Let $G= \mathbb{Z}_m \times \mathbb{Z}$ and $M$ be a oriented 3-manifold with G-action. i.e. There exists a map $f\colon M/G \to BG$, where $BG$ is classifying space.(...
- 698
12
votes
1
answer
535
views
4-manifolds in the 4-sphere such that it, *and* its complement have unsolvable word problem
In an earlier thread I had asked whether or not one can find a smooth 4-dimensional submanifold of $S^4$ whose fundamental group has an unsolvable word problem. The answer is yes, and the reference ...
- 44.4k
13
votes
6
answers
3k
views
Geometric flavored textbook on algebra
I am interested in topology, while I am not so comfortable with some
algebraic flavored textbook on algebra. Actually, it was not until I learned some topology that I began to understand some ...
- 139
22
votes
1
answer
1k
views
Word problem for fundamental group of submanifolds of the 4-sphere
Given any finitely-presented group $G$, there are a few equivalent techniques for constructing smooth/PL 4-manifolds $M$ such that $\pi_1 M$ is isomorphic to $G$. For most constructions of these 4-...
- 44.4k
7
votes
3
answers
750
views
Jordan Curve Homotopy
Does there exist a notion of Jordan curve homotopy?
In particular, suppose we have two Jordan curves $C_0 : S^1 \rightarrow \mathbb{R}^2$ and $C_1 : S^1 \rightarrow \mathbb{R}^2$. When does there ...
- 163
7
votes
1
answer
633
views
Intersection product in a manifold, taking values in one factor
In a joint paper that I am working on, we are interested in taking the intersection product $[X] \cap [Y]$ of the fundamental classes of two compact, oriented pseudomanifolds $X$ and $Y$ in a compact, ...
- 56.6k
32
votes
1
answer
1k
views
"Affine communication" for topological manifolds
There is a situation that comes up regularly in algebraic topology when giving proofs of facts about manifolds, like Poincare duality and the like. The typical sequence goes like this:
Prove ...
- 52.6k
7
votes
3
answers
444
views
An obstruction theory for promoting homotopy equivalences that are equivariant maps to equivariant homotopy equivalences?
Say I have a map of $G$-spaces $f : X \to Y$ and I know it is a homotopy-equivalence in the plain sense that there exists a map (maybe not equivariant) $g : Y \to X$ such that the two composites are ...
- 44.4k
23
votes
4
answers
5k
views
De Rham decomposition theorem, generalisations and good references
De Rham decomposition theorem states that every simply-connected Riemannian manifold $M$
that admits complementary sub-bundles $T'(M)$ and $T''(M)$ of its tangent bundle parallel with respect to the ...
- 28.9k
9
votes
1
answer
1k
views
How does the Lefschetz-Poincare dual torsion linking pairing on manifolds with boundary interact with the maps of the long exact sequence of the manifold-boundary pair?
I'm wondering if anyone can point me to a reference on how the various
Lefschetz-Poincare dual torsion pairings of a manifold with boundary fit
together.
To explain in more detail, consider a ...
- 5,509
8
votes
2
answers
949
views
Can we make rigorous the 'obvious' characterisation of singular homology?
It is a well known and often touted fact that the singular homology groups 'count the k- dimensional holes' in a space (see: How does singular homology H_n capture the number of n-dimensional "...
- 3,230
8
votes
2
answers
4k
views
tangent sphere bundle over sphere
are there some general description about tangent sphere bundle over sphere?
(it is a special $S^{n-1}$bundle over $S^n$)
say for n=1,it is trivial,$S^0\times S^1$,for n=2,it is $SO(3)\cong \mathbb{R}...
- 81
21
votes
2
answers
3k
views
Does this approach for the Poincaré conjecture work?
Several months ago a paper was posted at
http://arxiv.org/abs/1001.4164
called "Another way of answering Henri Poincaré's fundamental question." The author gave a talk on it today at my institution. ...
18
votes
2
answers
790
views
The kernel of the map from the handlebody group to Outer automorphisms of a free group
Let $K$ be a compact oriented 3-dimensional handlebody of genus $g$. The group $H_g$ of isotopy classes of diffeomorphisms of $K$ is called the handlebody group. (It embeds as a subgroup of the ...
- 6,229
2
votes
2
answers
2k
views
Group action, Fixed point set and Orbit Space
I want to know to what extent is the group action determined by its fixed point data and orbit data, i.e. if $G$ acts on $M$ in two ways with the same fixed point set and orbit space, on what ...
- 257
10
votes
3
answers
807
views
Is every (finite) group action on R^n by diffeomorphisms conjugate to a linear action?
I want to know if every smooth (finite)group action on $\mathbb{R}^n$ is conjugate to some linear action.Thank you!
- 257
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
6
votes
2
answers
3k
views
Classification of mapping tori
Assume $M$ is a topological space and $f\in \operatorname{Homeo}(M)$, then $f$ obviously plays a significant role in the torus bundle
$$M_f = M\times I/\{(x,0)\sim (f(x),1)\mid x\in M\}.$$
Hence ...
- 257
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
6
votes
1
answer
637
views
Rational homotopy type of a complement
Let $X$ and $X'$ be smooth closed manifolds. Take closed subpolyhedra $D\subset X$ and $D'\subset X'$ (with respect to some triangulations) and let $f:X\to X'$ be a homotopy equivalence such that $f(D)...
- 23.5k
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
10
votes
1
answer
847
views
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
12
votes
1
answer
1k
views
classification of smooth involutions of torus
Let $\mathbb{Z}_2=\{1,g\},T^2=\{(e^{i\theta_1},e^{i\theta_2})\}$ and place $T^2$ in $\mathbb{R}^3$ as the locus of the rotation of $2\pi$ rads of the circle$\{(y,z)|(y-2)^2+z^2=1\}$ around $z$ axis.
...
- 291
13
votes
3
answers
966
views
Rational homotopy theory of a punctured manifold
Let $M$ be a smooth simply connected manifold and let $N$ be $M$ minus a point. Is it possible to construct an explicit Sullivan model for $N$ (i.e. a commutative differential graded algebra (cdga) ...
- 23.5k
17
votes
2
answers
2k
views
Involutions of $S^2$
are there some complete results on the involutions of 2 sphere?
at least I have three involutions:
(let $\mathbb{Z}_2=\{1,g\}$,and $S^2=\{(x,y,z)\in\mathbb{R}^3|x^2+y^2+z^2=1\}$)
1.$g(x,y,z)=(-x,-y,-...
- 291
22
votes
5
answers
4k
views
Why is complex projective space triangulable?
In an exercise in his algebraic topology book, Munkres asserts that $\mathbf{C}P^n$ is triangulable (i.e., there is a simplicial complex $K$ and a homeomorphism $|K| \rightarrow \mathbf{C}P^n$). Can ...
- 4,254
16
votes
10
answers
3k
views
Orbifold fundamental group in terms of loops?
In chapter 13 in his notes on 3-manifolds, Thurston defines the orbifold fundamental group to be the group of deck transformations of the universal cover of the orbifold. He also makes a statement "...
- 13.6k