All Questions
Tagged with gt.geometric-topology at.algebraic-topology
1,145 questions
16
votes
0
answers
325
views
Rational equivalence of smooth closed manifolds
All spaces below will be assumed simply connected. A continuous map is a rational equivalence if it induces an isomorphism of the rational homology groups. Two spaces are rationally equivalent if they ...
- 23.5k
1
vote
2
answers
334
views
Examples of finite polyhedra with finitely generated simple fundamental group
For $n\geq 2$, $P\mathbb{R}^n$ is a simple example of finite polyhedron with finitely generated simple fundamental group. I was wondering if someone could give me an example of a finite polyhedron ...
- 25k
3
votes
1
answer
125
views
Cohomology of the coned off space
Let $X$ be a compact manifold with boundary $\partial X$ with $
\dim X\setminus \partial X=n$. Moreover, $X$ and $\partial X$ are both aspherical. Then what's the $H^n(X\cup_{\Sigma\subset \partial X} ...
- 5,489
3
votes
1
answer
507
views
Handlebody decomposition of $L(2,1)\times S^1$
I wish to know the handlebody decomposition of $L(2,1)\times S^1$ in terms of Kirby diagrams, where $L(2,1)\cong RP^3$. And if possible, is there a general recipe for getting the handlebody ...
- 244
6
votes
0
answers
374
views
Singular homology using singular cubes
When singular homology is defined using cubes instead of simplices it is important to factor out the degenerate cubes in the course of building the singular chain complex. If you omit this step it is ...
- 63.9k
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 ...
- 4,713
2
votes
1
answer
267
views
on second cohomology of $S^1$-manifold
Let $M$ be a closed oriented real manifold with a free smooth circle action. Denote $BS^1$ to be the classifying space of principal circle bundles and $ES^1\rightarrow BS^1$ to be the universal ...
- 113
8
votes
0
answers
290
views
When does the tangent microbundle of a closed orientable topological $4k$-manifold have a trivial rank 2 subbundle?
$\DeclareMathOperator{\Top}{Top}
\DeclareMathOperator{\co}{H}$Let $M$ be a closed orientable connected topological manifold of dimension $4k$ with $k > 1$. It is known (David Frank, On the index of ...
- 12.9k
13
votes
0
answers
223
views
Examples of manifolds with first nontrivial SW-class in degree 16 or bigger
As a module over the Steenrod algebra, $H^{\ast}(BO;\mathbb F_2) = \mathbb F_2[w_1, w_2, w_3, \dots]$ is generated by $w_{2^t}, t \geq 0$. Thus, the first nontrivial SW-class of any vector bundle $\xi ...
- 11.9k
6
votes
1
answer
414
views
Do $h$-cobordism groups arise from a 'Thom-like' spectrum?
Not thinking about $h$-cobordism, one usually defines a cobordism between manifolds, realizes it is an equivalence relation, chooses an appropriate class of structured manifolds (framed, unoriented, ...
- 609
22
votes
2
answers
978
views
Fixed-point free diffeomorphisms of surfaces fixing no homology classes
One of my graduate students asked me the following question, and I can't seem to answer it. Let $\Sigma_g$ denote a compact oriented genus $g$ surface. For which $g$ does there exist an orientation-...
- 28.2k
2
votes
1
answer
311
views
Vanishing cycles exact sequence for degeneration of curves
Let $f : X\rightarrow D$ be a family of smooth projective curves over the complex unit disk $D$, degenerating to a nodal curve $X_0$ above $0\in D$.
Let $\eta\in D - \{0\}$ be a general point, and let ...
- 1,803
4
votes
1
answer
576
views
Reference request for Poincaré–Lefschetz duality as an intersection pairing
I believe the following is well known after talking to some experts, but I am unable to find a reference for the case with boundary.
Fix a field $F$ and an oriented $n$-manifold $(M,\partial M)$. We ...
- 5,839
2
votes
0
answers
118
views
Configurations of points in a spectrum
I am wondering if the following construction has appeared in the literature:
Let $X$ be a pointed space. Define the singular configuration space of $X$, $F^*(X,k)$ ,to be $\{(x_1,\dots,x_k)| x_i=x_j \...
- 5,839
0
votes
0
answers
194
views
Equivariant cohomology with discrete group action
As far as I know, the equivariant cohomology can be regarded as the generalisation of de Rham cohomology with group action on manifolds. From the literature, the group action is Lie group type. I am ...
- 63.9k
7
votes
2
answers
637
views
An algebraic proof of Mumford's smoothness criterion for surfaces?
(Disclaimer: I'm a beginner in this area, so welcome corrections.)
Let $(X,x)$ be a germ of a complex surface (i.e. locally the zero set of some holomorphic functions) and assume that $x$ an isolated ...
- 1,446
2
votes
1
answer
130
views
Gluing isotopic smoothings
Let $M$ be a topological manifold which can be written as $M = U \cup V$ where $U$ and $V$ are open. Suppose both $U$ and $V$ admit smooth structures. Also assume that on the overlap $U \cap V$ the ...
- 5,839
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 ...
- 56.6k
1
vote
0
answers
157
views
Decomposing the homology of a connected sum of surfaces in a way which highlights the combinatorics of gluing
For each $i = 1,2$ and $j = 1,\ldots,n$, let $C_{i,j}$ be a connected compact oriented surface with $k$ boundary components. For $i = 1,2$, let $C_i = \sqcup_{j=1}^n C_{i,j}$, and let $C$ be the ...
- 7,527
3
votes
0
answers
130
views
Functoriality of short exact sequence of fundamental groups induced by a Seifert fibered space
Let $M$ be a Seifert fibered space over an orbifold $B$. Assume that $B$ is good and has infinite orbifold fundamental group. Then it is well known that there is a short exact sequence. $$1\to {\Bbb Z}...
- 585
6
votes
0
answers
148
views
Is every geodesic space with non-positively curved metric triangulable?
Let $(X, d)$ be a geodesic space which is also a closed manifold. Suppose $d$ is a local $CAT(0)$ metric. Here is the question:
Is it true that $X$ admits a triangulation? (No requiring the ...
- 2,623
2
votes
1
answer
177
views
Isotopic homeomorphisms of surface induces same map on the space of ends
Let $\Sigma$ be a non-compact orientable connected two-manifold without boundary. Let $f,g\colon \Sigma\to \Sigma$ be two homeomorphisms. Suppose there is a homotopy $H\colon \Sigma\times [0,1]\to \...
- 28.2k
7
votes
0
answers
265
views
Homotopy type of space of embeddings of a disk
Let $M^n$ be a smooth $n$-dimensional manifold and let $\mathbb{D}^n$ be the open unit disk in $\mathbb{R}^n$. Consider the space $\text{Emb}(\mathbb{D}^n,M^n)$ of smooth embeddings of $\mathbb{D}^n$ ...
- 44.8k
19
votes
6
answers
3k
views
Diffeomorphism of 3-manifolds
Surgery theory aims to measure the difference between simple homotopy types and diffeomorphism types. In 3 dimensions, geometrization achieves something much more nuanced than that. Still, I wonder ...
- 1,446
27
votes
6
answers
4k
views
Failure of smoothing theory for topological 4-manifolds
Smoothing theory fails for topological 4-manifolds, in that a smooth structure on a topological 4-manifold $M$ is not equivalent to a vector bundle structure on the tangent microbundle of $M$. Is ...
- 1,446
5
votes
1
answer
244
views
Coefficient of the top Pontryagin class in $L$-genus
The $L$ genus can be expressed as combinations of the Pontryagin classes with the first few terms as follows:
$$L_1=\frac{1}{3}p_1,$$
$$L_2=\frac{1}{45}(7p_2-p_1^2),$$
$$L_3=\frac{1}{945}(62p_3-...
- 8,167
5
votes
1
answer
184
views
Fundamental groups of primitive non-algebraic compact Kähler manifolds
Call a compact topological manifold $M$ primitive if there is no Serre fibration $M\to B$ where $B$ is a CW complex of dimension $0<d<\mathrm{dim}(M)$.
Given a Kähler group $G$ does there exist ...
- 1,170
3
votes
0
answers
137
views
Intersection number for 4 manifold with boundary
Let $X$ be a closed oriented smooth $4$-manifold. Suppose there is an embedding $\Sigma\to X$, it is known that the self-intersection number satisfies $[\Sigma]\cdot [\Sigma]=\pm\int_\Sigma c_1(N)$, ...
- 1,915
2
votes
0
answers
615
views
Fiber bundle orientability vs manifold orientability
This question seems like a pretty straightforward generalization of a result from vector bundles but its been on MSE for over a week with no answers so I'm reposting https://math.stackexchange.com/...
- 4,081
4
votes
1
answer
280
views
“Combinatorial” moves between cell complexes
EDITED:
A pair of finite simplical complexes are equivalent if and only if they are related by a finite sequence of the Pachner moves.
Is there a similar thing on finite cell complexes? That is, are ...
- 504
14
votes
1
answer
860
views
Mapping torus of Klein bottle
This got 5 upvotes but no answers on MSE (Mapping torus of Klein bottle), so I'm cross-posting to MO:
The mapping torus of a Klein bottle $ K $ is a compact flat 3 manifold.
The mapping class group of ...
- 12.9k
8
votes
1
answer
399
views
Universal cover with one end
Suppose that $M$ is a non-compact manifold of finite topological type with one end which is the universal cover of some closed manifold $N$.
Is $M $ necessarily homeomorphic to the total space of some ...
- 19.4k
30
votes
1
answer
1k
views
Are homeomorphic representations isomorphic?
Let $G$ be a finite group. Let $V_1, V_2$ be two finite-dimensional real representations. Suppose $f: V_1 \to V_2$ is a $G$-equivariant homeomorphism. Can one conclude that $V_1$ and $V_2$ are ...
- 12.9k
5
votes
1
answer
258
views
Nondegeneracy of kernel of map on homology induced by covering of surfaces
Let $f:X \rightarrow Y$ be a finite covering map between compact oriented surfaces and let $K$ be the kernel of the induced map $f_\ast: H_1(X) \rightarrow H_1(Y)$. Here homology has rational ...
- 38k
21
votes
1
answer
754
views
Is $\mathbb{CP}^3$ minus two points the universal cover of a compact manifold?
After reading some recent questions on mathoverflow about universal coverings, I am curious about the following:
Is it possible to construct a closed $6$-manifold $M$, with universal cover ...
- 7,842
5
votes
4
answers
3k
views
integral or rational cohomology of real grassmannians
I have obtained that the cohomology rings
$$
H^*(G_k(\mathbb{R}^\infty);\mathbb{Z}_2)=\mathbb{Z}_2[w_1,\cdots,w_k].
$$
Also
$$
H^*(G_k(\mathbb{R}^m);\mathbb{Z}_2)=\mathbb{Z}_2[w_1,\cdots,w_k]/(\bar ...
- 838
19
votes
1
answer
862
views
Diffeomorphism groups of h-cobordant manifolds
Do we have specific examples of h-cobordant smooth manifolds $M$ and $M'$ such that $\operatorname{BDiff}(M) \not \simeq \operatorname{BDiff}(M')$? Perhaps something can be said in terms of K-theory ...
- 18.2k
3
votes
0
answers
158
views
What is the meaning of local inertia conjugation property?
In Hatcher, Allen; Lochak, Pierre; Schneps, Leila, On the Teichmüller tower of mapping class groups, J. Reine Angew. Math. 521, 1-24 (2000). ZBL0953.20030., we have:
Abstract. Let $\widehat{G T}^{1}$ ...
- 63.9k
1
vote
1
answer
147
views
Transversal pre-image of a small enough trivial tubular neighborhood contains a trivial tubular neighborhood
A similar post on MSE without answer.
Let $f\colon M'\to M$ be a smooth map between two orientable closed smooth manifolds and $S$ be a smoothly embedded closed orientable submanifold of $M$ of co-...
- 7,842
5
votes
1
answer
376
views
Extension of covering map
The question is the following: suppose I have manifolds $N$ and $M$ both with boundary, and I have a covering map $\phi$ from $\partial N$ to $\partial M.$ The question is: when is there a covering ...
- 68.9k
11
votes
0
answers
335
views
Isotopy on embedded 3-manifolds in 4-manifolds
Suppose that $X$ be an oriented, closed four-manifold. Suppose that $Y$ is an oriented, closed three-manifold smoothly embedded in $X$. Suppose also that $f:X \to X$ is a diffeomorphism that fixes $Y$...
- 28.2k
3
votes
0
answers
113
views
Extending good covers of $\partial M$ to $M$
Suppose $M$ is an $n$-dimensional manifold with boundary with a free action of a finite group $G$. Suppose one has an equivariant collar $c: \partial M \times [0,1) \rightarrow M$. An open cover is ...
- 5,839
6
votes
0
answers
162
views
Uniqueness of normal microbundle of a smooth embedding
Suppose $M$ is a topological manifold and $\iota: N\hookrightarrow M$ be a submanifold. A normal microbundle of $N$ consists of an open neighborhood $U$ of $N$ and a retraction $\pi: U \to N$ such ...
- 803
25
votes
1
answer
1k
views
Is there a $4$-manifold which Immerses in $\mathbb{R}^6$ but doesn't Embed in $\mathbb{R}^7$?
I'm interested in both version of the question in the title, i.e. in the topological category and in the smooth category. By a topological immersion I mean a local embedding. I was asking in ...
- 439
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 ...
- 2,821
6
votes
1
answer
644
views
Torus bundles and compact solvmanifolds
I asked this question on MSE 9 days ago and it got a very helpful comment from Eric Towers providing the Palais Stewart reference, but no answers. So I'm crossposting it here.
Let
$$
T^n \to M \to T^m ...
- 4,081
6
votes
1
answer
324
views
Computation of $\pi_1$ for a Mazur manifold and its boundary
If we attach a $4$-dimensional $1$-handle $D^1 \times D^3$ to a $4$-dimensional $0$-handle $B^4$, we obtain $S^1 \times D^3$. The null homologous knot in $S^1 \times S^2$ indicated in the picture ...
- 1,292
5
votes
0
answers
132
views
geometry and connected sum of aspherical closed manifolds
Let $ G $ be a Lie group with finitely many connected components, $ K $ a maximal compact subgroup, and $ \Gamma $ a torsion free cocompact lattice. Then
$$
\Gamma \backslash G/K
$$
is an aspherical ...
- 4,081
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 ...
- 35.5k
7
votes
2
answers
617
views
Status of the Hopf-Thurston sign conjecture in dimension 4
A famous conjecture in topology asserts:
The Euler characteristic of a closed aspherical $2n$-manifold $M$ satisfies $(-1)^n\chi(M) \geq 0$.
This was conjectured by Hopf for manifolds with non-...
- 3,451