All Questions
80 questions
66
votes
8
answers
10k
views
What are the open subsets of $\mathbb{R}^n$ that are diffeomorphic to $\mathbb{R}^n$
I would like to know if there is a known necessary and sufficient
property on an open subset of $\mathbb{R}^n$ to be diffeomorphic to $\mathbb{R}^n$ :
For example :
Are all open star-shaped subsets ...
- 677
30
votes
4
answers
3k
views
Is the space of diffeomorphisms homotopy equivalent to a CW-complex?
Clarification: My question concerns the homotopy type of the space of $C^k$ diffeomorphisms with the compact-open $C^k$ topology, where $0< k \leq\infty$. I have stated my question below with $k=1$ ...
- 6,210
28
votes
3
answers
2k
views
Does a *topological* manifold have an exhaustion by compact submanifolds with boundary?
If $M$ is a connected smooth manifold, then it is easy to show that there is a sequence of connected compact smooth submanifolds with boundary $M_1\subseteq M_2\subseteq\cdots$ such that $M=\bigcup_{i=...
- 18.7k
26
votes
6
answers
3k
views
How to get convinced that there are a lot of 3-manifolds?
My question is rather philosophical : without using advanced tools as Perlman-Thurston's geometrisation, how can we get convinced that the class of closed oriented $3$-manifolds is large and that ...
- 2,696
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
25
votes
2
answers
844
views
Which homotopy classes $S^3 \to S^2$ lift to embeddings $S^3 \to S^2 \times D^3$?
The question is, for a smooth embedding
$$f : S^3 \to S^2 \times D^3$$
one can compose the map $f$ with projection $\pi : S^2 \times D^3 \to S^2$, giving the map $\pi \circ f : S^3 \to S^2$.
Which ...
- 44.4k
20
votes
3
answers
2k
views
Homotopy groups of spaces of embeddings
Let $\mathrm{Emb}(M, N)$ be the space of smooth embeddings of a closed manifold $M$ into a manifold $N$ equipped with smooth compact-open topology.
Question 1. Are there conditions ensuring that ...
- 29.1k
20
votes
1
answer
571
views
Can every manifold be dominated by a parallelizable one?
A closed, oriented $d$-manifold $M$ is said to dominate another such manifold $N$ if there exists a map $M \to N$ of non-zero degree. (This notion should not be confused with the unrelated concept of ...
- 11.9k
19
votes
0
answers
649
views
Bernoulli & Betti numbers (of manifolds) and the prime 34511
The purpose of this question is to resolve a mystery surrounding the prime 34511 that has got me bogged down for a while now. If you only care about the number theory and not the motivation coming ...
- 11.9k
18
votes
1
answer
1k
views
Wu formula for manifolds with boundary
The classical Wu formula claims that if $M$ is a smooth closed $n$-manifold with fundamental class $z\in H_n(M;\mathbb{Z}_2)$, then the total Stiefel-Whitney class $w(M)$ is equal to $Sq(v)$, where $v=...
- 1,329
18
votes
1
answer
1k
views
Is the restriction map for embeddings of manifolds with boundary a fibration?
Let $M$ and $W$ be smooth manifolds (possibly with boundary) and $V\subseteq W$ a submanifold. We have a map between embedding spaces
$$Emb(W,M)\rightarrow Emb(V,M)$$ given by restriction.
Richard ...
- 2,974
17
votes
2
answers
1k
views
Homotopy groups of Diff(X) and Homeo(X)
For a compact closed smooth manifold $X$, the group Diff(X) has a natural homomorphism $\Phi$ to the homeomorphism group Homeo(X). If $X$ has dimension at least $5$, I'm looking for some general ...
- 19.4k
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
15
votes
1
answer
541
views
Where is the Steenrod Realization problem at?
I'm wondering if there is a more modern reference out there for the Steenrod Realization Problem than the book of Connor and Floyd?
Realizing homology classes in a manifold via embedded submanifolds, ...
- 44.4k
14
votes
1
answer
826
views
An almost complex structure on $S^2\times ...\times S^2 / \mathbb{Z_2}$
Consider the product of $2n$ two-spheres $X_n=(S^2)^{2n}$. This manifold admits an orientation preserving involution that preserves the product structure and acts as the (orientation reversing) ...
- 14.3k
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 ...
- 4,081
13
votes
3
answers
1k
views
How to disjoint two cycles with zero intersection?
Suppose that $M^n$ is a smooth connected orientable manifold and $Z^k$ with $Z^{n-k}$ are two real cycles in $M^n$ with zero index of intersection $Z^k\cdot Z^{n-k}=0$ (these cycles are submanifolds ...
- 14.3k
13
votes
1
answer
580
views
Identifying two definitions of orientation on a vector space
Let $V$ be an $n$-dimensional real vector space. Here are two definitions of an orientation on $V$:
A generator of the $1$-dimensional vector space $\wedge^n V$, up to multiplication by positive ...
- 133
12
votes
3
answers
849
views
$A_{\infty}$-structure on closed manifold
Is there an exmaple of a closed smooth connected manifold $M$ having a structure of $A_{\infty}$-space (with unit) but $M$ is not homeomorphic to a compact connectd Lie group as space ?
Edit: First, ...
- 1,974
12
votes
3
answers
1k
views
Fixed point set of smooth circle action
Suppose $M$ is a connected closed smooth $d$-dimensional manifold, and suppose $S^1 = SO(2)$ acts smoothly on $M$. Then the fixed point set $Y = M^{S^1}$ will be a submanifold of $M$ of even ...
- 11.9k
12
votes
0
answers
408
views
The $\infty$-category of $n$-manifolds and open embeddings determined homotopically from that of topological manifolds?
Let $\mathrm{Diff}_n$, $\mathrm{PL}_n$, $\mathrm{Top}_n$ denote the $\infty$-categories of $n$-manifolds which are respectively smooth/PL/topological, and open embeddings (for instance by taking the ...
- 7,789
11
votes
1
answer
388
views
Twisting bordism classes
Let $X$ be a reasonable topological space (I'd be happy to assume that $X$ is a smooth closed manifold) and let $f\colon M^n \rightarrow X$ be a continuous map from a smooth oriented $n$-manifold $M^n$...
- 113
11
votes
1
answer
454
views
Is there a closed 5-manifold $M$ with $w_1(M)w_2(M)\ne 0$?
I'm trying to find generating manifolds for the cobordism group $\mathit{MO}_5(K(\mathbb Z/2, 2))\cong (\mathbb Z/2)^4$, which can be represented as the cobordism group of closed 5-manifolds $M$ ...
- 6,881
11
votes
0
answers
650
views
Triangulation of manifolds with corners
Let's begin with some definitions:
A (smooth) manifold with corners is a Hausdroff (and second countable if you want) space that can be covered by open sets homeomorphic to $\mathbb R^{n-m} \times \...
- 909
10
votes
2
answers
709
views
4-dimensional cohomology $\mathbb{CP}^2$'s
Let $M$ be a closed, smooth $4$-manifold with integral cohomology ring isomorphic to that of $\mathbb{CP}^2$, is it diffeomorphic to it?
- 6,995
10
votes
2
answers
703
views
When does an even-dimensional manifold fiber over an odd-dimensional manifold?
Are there simple necessary and sufficient conditions for an (oriented) even-dimensional compact smooth manifold to fiber over an (oriented) odd-dimensional manifold (with oriented fibers)?
For ...
- 54.6k
10
votes
1
answer
635
views
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
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
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
9
votes
1
answer
5k
views
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
9
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
322
views
Torsion and nilpotence of framed manifolds under the Pontryagin-Thom map
The framed Pontryagin Thom construction produces a graded ring isomorphism from the framed cobordism ring $\Omega^{fr}_*$ to the stable ring of homotopy groups of spheres $\pi_*(\mathbb{S})$. I have a ...
9
votes
2
answers
755
views
Is limit of null-homotopic maps null-homotopic?
The question is motivated by my failed comment to this one.
Let $M$ and $N$ be path connected locally compact, locally contractible metric spaces (you may assume that they are manifolds).
Let $\...
- 5,529
9
votes
0
answers
287
views
Rational cobordism classes of manifolds fibered over spheres
Let us fix positive integers $k, m$. Let $A^k_{4m} \subset \Omega^{\text{SO}}_{4m} \otimes \mathbb Q$ be the subgroup generated by oriented cobordism classes of manifolds fibered over $S^k$.
The ...
- 11.9k
8
votes
1
answer
387
views
Outer automorphism group of Brieskorn homology sphere?
In this post, it is discussed how a Brieskorn homology sphere $\Sigma(a_1,a_2,a_3)$ with $\displaystyle \frac{1}{a_1}+ \frac{1}{a_2}+ \frac{1}{a_3} < 1$ is an aspherical manifold with a ...
- 309
8
votes
1
answer
379
views
Existence of certain "nondegenerate" function and manifold topology
Let $M$ be a smooth manifold without boundary, not necessarily compact.
Let $f$ be a real-valued smooth function on $M\times M$. We say $f$ is good if for any point $(x,y)\in M\times M$ with local ...
- 783
8
votes
1
answer
391
views
Homotopy type of diffeomorphism which are the identity on and near the boundary
Let $M$ be a compact manifold with boundary. Denote by $Diff(M), Diff_\partial(M)$ and $Diff_{U\partial}(M)$ the groups of diffeomorphisms of $M$ and the subgroups of the ones that are the identity on ...
- 81
8
votes
2
answers
633
views
Construction of exotic spheres that do not bound parallelizable manifolds
There are at least two ways to construct homotopy spheres that bound parallelizable manifolds, namely Milnor's plumbing construction and Brieskorn's method of singularities, and each of these methods ...
- 985
8
votes
0
answers
118
views
Defining convex sums locally on the sphere?
$S^1$ and the torus $T^2$ are spaces in which convex combinations don't make sense globally but do locally. Despite their standard representations in $\mathbf{R}^2$ and $\mathbf{R}^3$ respectively not ...
- 637
8
votes
0
answers
125
views
Relating bordism generators in d and d+2 dimensions --- an explicit example
This is an attempt to make my relation between bordism invariants in $d$
and $d+2$ dimensions, following a previous attempt more explicit. This counts as a different question, since some more specific ...
- 2,147
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
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
6
votes
1
answer
442
views
Are framed manifolds cubulatable?
Let's say an $n$-manifold is cubulated if it is glued out of cubes $[0,1]^n$ in a way that looks locally like the standard cubulation of $\mathbb R^n$. For instance, the face $[0,1]^{k-1} \times \{1\} ...
- 54.6k
6
votes
2
answers
1k
views
A question about Dehn surgery and Brieskorn homology 3-spheres
I have been learning about Brieskorn homology 3-spheres $\Sigma(a_1,...,a_n)$ and Seifert manifolds. My reference is the first few pages of Saveliev's "Invariants of Homology 3-spheres."
If I ...
- 1,331
6
votes
1
answer
222
views
Stable smoothing of topological manifolds relative to an embedding
Let $M$ be a topological manifold. We know that $M$ is stably smoothable if and only its tangent microbundle, up to stabilization, admits a reduction to vector bundle.
Now I wonder if there is a ...
- 803
6
votes
1
answer
293
views
Riemann-Hurwitz for real maps
Let $S$ be a (compact, connected) Riemann surface of genus $g$ and $f: S\to \mathbb CP^1$ be a degree $d$ meromorphic function. Then the Riemann-Hurwitz formula tells us that the number of ...
- 14.3k
6
votes
1
answer
489
views
Equivariant handle decompositions
Suppose I have some smooth closed high-dimensional manifold $M$ acted on smoothly by a finite group $G$. By a metric averaging procedure, we can equip $M$ with a smooth Riemannian metric so that $G$ ...
- 18.7k
6
votes
0
answers
634
views
Quotient space, a fundamental group, and higher homotopy groups 2
Previously, I ask for comments/suggestions on setting up the calculation in Quotient space, homogeneous space, and higher homotopy groups. There, however, I was looking for whatever methods and tools ...
- 10.5k
6
votes
0
answers
231
views
Does $\#_n S^2×S^1$ really admit a map of non-zero degree from $B×S^1$?
In Proposition 4 on page 6 of this paper, http://arxiv.org/abs/1202.6302, the authors claim to produce a degree 2 $\pi_1$-surjective map $f$ between $M=S^1 \times \Sigma_2$ and $N=\#_2 S^2 \times S^1$ ...
- 61
5
votes
2
answers
520
views
What 3-manifolds can be obtained by gluing $ S^1 \times P $ and two copies of $S^1 \times D^2$
Let P denote the pair of pants e.g. a sphere minus three small discs $D_1$,$D_2$,$D_3$ about marked points $x_1,x_2,x_3$. I then consider $P \times S^1$. We have boundary components $T_1$,$T_2$,$T_3$. ...
- 1,331