All Questions
Tagged with gt.geometric-topology at.algebraic-topology
1,145 questions
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Meaning of Regular Neighborhood for Homology Basis Curves in $S_{g,2}$
I have been trying to understand the meaning of the expression "regular neighborhood" in the context described below, but I'm stuck:
We have a collection of curves $c_i$ for $i=1,2,..,n$ embedded in ...
- 1
1
vote
2
answers
410
views
Has this kind of question in topology a special name?
Consider a space $X$ and the group $Homeo(X)/\sim$ of homeomorphisms on $X$ modulo homotopies which are homeomorphisms in each step. One could also consider diffeomorphisms on $X$ or whatsoever.
Have ...
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2
votes
1
answer
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Does $S^2$ have a trivial normal bundle in any closed orientable manifold?
We know that the middle circle $S^1$ in Mobius band has a nontrivial normal bundle. Now consider the higher dimensional case. Let $M$ be a $n$-dimensional ($n\geq5$) closed orientable manifold and ...
- 1,373
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Codimension zero immersions
Given an immersion of the n-1-sphere into a (closed) n-manifold, when does it extend to an immersion of the n-disk?
Remark: If the sphere had dimension k smaller than n-1, then such an immersion ...
- 10.4k
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Generalizations of Dehn-Nielsen-Baer
For a manifold $M$, define the mapping class group $Mod(M)$ to be the set of self-diffeomorphisms of $M$, modulo isotopy. In symbols, $Mod(M) = \pi_0 Diff(M)$. Of course, every self-diffeomorphism ...
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2
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2k
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Why is the mapping class group of hyperbolic manifolds finite?
Hi! I'm trying to understand why a hyperbolic n-manifold has finite mapping class group if $n \geq 3 $. In books I'm reading it's said it's a consequence of Mostow's rigidity theorem:
"If M and N are ...
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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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1
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Cohomology of the infinite loop space of the affine grassmanian (as in the generalized Mumford conjecture)
I've been reading Hatcher's survey "A short exposition of the Madsen-Weiss theorem". In it, he outlines a nice proof of the "generalized Mumford conjecture", which asserts that the stable cohomology ...
- 71
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votes
2
answers
521
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space of homotopy equivalences of $S^1$
Does the space of homotopy equivalences of $S^1$ deformation retract onto the space of homeomorphisms of $S^1$? If so, does anyone have a reference?
I found that Kneser proved that $Homeo(S^1)$ ...
- 53
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6
answers
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Is there a classification of open subsets of euclidean space up to homeomorphism?
I hope this question is reasonable enough to have a well known answer. i.e either there is a simple invariant (like the homotopy groups) that characterizes the homeomorphism type of such set among ...
- 2,027
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0
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430
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The signature of a mapping torus
Consider a manifold $M$ of dimension $4k + 2$, $k$ an integer. Pick a diffeomorphism $\phi$ of $M$ and construct the mapping torus $T$ of $\phi$. Suppose that there is a $4k+4$ dimensional manifold $B$...
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1
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801
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Can every 3-manifold be triangulated? [closed]
One of my classmates was telling me that it is an open question whether every 3-manifold can be triangulated. This was rather surprising. He said that the question as far as he remember is settled ...
- 87
49
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4
answers
7k
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Elegant proof that any closed, oriented 3-manifold is the boundary of some oriented 4-manifold?
I'm looking for an elegant proof that any closed, oriented $3$-manifold $M$ is the boundary of some oriented $4$-manifold $B$.
- 1,709
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2
answers
728
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Euler class of S^1-orbibundle
Given a topological space $M$ with a locally free $S^1$ action on it, assume the slice representation holds,(this is often the case, e.g. M is a smooth manifold) then this will make $M$ a principal $S^...
- 73
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System dynamic of space euclidean and hyperbolic tilings
Theorem 2.9. (Rudolph [Rud89]) Suppose $X_{T}$ is a finite local complexity (FLC)
tiling space. Then $X_{T}$ is compact in the tiling metric d. Moreover, the action $T$ of
$R^{d}$ by translation is on ...
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1
answer
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Spin TQFT's in dimensions (1+1)
I don't seem to be able to find anything written about Spin TQFT's in dimension (1+1). Does anyone know any references? Or is there some reason it is uninteresting?
- 18.2k
1
vote
3
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549
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Regular homotopy invariance of Wall's self-intersection form.
This is the mirror of previous post.
For an immersion $f\colon N^n\to M^{2n}$ with fixed lift $\tilde{f}\colon N\to \tilde{M}$ and $N,M,\tilde{M}$ are oriented, first we define a unordered double ...
- 115
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4
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Explicit constructions of K(G,2)?
Recall that an Eilenberg-Maclane space $K(G, n)$ is characterized by $\pi_i(K(G,n)) = G$ if $i=n$ and is trivial otherwise. (Of course $G$ should be abelian if $n>1$.)
Let $G$ be a finite abelian ...
- 12.8k
9
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1
answer
778
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Low degree cohomology of Eilenberg-MacLane space K(G,2)?
Recall that an Eilenberg-Maclane space $K(G, n)$ is characterized by $\pi_i(K(G,n)) = G$ if $i=n$ and is trivial otherwise. (Of course $G$ should be abelian if $n>1$.)
I'm aware that computing $H^...
- 12.8k
3
votes
3
answers
316
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Existence of sequence of examples of braking 'Cancellation law in homeomorphic products'
I know there are manifolds (with or without boundary) $A$ and $B$ such that $A\times C$ is homeomorphic to $B\times C$ but $A$ is NOT homeomorphic to $B$.
My question is (in the Diffeomorphism ...
- 1,101
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1
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817
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Generalization of Moise's theorem
I am looking for a generalization of Moise's theorem, which the few professors that I asked treat as a "known geometric fact" but none could find a reference to an article proving it.
The claim is ...
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2
answers
658
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Do homeomorphisms of boundary components of 3-manifolds extend to the manifold?
The question that I'd like to answer can be generalized to the following: if $M$ is an orientable 3-manifold and $F$ is a boundary component of $M$ (which may have other boundary components), can an ...
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1
answer
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Why is Casson's invariant worth studying?
Hi everybody! I am reading some papers about Casson's invariant for (integral) homology 3-spheres...as the wiki says "Informally speaking, the Casson invariant counts the number of conjugacy classes ...
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2
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On the fundamental group of a finite CW complex
So let $X$ be a finite CW complex which is connected.
Q1: Is $\pi_1(X)$ necssarily a finitely presented group?
If the answer is yes, then how does prove it. I've tried to prove it using
an ...
- 7,609
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2
answers
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What characteristic class information comes from the 2-torsion of $H^*(BSO(n);Z)$?
This is just a general curiosity question:
In the standard textbook treatments of characteristic classes, and in particular the treatment of universal Pontrjagin classes, it's standard to consider $H^...
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2
answers
1k
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Is there a discrete Cerf theory?
Towards the end of the 1990's, Robin Forman developed a discrete version of Morse theory, which concerns maps from a simplicial complex to $\mathbb{R}$ satisfying a combinatorial analogue to the ...
- 22.1k
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5
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What is the intuition behind the Freudenthal suspension theorem?
The Freudenthal suspension theorem states in particular that the map
$$
\pi_{n+k}(S^n)\to\pi_{n+k+1}(S^{n+1})
$$
is an isomorphism for $n\geq k+2$.
My question is: What is the intuition behind the ...
- 727
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2
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3k
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First Chern class of a flat line bundle
A referee asked me to include a reference or proof for the following classical fact. It's not hard to prove, but I'd prefer to just give a reference -- does anyone know one?
Let $X$ be a nice space (...
- 44.8k
3
votes
2
answers
486
views
Poset fiber theorems under a special assumption on the poset map?!
Hey everyone, I am facing the following problem:
Say that a (order-preserving) poset map $f:P\to Q$ has property $(\star)$ if for all $q_1,q_2\in Q$ with $q_1\leq q_2$ and every $p_2\in f^{-1}(q_2)$ ...
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Homotopy type of the plane minus a sequence with no limit points
It is well known that the plane minus $n$ points is homotopy equivalent to a wedge of circles and hence its fundamental group is free on $n$ letters.
Question: Is the plane minus an infinite sequence ...
- 1,329
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3
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2k
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Identifying the orientation bundle uniquely
A nonorientable surface $S$ is homeomorphic to the $k$-th connected sum
$\mathbb{R}P^2 \sharp \ldots \sharp \mathbb{R}P^2$.
For each nonorientable surface $S$ there exists an oriented $2$-fold ...
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answer
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Orientation of a "glued"-manifold
Im wondering if there's a short way to prove that when two manifolds with diffeomorphic boundaries are glued together along the boundaries the orientations of these must be inverse to each other. That ...
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1
answer
640
views
Length of shortest possible knot
Consider a line L in R^3 in the shape of a trefoil knot. Consider the surface S that is the union of all unit circles that have centers on this line and whose tangent vectors are all perpendicular to ...
- 453
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2
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426
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Are there oriented $4k+2$ manifolds such that $im(H_{2k+1}(M; Z/2)\to H_{2k+1}(M, \partial M; Z/2))$ has odd dimension?
The following fairly specific question comes up in a bordism computation I'm trying to do:
Are there compact $\mathbb Z$-oriented $4k+2$ dimensional manifolds with boundary $M$ such that $im(H_{2k+1}(...
- 5,489
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votes
1
answer
958
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When does an antipodal map on a manifold extend to the antipodal map on a spheres
So I have been mulling the following question over in my head for awhile now, and want to see if anyone else might have any ideas.
Begin with $M$ a manifold and suppose that $M$ has an antipodal map $\...
- 5,191
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votes
5
answers
3k
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Can surfaces be interestingly knotted in five-dimensional space?
It's possible this question is trivial, in which case it will be answered quickly. In any case, I realized that it's a basic question the answer to which I should know but do not.
Everybody loves ...
- 54.6k
8
votes
1
answer
679
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topological type of smooth manifolds with prescribed homotopy type and pontryagin class
Can someone help explain the following result:
If the dimension is at least five, there are at most finitely many different smooth manifolds with given homotopy type and Pontryagin classes.
Thank ...
- 179
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2
answers
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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
13
votes
2
answers
791
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"C choose k" where C is topological space
One day I read a generating function in a paper. For any "sufficietly nice topological space", $C$:
$$ \sum_{l \geq 0 } q^{2l}\chi(\mathrm{Sym}^l[C]) = (1 - q^2)^{-\chi(C)} = \sum_{l \geq 0} \binom{...
- 22.8k
6
votes
0
answers
312
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homotopy domination that splits a non-split epimorphism and still wants to be a homotopy equivalence
Can a homotopy domination by a space supporting a free action of $G$ be promoted to a homotopy equivalence with such a space? As stated, this is not a serious question (multiply by an $EG$). But with ...
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3
votes
1
answer
552
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Is the Action of the mapping class group transitive on embedded arcs?
Let S be a surface of genus g with some parked points (n of them). Assume $n \geq 2$ and fix two of the marked points. Consider the set of embedded arcs going between these two special points. The ...
- 27.5k
10
votes
1
answer
544
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!
- 5,489
1
vote
1
answer
304
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good perspective in viewing manifolds of infinite dimension
Borel conjectued aspherical closed manifolds are topologically rigid.(i.e.a homotopy equivalence between two aspherical manifolds is homotopic to a homeomorphism).
now,soppuse M is a K(G,1) space,
it ...
- 179
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2
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fundamental group and complete invariant of irreducible 3-manifolds
I heard that,by Perelman's work,we can get that the fundamental group is a
complete invariant of irreducible 3-manifolds (except for lens spaces).
can someone help explain this.Thank you!
- 179
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votes
4
answers
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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
7
votes
1
answer
364
views
Aspherical homotopy orbit space of configurations on the 2-sphere
The group SO(3) acts naturally on $S^2$ and thus on $Conf(S^2, q)$, the configuration space of $q$ distinct points on the 2-sphere, via the diagonal action on $S^2 \times...\times S^2$. This is a ...
- 2,734
7
votes
1
answer
456
views
Space-discriminating injective curve
Let $f\colon \mathbb R^1\to \mathbb R^3$ be a continuous and injective map. Is $\mathbb R^3\setminus f(\mathbb R^1)$ a path-connected space?
- 5,053
12
votes
4
answers
832
views
$S^n \to S^m \to B$ bundle: possible?
Sphere bundles and bundles over spheres are everywhere and are excellent things to get one's hands dirty with.
(1a) But when can we have a bundle $S^n \to S^m \to B?$ It seems like requiring the ...
- 2,734
5
votes
1
answer
492
views
Pair consisting of a compact manifold and Morse function
Consider the following situation:
Let $M$ and $M'$ be two closed manifolds and suppose $f:M\to \mathbb{R}$ and $f':M'\to \mathbb{R}$ are smooth morse functions on $M$ and $M'$ respectively. We say ...
- 2,299
20
votes
2
answers
2k
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Simple curves on non-orientable surfaces.
Given an element in the (first) homology group of a surface, I would like to know if it can be represented as a simple closed curve. For orientable surfaces, this is well-known, but I wasn't able to ...
- 32.1k