All Questions
9,056 questions
2
votes
1
answer
329
views
Model categories and cellular maps
A question came up on MSE and it generated, for me, the following question:
When looking at the maps of CW/cell/simplicial complexes do the cellular/simplicial maps have a model theoretic ...
- 18.8k
29
votes
10
answers
3k
views
Are infinite dimensional constructions needed to prove finite dimensional results?
Infinite dimensional constructions, such as spaces of diffeomorphisms, spectra, spaces of paths, and spaces of connections, appear all over topology. I rather like them, because they sometimes help me ...
CommunityBot
- 1
5
votes
0
answers
583
views
Cohomology of Real algebraic Varieities
I understand Serre's GAGA theorem as saying that equations over algebraically closed fields can be studied equally from the algebraic and analytic points of view, at least with respect to cohomology.
...
CommunityBot
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11
votes
0
answers
1k
views
the first cohomotopy group of a P-adic solenoid
Greetings.
I have a problem concerning the definition of the first cohomotopy group of P-adic solenoid. In [1], E. Spanier defines group operation on the set of homotopy classes of maps from X to n-...
CommunityBot
- 1
0
votes
3
answers
548
views
Compact Hypersurfaces Bounding Compact Domains
The following statement seems to be taken as given in papers I'm reading:
Let $\mathcal{M}^n$ be a compact, embedded hypersurface in $\mathbb{R}^{n+1}$. Then $\mathcal{M}$ is the boundary of some ...
- 9,389
1
vote
1
answer
154
views
undecidability in the dynamics of functions $f: \Sigma^* \rightarrow \Sigma^*$
Does anyone know any undecidable problems in the dynamics of functions (not necessarily monoid homomorphisms or anything) from \Sigma^* to \Sigma^* where \Sigma is a finite set? In particular, I'm ...
- 236k
11
votes
4
answers
1k
views
Equivariant singular cohomology
One can define the $G$-equivariant cohomology of a space $X$ as being the ordinary singular cohomology of $X \times_G EG$ --- I think this is due to Borel? (See e.g. section 2 of these notes)
...
- 30.4k
1
vote
3
answers
312
views
Lifting relative homotopies between maps $X\to Y/B$ when $B$ is contractible
Let $(X,A)$ be a CW-pair, $Y$ a CW-complex, and $f,g:X\to Y$ homotopic maps such that $f_{|A}=g_{|A}$. Even though $f$ and $g$ are homotopic, they do not have to be homotopic relative $A$. (...
- 55.9k
10
votes
0
answers
463
views
Bicommutative Hopf algebras have internal hom objects. What are they?
Let $k$ be a field and $\mathcal H$ the category of conilpotent bicommutative Hopf algebras over $k$. (Conilpotent means that every element eventually becomes zero if you apply the reduced ...
- 33.8k
6
votes
0
answers
367
views
Does every exact six-term sequence arise as the K-theory of a locally compact pair?
Consider six countable Abelian groups and six group homomorphims as in the following diagram
G → H → I
↑ ↓
L ← K ← J
Assume that the resulting sequence is exact ...
- 3,184
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 $p_2\in f^{-1}(q_2)$ ...
- 18.8k
11
votes
2
answers
657
views
On-the-nose commutative cup product $\Longrightarrow$ characteristic $0$?
I was discussing with a student today the nature of the non-commutativity of cup-product at the level of cochains. In trying to explain what happens, I came up with the following statement.
...
- 52.6k
10
votes
4
answers
1k
views
Singular complex = cohomology ring + Steenrod operations?
Fix a prime $p$ and consider everything mod $p$. Steenrod operations arise somehow from the loss of information passing from the singular complex of a space to its cohomology ring. Are they exactly ...
CommunityBot
- 1
16
votes
1
answer
1k
views
Which cohomology theories are real- and complex-orientable?
A complex-oriented cohomology theory $E^*$ is a multiplicative cohomology theory with a choice of Thom class $x\in\tilde{E}^2(\mathbb{C}P^\infty)$ for the universal complex line bundle (which can be ...
- 56.9k
15
votes
3
answers
3k
views
Why is the dual of a torus the same as its fundamental group?
The set of continuous homomorphisms from a torus ${\mathbb T}^n = ({\mathbb R}/{\mathbb Z})^n \to {\mathbb R}/{\mathbb Z}$ can be identified with ${\mathbb Z}^n$ if we assign to each $k = (k_1, \ldots ...
- 17.5k
8
votes
1
answer
469
views
Computing H^2(X, T_X(-\log D))
Let $X\subset \mathbb P^3$ be a smooth variety and $D \subset X$ be an effective, reduced, irreducible divisor. My question is the following.
If I know the defining equations of $X$ and $D$ then is ...
CommunityBot
- 1
21
votes
3
answers
2k
views
Cohomology of fibrations over the circle: how to compute the ring structure?
This question is inspired by Cohomology of fibrations over the circle Moreover, it can be considered a subquestion of the above, but somehow it seems to me that some of the more interesting points ...
CommunityBot
- 1
2
votes
0
answers
148
views
Is the homotopy of a primitively generated Hopf algebra still primitively generated?
Let $A=\oplus A_n$ be a primitively generated graded Hopf algebra, where each $A_n$ is a simplicial group. This allows us to define the homotopy group $\pi_*(A)$.
Question: is the graded Hopf algebra ...
- 681
10
votes
3
answers
2k
views
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 ...
CommunityBot
- 1
9
votes
1
answer
514
views
Models for P map in EHP sequence
The E and H maps in the EHP sequence have models that aren't too hard to define. To review, the E map is induced on homotopy by $E: \Omega^n S^n \to \Omega^{n+1} S^{n+1}$ sending a map to its ...
- 18.8k
6
votes
2
answers
462
views
need references regarding the elementary theory of free semigroup and free abelian groups
Recently, I read that two free abelian groups $S$ and $T$ have the same elementary theory if and only if rank$S$=rank$T$. Does anyone have a reference with a proof of this? Also, what is known about ...
CommunityBot
- 1
17
votes
1
answer
683
views
Ordinal-indexed homology theory?
Back when I was a grad student sitting in on Mike Freedman's topology seminar at UCSD, he posed the following question. Does there exist a good homology theory $H_{\alpha}(X)$ where $\alpha$ is an ...
- 12.8k
9
votes
3
answers
1k
views
Can homologous submanifolds be connected by an immersed manifold with boundary?
Supposed I have an n-dimensional manifold M with a k-dimensional submanifold that is homologous to zero (or, equivalently, two homologous submanifolds). Can I always construct a k+1-dimensional ...
- 35.9k
7
votes
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 ...
- 1,793
3
votes
0
answers
423
views
Cohomologies associated to residually torsion-free nilpotent groups
This question is related to my previous question: Relationship between the cohomology of a group and the cohomology of its associated Lie algebra.
A group $G$ is ${\it residually \ torsion \ free \ ...
CommunityBot
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15
votes
2
answers
2k
views
Topological vs pro fundamental groups
Consider the following two structure-adding refinements of the fundamental group of a topological space:
the set $\pi_1(X)$ inherits a quotient topology from the compact-open topology of $X^{S^1}$, ...
CommunityBot
- 1
2
votes
1
answer
277
views
computing homotopy type
I'm trying to construct a computer algorithm that decides, whethear a map is homotopically trivial and came to the following question: Let U by a connected closed hypersurface of dimension n-1 that is ...
- 44.4k
8
votes
4
answers
6k
views
How to triangulate real projective spaces (as simplicial complexes in Mathematica)?
Hello!
I have written a program in Mathematica 7, which calculates for a (finite abstract) simplicial complex all its homology groups. I would really like to test it on the projective spaces, but ...
- 1,589
2
votes
2
answers
456
views
Algorithm that decreases the size of the simplicial triangulation
Hello!
Let X be a topological space. We are considering only abstract simplicial complexes, i.e. a finite list of finite lists of integers. Is there any algorithm (more or less efficient?), that ...
- 4,254
8
votes
3
answers
1k
views
Computing H_2 from pi_1=Z and pi_2
(Related question: What part of the fundamental group is captured by the second homology group?)
Suppose I have a path-connected space $X$ for which $\pi_1(X)=\mathbb{Z}$. Suppose I know $\pi_2(X)$ ...
CommunityBot
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25
votes
4
answers
4k
views
How canonical is cofibrant replacement?
Quillen's original definition of a model category included noncanonical factorization axioms, one being that any map can be factored into a cofibration followed by an acyclic fibration. More recent ...
- 111
3
votes
1
answer
298
views
Eilenberg-Mac Lane spaces for groups that can't see $p$-groups
All groups here are abelian and $p$ is a prime number; I'll say $P$ is a $p$-group if every element
of $P$ has finite order which is a power of $p$.
Suppose $\mathrm{Hom}(G,P) = 0$ for every $p$-...
- 55.9k
8
votes
2
answers
370
views
Spectral techniques for genus of a graph
A generic question:
are there any spectral techniques to estimate the genus of a graph? I am interested in complete balance multipartite graph.
- 16.9k
3
votes
1
answer
1k
views
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 ...
- 56.6k
2
votes
2
answers
390
views
vanishing of $\pi_2$ and $H_2$
I am looking for an "easy" proof of the following statement: Suppose that $X$ is a simply connected space for which $\pi_2(X)=0$. Then $H_2(X)=0$ as well.
I know that one can use the Hurewicz ...
- 27.2k
5
votes
2
answers
754
views
explicit linear representations of fundamental groups of surfaces
I am looking for an explicit representation of the fundamental group of a closed orientable surface of genus >1. I guess they should be abundant in degree 2. Did anyone see the explicit matrix ...
- 8,803
8
votes
0
answers
205
views
Characteristic classes from moduli of alternating forms
Suppose, just for example, that you have a smooth manifold $M$ of dimension greater than $8$,and a cohomology class $q$ in $H^3(M,{\Bbb R})$. Suppose further that one can represent $q$ with a ...
- 17.6k
2
votes
2
answers
854
views
Fundamental group of a product of two curves
Let $S$ be a complex surface whose fundamental group is isomorphic to the fundamental group of a product of two curves of genera $>1$. Does $S$ have to be a product of two curves?
- 5,050
16
votes
3
answers
3k
views
smooth sections of smooth fiber bundles
A maybe trivial question about fiber bundles (I'm not an expert, and I didn't find quickly an answer looking here and there). Suppose you are given fiber bundle $p\colon E\to M$, where
$E,M$ are ...
- 4,306
0
votes
1
answer
305
views
Are braid links proper links?
Are braid links proper links? Or are the concepts involved unrelated?
- 4,404
5
votes
0
answers
571
views
Are there cohomology classes on a hyperkähler manifolds which pull back to the Stiefel-Whitney classes on every Lagrangian submanifold?
This is a bit of a stab in the dark but I was wondering if anyone has defined cohomology classes on a hyperkähler manifold which pull back to the Stiefel-Whitney classes on any submanifold which is ...
- 44.7k
7
votes
2
answers
426
views
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 $im(H_{2k+1}(...
- 55.9k
11
votes
1
answer
594
views
co-$A_\infty$ spaces
A co-$A_n$ space is a based space $Y$ equipped with a co-action by the Stasheff associahedron operad $K_\bullet$. This means that $Y$ is comes with certain maps $c_n: Y \times K_n \to Y^{\vee n}$, $n =...
CommunityBot
- 1
51
votes
3
answers
12k
views
Spaces with same homotopy and homology groups that are not homotopy equivalent?
A common caution about Whitehead's theorem is that you need the map between the spaces; it's easy to give examples of spaces with isomorphic homotopy groups that are not homotopy equivalent. (See Are ...
CommunityBot
- 1
3
votes
1
answer
958
views
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 $\...
- 18.8k
4
votes
1
answer
473
views
Cohomological dimension of a group, fibration and local coefficients
Hello,
I want to show that the cohomological dimension (say over Z or R) of some group $K$ is 1. $K$ occurs in an exact sequence $1 \to K \to \pi_1(X) \to \pi_1(C) \to 1$, where $\pi_1(X)$ has ...
- 8,717
9
votes
1
answer
551
views
Extreme rigidification of homotopy self-equivalences
Suppose $X$ is a CW-complex. The monoid of homotopy self-equivalences $M = hAut(X)$ is the subspace of $Map(X,X)$ consisting of those maps with a homotopy inverse. It is a union of path components. ...
- 18.8k
4
votes
1
answer
195
views
Contractible space of maps between Eilenberg-Mac Lane spaces, 2
Let $G$ and $H$ be torsion abelian groups.
Are the following are equivalent:
$\mathrm{Hom}(G, H) = 0$
$\mathrm{map}_*( K(G, m), K(H,n)) \sim *$ for all $n, m\geq 1$
?
Clearly (2) implies (1).
- 12.5k
4
votes
1
answer
439
views
Contractible space of maps between Eilenberg-Mac Lane spaces
Suppose $A$ is an abelian torsion group, with no elements of order $p$, and let
$P$ be an abelian $p$-group (i.e., the order of each element is a power of $p$).
It sure seems to me that
$$
\mathrm{...
- 18.8k
24
votes
5
answers
3k
views
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 ...
- 5,575