All Questions
9,056 questions
3
votes
1
answer
210
views
Classifying space of variant on category of simplices
This question might have an easy answer, but my research is far from the region of topology that makes use of classifying spaces of categories, so I can't find it.
For (possibly infinite) integers $0 ...
19
votes
0
answers
773
views
Folk Functorial Figuring
In the CRM Proceedings & Lecture Notes Volume 50 "A Celebration of the Mathematical Legacy of Raoul Bott" Herbert Shulman writes (p. 48):
"[Bott] taught many of us to think functorially, like ...
- 2,684
2
votes
0
answers
179
views
About Thom Theorem (representation submanifold for $H_{n-2}(M^n)$)
Recall Thom theorem : If $M^n$ is a smooth orientable closed manifold then any homology class in $H_{n-2}(M)$ is represented by the fundamental class of a smooth submanifold.
And in the Harper and ...
- 1,100
2
votes
0
answers
176
views
Two questions on axiomatic homology
1) Given the Eilenberg-Steenrod axioms, there are several Mayer-Vietoris type sequences that can be deduced. The most general form seems to be
$$\rightarrow H_n(X \cap Y, A \cap B) \rightarrow H_n(X,...
- 245
4
votes
1
answer
497
views
Riemann Existence Theorem for Real Curve
By real curve, we mean a Riemann surface $X$ together with an anti-holomorphic involution
$\sigma : X\rightarrow X$. Let $S$ be a finite subset of $X$. For each point $x\in S$, we associate a positive ...
- 135
2
votes
1
answer
218
views
Shrinkable maps and universal weak equivalences
Recall that a morphism $f:X \to B$ is called shrinkable is there exists a section $s:B \to X$ together with a homotopy $$H:I \times X \to X$$ from $sf$ to $id_X$ over $B,$ i.e. for all $t$, the map $$...
- 15.5k
10
votes
0
answers
235
views
Computation of stable cohomology ring of SL_n(Z) using algebraic topology
It is known that $H^k(SL(n,\mathbb{Z}))$ is independent of $n$ for $n \gg k$, so we can define a stable cohomology ring
$$V = \text{lim}_{n \rightarrow \infty} H^{\ast}(SL(n,\mathbb{Z});\mathbb{R}).$$...
- 101
2
votes
0
answers
124
views
Reasoning about "approximately" associative structures and "almost monoids".
If $(M,+)$ is a monoid then it obeys the laws:
$$m_1 + 0 = 0 + m_1 = m_1$$
$$m_1+(m_2+m_3)=(m_1+m_2)+m_3$$
But what if I have a structure $(A,+)$ that is almost a monoid, but not quite. For example,...
- 21
0
votes
0
answers
292
views
Is $GL_{S^1}(H)$ (for $H$ Hilbert space ) path-connected?
Due to the negative answer to my last question I want to know if at least the following is true:
Let H be an infinite dimensional separable complex Hilbert space with $S^1$-action. Let $\text{Gl}_{S^...
- 1,282
4
votes
1
answer
360
views
An analogue of Lefschetz hyperplane theorem for complements to subvarieties in $\mathbb C^n$ ?
Let $V^{2k}$ be a complex subvariety of dimension $2k$ (real dimension $4k$) in $\mathbb C^n$. Let $A$ be a complex $n-k$ dimensional plane in $\mathbb C^n$.
Question. Is it true that the inclusion $...
- 14.3k
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
314
views
Homology of a complex projective conic
Let $Q$ be a smooth conic (the zero set of a homogeneous degree 2 polynomial)
in $\mathbb{P}^2(\mathbb{C})$ and let $j:Q\rightarrow\mathbb{P}^2(\mathbb{C})$ be the
immersion in the projective plane. ...
- 1,727
6
votes
0
answers
398
views
Differential forms on the simplex which are "constant towards the boundary"
Let $\Delta^k$ the standard $k$-dimensional simplex, $\Delta^k=\{(x^0,\dots,x^k)\in \mathbb{R}^{k+1}|\sum_{i=0}^kx^i=1\}$, and let $\Omega^\bullet(\Delta^k)$ be the de Rham complex of smooth ...
- 6,777
0
votes
1
answer
285
views
A Nomenclature Issue : Imprimitive Semigroup?
The following question was asked by me on the forum sci.math.research,
“An imprimitive group is a transitive permutation group with a non-trivial
equivalence relation compatible with the action of ...
- 113
5
votes
1
answer
383
views
Killing Chern classes
Let $G$ be a compact connected Lie group and let $E\to B$ be a principal $G$-bundle. Suppose $a$ is a rational cohomology class of $E$ such that its pullback $b$ under an orbit inclusion map $G\to E$ ...
- 23.5k
4
votes
1
answer
328
views
Algebraic K-groups and braids
This is (I think) a reference request:
Are there calculations of any algebraic K-groups for the (group ring of) the Artin braid groups?
- 1,180
5
votes
0
answers
263
views
Coloring $\mathbb{Z}^k$ and a fixed point theorem
This is potentially another approach to this question. I put it as an update there, but perhaps it would be better to post it separately. If we color $\mathbb{Z}^k$ with the $\ell_\infty$ metric in ...
user6976
2
votes
1
answer
270
views
homotopy groups for good rings
I think this question should already be abound in literature but the only place I find is from this article:
http://math.uchicago.edu/~lxiao/files/notes/Fundamental%20Groups.pdf
which seems to be ...
- 799
0
votes
1
answer
94
views
What is the way to after finding Cohen-Macaulay semigroup as ring of monomial?
http://people.missouristate.edu/lesreid/reu/2007/PPT/robin.ppt
it said missing part inside and not missing part outside is non Cohen-Macaulay semigroup
which determine whether is missing in the most ...
- 1
2
votes
1
answer
307
views
Name for a kind of topological property?
What should I call a property (P) of (open) subspaces of a space $X$ such that:
If $U$ satisfies (P), then so does every open subset $V\subset U$
If {$U_i$} is a pairwise disjoint collection of ...
- 12.5k
4
votes
0
answers
732
views
Spectral sequence for reduced homology
In the Serre spectral sequence, is it true that we can replace homology by reduced homology? That is:
If $f:X\rightarrow B$ is a Serre fibration,with $F$ the fiber, then if
$\tilde E^2_{pq}=\tilde H_p(...
- 1,499
0
votes
0
answers
127
views
How to compute the Betti numbers of S-D for a surface S and a divisor D?
Let S be a projective non-singular surface and D a Cartier divisor which has a smooth representative. Can the Betti numbers of S-D be represented by the Betti numbers of S and D? In a paper $b_i(S-D)=...
- 1
2
votes
1
answer
215
views
$b_2$ of the blow up of a complex $3$-fold in a curve
Suppose that $V$ is a complex analytic manifold of dimension 3 with mild singularities, say it is an orbifold (i.e. has only quotient singularities). Let $C$ be a complex irreducible curve in $V$. ...
- 14.3k
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 ...
- 6,162
7
votes
0
answers
370
views
A question about a blue fan and a red fan and their common refinement
Is the following conjecture true?
Conjecture: Let $M_1$ be a red map and let $M_2$ be a blue map drawn in general position on $S^n$, and let $M$ be their common refinement. There is a vertex $w$ of $...
- 24.7k
0
votes
1
answer
485
views
What is a right-handed Dehn twist of a cut curve of a Riemann surface?
Let $\Sigma_g$ be a Riemann surface of genus g, and $C$ is a cut curve of $\Sigma_g$, i.e. an oriented simple close curve.
What is a right-handed Dehn twist of $C$ of $\Sigma_g$?
- 471
2
votes
0
answers
338
views
Do non-ordinary Bredon cohomology theories extend?
As shown by Lewis, May, and McClure (MR0598689), the ordinary equivariant Bredon cohomology theory $H^*_G(-; M)$ extends to an $RO(G)$-graded cohomology theory precisely when the coefficient system $M$...
- 865
0
votes
1
answer
305
views
Are braid links proper links?
Are braid links proper links? Or are the concepts involved unrelated?
- 63
2
votes
0
answers
430
views
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 $B$...
- 1,582
3
votes
0
answers
228
views
Extension of homeomorphism of boundaries to a homeomorphism of a cobordism
Suppos we have a cobordism $(M, \partial_{-}M, \partial_{+}M)$, where $M$ is a oriented compact (topological) 3-manifold.
Assume we have orientation preserving homeomorphism $f_{\pm}: \Sigma \to \...
- 93
4
votes
0
answers
314
views
Combining Lefschetz numbers with Euler classes
Given an $n$-manifold $M$ (say), we can talk about its Euler characteristic
$\chi(M)$.
This can be generalized to the Euler number of any $n$-dimensional
bundle ${\mathcal V}$. Or indeed, the Euler ...
- 27.9k
2
votes
1
answer
424
views
Cohomology groups of an intersection
Suppose that $A$ is an affine algebraic variety and $P$, and $Q$ are subvarieties. It is easy
to see that the coordinate rings $C[P]$ and $C[Q]$ of $P$ and $Q$ are modules over $C[A]$ and
$C[P]\...
- 3,474
1
vote
0
answers
75
views
Twisted calibrations and sufficient conditions on homology of sub-manifolds
I think my question is somehow easy to solve, but I'm not very familiar with algebraic topology, so I'm not able to figure out the solution for myself. I'm working on a problem in metric geometry and ...
- 111
4
votes
0
answers
382
views
Exercise concerning locally constant presheaves [closed]
Let $\mathscr{F}$ be a presheaf of abelian groups on some topological space $X$. We say that $\mathscr{F}$ is locally constant if there exists an open cover $\mathcal{U}$ of $X$ (i.e. $X=\bigcup_{U\in\...
- 4,902
4
votes
0
answers
264
views
Infinity-groupoid on the etale site of a scheme.
Let $X$ be a topological space. We may naturally associate to $X$ a simplicial set which is a Kan complex. This in turn gives rise to the fundamental $\infty$-groupoid associated to $X$. From this ...
- 491
-1
votes
1
answer
263
views
Is $X$ homeomorphic to $S^1 \times Y$? [closed]
Is it true that when the first fundamental group of a topological space $X$ is isomorphic to $\mathbb{Z}$ then $X$ is homeomorphic to $S^1 \times Y$ where the first fundamental group of $Y$ is ...
- 1,661
8
votes
0
answers
370
views
Dualizing complex of the product of two locally compact spaces
Hello!
In the setting of locally compact Hausdorff spaces, I would like to understand the relation between the exterior product ${\mathbb D}_X\boxtimes{\mathbb D}_Y$ of the dualizing complexes of two ...
- 2,756
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 \ ...
- 373
2
votes
0
answers
251
views
Knots that turn around an axis [closed]
Take a thick cord (the alim cord of your laptop or your mouse cord for example) and wrap it around your hand (or finger) turning always in the same direction but possibly knotting it. Then try to ...
- 3,033
9
votes
0
answers
700
views
Is a functor which is a sheaf for open covers and finite closed covers automatically a sheaf for covers by simplices?
Suppose $F:Top^{op}\rightarrow Set$ is a functor which is a sheaf for open coverings as well as for finite closed coverings, i.e. such that, apart from the usual sheaf property we also have that for ...
- 12.3k
0
votes
0
answers
72
views
Decomposition results for locally commutative semigroups
Every finite abelian group is the direct product of its cyclic groups of prime order, and every commutative monoid divides a product of its cyclic submonoids. Could these results generalized to ...
- 798
10
votes
0
answers
325
views
H-space structure on the Calkin algebra
By the Atiyah-Jänich theorem the K-group $K^0(X)$ for a compact space $X$ may be represented as $[X, U(Q)]$, where $Q = B(H)/K(H)$ is the Calkin algebra and $H$ is a separable infinite dimensional ...
- 7,637
7
votes
2
answers
268
views
What are the fibres of a representable simplicial sheaf (in the Nisnevich topology)
Let $k$ be a field and $X$ a smooth $k$-scheme. We consider now the pointed (constant) simplical Nisnevich-sheaf $X_{+}$ that is represented by $X$. Let now $\nu\in U$ be a point of a smooth scheme, ...
- 71
0
votes
0
answers
236
views
Topological K-theory of Bohr compactification of real numbers
I am interested in the
K-theory of the Bohr compactification $\mathbb{R}_B$ of the real numbers.
Do we have $K_0(C(\mathbb{R}_B))$ isomorphic to $K_1(C(\mathbb{R}_B))$ ?
More generally, what do we ...
- 357
10
votes
0
answers
484
views
Applications of sheaf theory to the computation of invariants of LS-category type
I would like to know if sheaf theory can be applied to a particular class of questions in topology.
The Schwarz genus (also known as sectional category) of a continuous map $p\colon\thinspace E\to ...
- 35.9k
1
vote
1
answer
245
views
Simplified Jones trace invariant for links
Jones (1985) defines a simplified trace invariant for knots by $W_K(t)=\frac{1-V_K(t)}{(1-t^3)(1-t)}$. Then, e.g., the Arf invariant for $K$ is $Arf(K)=W_K(i)$. Does this work for oriented links as ...
- 63
2
votes
2
answers
463
views
homotopy type of complement of subspace arrangement
I am studying the homotopy type of a space,and i hope it would be a $K(\pi,1)$ space.
now i have find its covering,once we can say the covering is $K(\pi,1)$,so is the space
itself.and the covering is ...
- 157
10
votes
0
answers
227
views
Spaces with free MU-homology
Let $E$ be a homology theory whose coefficients $E_*(pt)$ are concentrated in even dimensions. This could be complex bordism $MU$, but also complex $K$-theory, $BP\langle n \rangle$, $E(n)$, ...
Let ...
- 12.1k
5
votes
0
answers
440
views
Reference for homotopy orbits of pointed spaces
Can someone point me to a good (hopefully simple and brief) place to read about the basics
of homotopy orbits for pointed spaces?
More detail:
As I understand it, in the unpointed case,
we use the ...
- 12.5k
2
votes
1
answer
215
views
Second cohomology group with finite coefficients of the product of two varieties
This is surely well known. Let $X$ and $Y$ be smooth, projective, connected complex varieties. Then
$$H^2(X\times Y,Z/n)=H^2(X,Z/n)\oplus H^2(Y,Z/n) \oplus (H^1(X,Z/n)\otimes H^1(Y,Z/n))$$
for any $n&...
