All Questions
8,725 questions
3
votes
1
answer
292
views
Cartesian-closed category of spaces with the Whitehead property?
I'm not sure if this is standard, but we'll call the property that every weak homotopy equivalence is an honest homotopy equivalence the Whitehead property (from Whitehead's theorem for CW complexes). ...
- 19.6k
2
votes
0
answers
512
views
Lifting criteria of covering space by using homology condition
Let $\pi\colon\tilde{X}\to X$ be a p-fold (regular) cyclic covering(p:prime) and $\mathcal{A} = \mathrm{Im}(\pi_* )$, where $\pi_* \colon H_1(\tilde{X};\mathbb{Z}_p) \to H_1(X;\mathbb{Z}_p)$ is ...
- 698
7
votes
1
answer
282
views
Can you construct a mapping space from local data? (looking for reference)
I'd to know if/where there is a reference for the following construction.
Let C_*(maps(M, T)) denote the singular chains on the space of continuous maps from an n-...
- 12.8k
1
vote
1
answer
156
views
Sufficient Conditions for Free Indecomposability
An interesting fact was relayed to me in another question of mine that
If $M$ is any closed manifold with universal cover homeomorphic to $R^n$ for $n>1$ then $\pi_1(M)$ is freely ...
- 726
10
votes
0
answers
362
views
Question about $A_\infty$ maps
Given $A_\infty$-spaces $X$ and $Y$, Boardman and Vogt defined an $A_\infty$-map from $X$ to $Y$ to be a map $f: X \to Y$ of underlying based spaces and an $A_\infty$-structure on the reduced mapping ...
- 18.9k
1
vote
0
answers
153
views
Pushout of the skeleton of homotopy colimit of diagrams
First of all let me define what homotopy colimit i'm talking about. Let F be a functor from a small category to the category of simplicial set, the homotopy colimit is the simplicial realization of ...
- 111
2
votes
1
answer
378
views
How can I show that the map L-->K(\pi_n(L),n) representing the fundamental class of an (n-1)-connected space is an isomorphism on \pi_n?
As an exercise, I'm trying to show that for an $(n-1)$-connected space $L$ with $\pi=\pi_n(L)$, the map $\iota_L:L\rightarrow K(\pi,n)$ associated to the fundamental class $\iota_L\in H^n(L;\pi)$ ...
- 6,069
1
vote
1
answer
304
views
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
2
votes
0
answers
486
views
Casson Gordon paper - Cobordism of classical knots
It is given in Progress in mathematics 62, Guillou and Marin book. In the proof of Lemma 4, They choose $\alpha$ and $r\in \mathbb{N}$ such that $h^r_*\colon H_1(X;Z_p)\to H_1(X;Z_p)$ satisfies $h^r_*(...
19
votes
0
answers
504
views
Other examples of computations using transfer of structure from the chains to the homology?
There is a `long' history of transfer (up to homotopy!) of algebraic structure from a dg _ algebra A to its homology H(A) (e.g. Kadeishvili for the associative case and Heubschmann for the Lie case). ...
- 3,880
0
votes
0
answers
68
views
can we say fixed point existance of a set valued map over a compact set is homotopy invariant?
Consider two set valued maps over different compact sets as $F(\mathbf{x}):D\rightarrow\rightarrow D$, $G(\mathbf{x}):E\rightarrow\rightarrow E$ where $D,R\subset Y$. Assume there is a homotopy pair $(...
- 383
1
vote
0
answers
267
views
subset embedding gives trefoil knot [closed]
Let $X$ be a topological space and $E_n(X)$ the space of finite sets of cardinality $\leq n$.
It is a theorem of Bott that $E_3(S^1)=S^3$. What is the idea to show that
the embedding $S^1\...
- 11
2
votes
0
answers
152
views
Does the ordinary cokernel respect weak homotopy equivalences?
Let $M_1\subset M_2$ and $K_1\subset K_2$ be inclusions of simplicial monoids. If there is a weak equivalence $f:M_2\to K_2$ that restricts to a weak equivalence $f|_{M_1}:M_1\to K_1$, does this ...
- 21
2
votes
1
answer
350
views
Commutativity of a diagram of boundary morphisms from the long exact sequence of homotopy groups of a fibration and its loop spaces
Let $f: X \to Y$ be a fibration of pointed Kan complexes, and let $F$ be the fiber.
Question: How do you prove that the following diagram of homotopy groups commutes?:
$\pi_n(Y) \to \pi_{n-1}(\...
- 155
3
votes
0
answers
446
views
When does the normal bundle of a submanifold of Euclidean space admit a flat connection?
Given a smooth submanifold of $R^n$, I was wondering if there is a reasonably simple criterion for deciding whether its normal bundle admits a flat connection. I am not ruling out monodromy in the ...
- 313
1
vote
0
answers
371
views
differential form of charge for pi_4(S^3) or pi_4(S^2)
How to write a 4-form of topological charge which would correspond to non-zero element of
the homotopy group $\pi_4(S^3)$ or $\pi_4(S^2)$ (both are equal to $Z_2$) ? ?
An example of such a mapping (...
- 11
3
votes
1
answer
265
views
Equivariant Surgery problem
I have a question about surgery.
Let $G= \mathbb{Z}_m \times \mathbb{Z}$ and $M$ be a oriented 3-manifold with G-action. i.e. There exists a map $f\colon M/G \to BG$, where $BG$ is classifying space.(...
- 698
0
votes
0
answers
198
views
Euler characteristic of a subset of cartesian product induced by a group action
let $X$ be a CW-complex on which a finite group $G$ acts.
define
$$F=\{ (x,gx)\;|\; x\in X ,g \in G \}$$
i want to compute the Euler characteristic of $F$. I wrote $$F=\cup_{g\in G}{F_g}\;\;,\;\; ...
- 1
5
votes
1
answer
283
views
how good an approximation to the equivariant derived category is given by the Grassmannian filtration of the classifying space?
So, let's say one has an action of $GL_n$ on an algebraic variety $X$ over a field $k$, and two objects $F,G$ in the equivariant derived category (i.e., the derived category of constructible sheaves ...
- 44.7k
6
votes
0
answers
360
views
The Space of Cellular Maps
Let $X$ and $Y$ be CW complexes.
Inside of the space of maps $\mathrm{map}(X,Y)$, we have the subspace $\mathrm{CW}(X,Y)$, consisting of just the cellular maps from $X$ to $Y$. The Cellular ...
- 12.5k
0
votes
1
answer
314
views
Homology of symmetric groups
Let $S_n$ denote the symmetric group on $n$ letters, and let $S_n(p)$ denote a Sylow $p$-subgroup. Why is the image of $H_i(S_n(p))$ in $H_i(S_n)$ the $p$-primary part of $H_i(S_n)$?
- 803
12
votes
0
answers
440
views
K-Weil cohomology theories?
I don't know very much about this stuff, so I'm a bit afraid that I'm being naive or stupid, and I apologize if I am --- but it seems to me that Weil cohomology theories, or at least the standard ...
- 21k
6
votes
0
answers
312
views
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 ...
- 5,575
4
votes
0
answers
140
views
Cartesian and strictly associative Path Object.
Let $X$ be a topological space. Is there a path object functor $\mathrm{P}:\mathbf{Top}\rightarrow \mathbf{Top}$, with fibration $ev_{0},ev_{1}: \mathrm{P}(X)\rightarrow X\times X$ such that:
(1) $\...
- 1,974
2
votes
1
answer
190
views
bundle over a chain
Consider the universal bundle $G\hookrightarrow EG\rightarrow BG$. Is it possible to get another bundle $EG|_{B}$ by restricting $EG$ over a smooth singular $k$-chain $B
\in H_k(BG)$ ($k\leq n$)? I ...
- 1,709
1
vote
0
answers
101
views
How can one compute the cohomology of $i'^*C$, for $i':\mathbb{A}^{N-1}\setminus \{0\}\to \mathbb{A}^{N}\setminus \{0\} $?
For an (etale or 'topological', constructible bounded) complex of sheaves $C$ on $X'=\mathbb{A}^{N}\setminus \{0\} $, $i'$ being the embedding $\mathbb{A}^{N-1}\setminus \{0\}\to \mathbb{A}^{N}\...
- 16.9k
8
votes
0
answers
438
views
Differential K-theory computation
I am trying to read about K-theory and differential K-theory. I understand that the K-theory of spheres can be computed explicitly (by getting to the stable range and using Bott periodicity and so on)....
- 3,383
6
votes
0
answers
532
views
Can one calculate Ext's between microlocalized perverse sheaves/D-modules using topology?
So, I know one really good technique for calculating Ext's between perverse sheaves/D-modules using topology: the convolution algebra formalism, worked out in great detail in the book of Chriss and ...
- 44.7k
3
votes
0
answers
251
views
Is an element of the cohomotopy space trivial?
Let as have a continuous map from a topological space X to an n-sphere, realized as a simplicial map between simplicial complexes (for example), i.e. represented by a finite set of data. Is there an ...
- 972
0
votes
2
answers
356
views
Can all induced maps be described categorically.?. (or at least as generally as possible)
Hi: I am new here. I went over the fAQ's, still, sorry if I break protocol.
I am pretty confused about induced maps in different areas of algebraic
topology; I do know how these induced maps are ...
- 1
0
votes
0
answers
185
views
the boundary homomorphism $[\Sigma S^{n-1},X]\to[S^n,X]$ is identity?
Given a Puppe sequence $\cdots \to S^{n-1} \to Y \to S^n(\simeq Y/S^{n-1}) \to \Sigma S^{n-1} \to \cdots$, where $Y=S^{n-1}\cup_{2\iota} e^n$ where $\iota:S^{n-1}\to S^{n-1}$ is identity,
we have a ...
- 699
3
votes
0
answers
122
views
Torsion of Bass, Farrell and Waldhausen nil groups
Let H be an infinite virtually cyclic group. If H is orientable (resp. non-orientable) the Farrell nil groups $N_n(\mathbb{Z}H,\alpha) $ (resp. the Waldhausen nil groups $N^W_n(\mathbb{Z}H;\mathbb{Z}[...
- 235
4
votes
0
answers
208
views
Mapping into a geometric realization.
Suppose $S$ is a simplicial set, $X$ is a space, and we are given a map
\[
f: \text{Sing}\,X\to S.
\]
When is is possible to produce a map $X\to |S|$?
We can take ...
- 569
2
votes
0
answers
526
views
How much of math could be taught without using mathematical notation? [closed]
Given that mathematics is not about number, and that it is not even about the cryptic notation used to describe mathematical problems, how much of mathematics could be taught without reference to ...
2
votes
0
answers
409
views
virtual bundle with compact support
A virtual bundle with compact support is a triple $E$={$E_1$,$E_2$,$a$},
where $E_1$ and $E_2$ are bundles over $M$ of the same dimension, $M$ is a closed manifold, $a$ is a bundle map from $E_1$ to $...
- 381
3
votes
1
answer
187
views
Examples of H-cogroups and a question about julia sets
The following questions should be very easy, but I need help. Notations are same as Bredon's geometry and topology book. This question is related to chapter VII.3.page 441.
$\nabla \colon X\vee X\...
- 244
3
votes
0
answers
236
views
Mapping from $X$ to $S^4$
I asked this question previously on stackexchange (https://math.stackexchange.com/questions/18668/mapping-from-x-to-s4) but could not get a solution. Any help is appreciated.
The question is (...
- 31
4
votes
1
answer
149
views
singular cohomological dimension
Let $X$ be a finite CW complex. Is there an integer $N,$ such that $H^i(X,F)=0$ for all $i>N$ and all abelian sheaves $F$ on $X?$ The cohomology is defined to be the derived functor of the global ...
- 4,265
1
vote
0
answers
45
views
Equivalence between the definitions of Reidemeister Coincidence number
Given two applications $f$ and $g$, denote by $R (f, g)$ the set of Reidemeister classes determined by $f$ and $g$ (according to the algebraic definition, on the induced on fundamental groups). And $\...
- 29
5
votes
1
answer
190
views
Adapting families of diffeomorphisms to an open cover
Has anyone seen the following result in the literature? I've asked a few experts but so far I've come up with nothing.
Given a manifold M and an open cover {U_i} ...
- 12.8k
0
votes
0
answers
109
views
finiteness of the dimensions of cohomologies of open subsets of a compact manifold
Let $M$ be a compact differentiable manifold which can be covered by two open subsets $U$ and $V$. Then $H_{\text{dR}}^n(M)$ is finite-dimensional for all $n$. But how about $U$, $V$ and $U\cap V$? ...
3
votes
0
answers
124
views
$B\Gamma$ constructed
In his pioneering paper on foliations, Haefliger obtains $B\Gamma_q$
as the representing space of a functor, NOT by construction analogous to the bar construction.
Who made that connection first? Bott?...
- 3,880
0
votes
1
answer
251
views
altering curvature on a tessellation representation of a compact surface
I have been reading about tessellation representations of compact surfaces, such as how the square tiling the plane represents the torus. For surfaces of genus > 1 (the ones that interest me), we ...
1
vote
0
answers
193
views
How to get countably many generators for $K_{j}^{G}(\beta G)$ ??
Hey
I am trying to find out how the Baum-Connes conjecture works over $GL(1)$ over local fiels.
I am just wondering if anybody knows how to get a countable many generators for in the L.H.S of the ...
- 85
8
votes
0
answers
340
views
Do p-compact groups have a sufficiently good notion of "flag variety" and "intersection cohomology"?
This is mostly an idle question, since I don't think I'd be able to do anything with a positive answer. But a positive answer would still be interesting, I think.
Background
An outstanding problem ...
- 118k
0
votes
0
answers
60
views
Relative homology of interlevel set
Let us consider a function $f:\mathbb{R}^3→\mathbb{R}$,
$f(x,y,z)=x^3+y^3+z^3-5yz$. Can anybody drop a hint how
to compute relative homology of interlevel sets with coefficients in $\mathbb{R}:
H_{\...
- 181
2
votes
0
answers
250
views
Examples of weakly nullhomologous cycles on smooth manifolds
Suppose X is a smooth manifold with homology groups H_p(X). For example let X be an open subset of Euclidean space.
What would be natural examples of cycles that are "weakly homologous to zero" but ...
0
votes
1
answer
219
views
Cofibrations of differential graded commutative algebras
Let $X$ a smooth manifold. Is the pullback morphism
$\Omega^\bullet(X)\to\Omega^\bullet(X\times \mathbb{R}^n)$ an acyclic cofibration of differential graded commutative algebras? I guess so, and even ...
- 6,777
7
votes
0
answers
297
views
Inner product on Hochschild homology in 2d TCFTs
This should be an easy question for some people. Take a compact $A(\infty)$ algebra with a cyclically symmetric non-degenerate inner product. In Kontsevich and Soibelman's article "Notes on $A(\infty)$...
- 3,758
2
votes
0
answers
129
views
spaces of projections
Let $\mathbb{K}$ be the compact operators on a separable infinite dimensional Hilbert space. Denote by $\mathcal{P}(\mathbb{K})$ the space of projections in $\mathbb{K}$. If I am not terribly wrong ...
- 7,637