All Questions
8,725 questions
5
votes
1
answer
580
views
Free actions of finite groups on products of even-dimensional spheres
Suppose a finite 2-group G acts freely on X = $\prod_{i=1}^k$ *S*$^{2n_i}$, a product of k even-dimensional spheres, k > 2. Is it possible for G to be non-abelian? What if we additionally assume that ...
3
votes
1
answer
233
views
Aspherical amalgamations without injective maps
The situation I find myself in is as follows: I have a CW complex $X$ which is covered by two subcomplexes $A$ and $B$ and I know that $A$, $B$ and $A \cap B$ are connected and aspherical. The term ...
- 1,680
4
votes
0
answers
505
views
Cohomology of a hypersurface in a projective bundle
Let $Y$ be a smooth hypersurface in the total space of a projective bundle $\pi:\mathbb{P}(\mathscr{E})\to B$ (here $\mathbb{P}(\mathscr{E})$ denotes the projective bundle of lines in the vector ...
- 673
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 $p_2\in f^{-1}(q_2)$ ...
- 937
12
votes
0
answers
472
views
What is the history of the notion of subdivision of categories?
A recent answer by Peter May prompts me to ask a question which I have been considering to ask for several months. (The reason why I have not asked it before is that it is not directly related to my ...
- 2,413
7
votes
1
answer
458
views
Reference for equivalent definitions of the genus
Let $X$ be a (edit: nonsingular) projective complex algebraic curve. The genus of $X$ can be defined as the dimension of the space of holomorphic $1$-forms on $X$, which in turn can be defined either ...
2
votes
0
answers
459
views
Cohomology theory associated to the spectrum BG
Hi, I've recently been interested in Stable Homotopy Theory and was reading this text to understand some basics: http://www.maths.ed.ac.uk/~aar/papers/carlmilg.pdf
Near the end of the text (p582) we ...
- 757
2
votes
1
answer
445
views
Homology and submanifolds...
I'm reading a paper which includes the following line, and I can't find a reference anywhere to the result the authors mention:
"Let M be a compact orientable embedded minimal hypersurface of a ...
- 1,123
7
votes
1
answer
597
views
Pontryagin product from an operad
For a topological group G, we have a Pontryagin product in homology by multiplying representative cycles. This gives the homology the structure of an associative graded algebra. Am I correct in ...
- 71
3
votes
1
answer
858
views
Any reason why K_23(Z) has order 65520?
I'm rereading my notes and they mention that $K_{23}(\mathbb Z) = \mathbb Z/(65520)$
This looks like a good point to stop and ask whether there is any explanation for this $K$-group of integers (23 ...
- 15.1k
2
votes
1
answer
271
views
Non-commutativity of certain Hopf spaces
How do one prove (or disprove) that $\Omega S^{2}$ and $\Omega(S^{2} \vee S^{2})$ are non commutative Hopf spaces?
I thought this is a question for math.stackexchange, but not many people even viewed ...
- 2,023
1
vote
1
answer
151
views
A formula for isotropy group $\pi_1(G_a)$
Let $G$ be a compact Lie group and $T$ be its maximal tours, and $a\in \mathfrak{g}^*$. and $G_a$ be the isotropy group of $G$ then $T\subset G_a$ and we know that $\pi_1(T)=\mathbb{Z}^n$. My ...
user21574
0
votes
1
answer
175
views
Subvarieties with different topology representing the same cycle
Let $X$ be a topological space, and $Y,Z$ be subspaces. My question is a bit vague and open-ended: when is it the case that, if $Y$ and $Z$ represent the same (nonzero) cycle in homology, then $Y$ and ...
1
vote
0
answers
297
views
Simple proof of an isomorphism theorem
I haven't references about a proof of this theorem:
Let $p: Y \rightarrow X$ be a fiber bundle. If for all $x \in X$ $p^{-1}(x)$ satisfies $H^{\ast}(p^{-1}(x)) \simeq \mathbb{R} $, then $p$ induces an ...
- 21
3
votes
0
answers
117
views
Why "non-linear similarity" is the same as equivalence of representations for connected Lie groups?
Let $G$ be a compact Lie group and $V$ a finite-dimensional orthogonal $G$-representation. Write $S^V$ for the quotient $D(V)/S(V)$, where $D(V)$ and $S(V)$ are the unit disk and sphere in $V$, ...
2
votes
0
answers
154
views
Action of Landweber-Novikov algebra on infinite polynomial ring
Denote $\text{Aut}(\hat{{\mathbb A}}^1)$ be the affine group over ${\mathbb Z}$ that sends some ring $R$ to the strict automorphisms of $R[[t]]$, i.e. those of the form $X\mapsto X + r_1 X^2 + r_2 X^3 ...
- 2,756
0
votes
1
answer
1k
views
About universal coefficient theorem
Let $(X,A)$ be a finite CW-pair $m=p^r$ for some prime $p$. Unspecified coefficient is in $\mathbb{Z}$.
From the universal coefficient theorem, We know that
$H^1(A;\mathbb{Z}_m)=\textrm{Hom} (H_1(A),...
- 698
6
votes
1
answer
779
views
primitive of an exact differential form with special properties
We were working on a smoothing problem and ran across the apparently simple following question:
X is a triangulated smooth manifold of dimension $n$, and $\alpha$ is an exact differential form of ...
5
votes
1
answer
550
views
Semicosimplicial totalization
Can somebody help me with a reference showing that the homotopy semicosimplicial totalization of a cosimplicial space is homotopy equivalent to its usual homotopy totalization? Is it because the ...
- 1,875
4
votes
1
answer
564
views
Why is the induced map between pullbacks (of inclusions) by a right fibration a deformation retract?
Let $X$ be a simplicial set. Let $X\to \Delta^n$ be a right fibration (has the right lifting property with respect to right horn inclusions), and let $$\Delta^{\{n-i\}}\hookrightarrow \Delta^{\{n-i,\...
- 19.6k
1
vote
1
answer
159
views
homology of $B S^{-1} S$ computation in the proof that $+ = Q$
Let $S$ denote the category of projective (left) $R$-modules with isomorphisms for arrows. We have that
$BS^{-1}S \sim B \text{GL}(R)^+ \times K_0(R)$
In proving this, in Srinivas' algebraic K-...
- 1,105
7
votes
1
answer
565
views
Reference for base change of cohomology pull-push for clean intersections.
Let $X$ be a compact oriented manifold, and $A$ and $B$ closed oriented submanifolds intersecting cleanly. Then I've always been under the impression that pushing forward a cohomology class from $A$ ...
- 44.7k
2
votes
0
answers
313
views
Transforming the Dirac Operator on $S^1$
This is related to my question https://math.stackexchange.com/questions/252742/transforming-the-dirac-operator-on-s1 on stack exchange which has not yet received an answer. For the purposes of this ...
- 1,010
3
votes
1
answer
458
views
Slick verification of the model category axioms for Spaces and SSets with the q-model structure?
We choose our category of spaces to be compactly generated weak Hausdorff spaces for convenience, denoted $CGWH$.
Questions:
1.) Is there any sort of slick argument to verify that CGWH with the ...
- 19.6k
4
votes
1
answer
527
views
Existence of homotopy inverses for co-H spaces
Suppose $c: X \to X\vee X$ is a co-$H$ structure on a based CW complex $X$.
Question: Under what circumstances can one find left (right) homotopy inverses for $c$?
Remarks: If $X$ is $1$-...
- 18.9k
5
votes
0
answers
717
views
Can Euler Class be defined by the Splitting Principle for Real Vector Bundles?
Let $M$ be a manifold and $S$ its sphere bundle with fiber $\mathbb{S}^n$. As we know, the notion of the Euler class is raised from the problem of finding a global form on $S$ which restricts on each ...
- 643
1
vote
1
answer
870
views
Simplicial set notation and vocabulary question.
Notation question:
What does $(\Delta^1)^{ \{1, \ldots,n-1 \}}$ denote? UPDATE: I (David Speyer) tried to fix the LaTeX. Please see if I got it right.
Vocabulary question:
Suppose $z:\Delta^{n+1} \...
3
votes
1
answer
149
views
Interesting examples of a 4-torsion X in a triangulated category such that $2 End(X/2X)\neq 0$?
It is well-known that for the sphere spectrum $S$ in the ('topological') stable homotopy category the object $S/2S$ i.e. the cone of $S\stackrel{\times 2}{\to}S$, is not $2$-torsion.
So I wonder ...
- 16.9k
0
votes
1
answer
156
views
Calculation of L2-dimension
For a group $G$, can we calculate $dim^{(2)}_{\mathcal{N}G}(\ell^2 G)$, where $\mathcal{N}G$ is the von Neumann algebra of $G$ and $\ell^2 G$ is the Hilbert space on $G$? I want to see whether this is ...
- 537
1
vote
0
answers
355
views
Induced model structure?
Let $H$ be a set with a binary operation $\cdot _H$ on it. To show that it is a group, one has to show that $\cdot _H$ is associative, find an identity element in $H$, and so forth; it might take ...
- 583
1
vote
0
answers
366
views
Question on Steenrod realizability problem
René Thom proved that for any topological space $X$, any integral homology class of $X$ (of any degree) has an odd multiple that is the pushforward of the fundamental class of a smooth compact ...
- 257
8
votes
2
answers
658
views
Cofinal inclusions of Waldhausen categories
Let $\mathcal{C}$ be a Waldhausen category. Suppose that $\mathcal{B}$ is a subcategory of $\mathcal{C}$, and that $\mathcal{B}$ is closed under extensions. If $\mathcal{B}$ is strictly cofinal in $\...
- 1,025
4
votes
0
answers
170
views
Homotopy unites of a differential graded algebra
I apologize in advance if the question is too basic.
Let $A$ be a differential graded algebra and $H^{0}A$ the 0-cohomology of $A$ (which is an ordinary ring) and $A^0$ is the 0-level of $ A$ . An ...
- 1,974
1
vote
0
answers
256
views
Cohomology with compact support and the nerve of a recouvrement
Let $X$ be a simplicial complexe and we assume it localy finite and finite dimensional. We suppose taht there exist a simplicial complexe $Y$ and a map assigning to each vertex $s\in Y$ a finite ...
- 933
11
votes
0
answers
203
views
Fundamental groups of reduced subgroup lattices
Let $G$ be a group. Its subgroup lattice, denoted $\Sigma G$, consists of all subgroups of $G$ partially ordered by inclusion. The topology of this poset is quite trivial, since it always has a ...
- 15.5k
5
votes
1
answer
232
views
Finite index subgroups of the mapping class group with geometric meaning
I have got a question that is perhaps not precise in a mathematical sense.
Is there a classification of all coverings of the moduli space of Riemann surfaces which are moduli spaces themselves, that ...
- 471
5
votes
0
answers
207
views
Refined cofinality theorem for homotopy limits of spaces
If $C$ is a simplicial model category, $F\colon I\longrightarrow J$ a functor between small categories, and $X\colon J\longrightarrow C$ a diagram, the canonical map $holim_JX\longrightarrow holim_IX\...
- 425
8
votes
0
answers
211
views
Fibrations of orthogonal G-spectra and fixed points
There are at least two fixed point functors that characterize stable equivalences of orthogonal G-spectra: the geometric fixed points and the naive fixed points of a fibrant replacement.
Is this true ...
- 425
4
votes
1
answer
2k
views
Fiber bundle = principal bundle + fiber?
This question is heavily related to this question.
Fix a sufficiently nice and connected topological space $B$ and let $FB$ be the category of fiber bundles over $B$. A morphism $f: (E\to B)\to (E'\...
- 1,085
14
votes
1
answer
961
views
Founding of homological without quite involving derived categories
I am looking at the foundations of homological algebra, e.g. the introduction
of Ext and Tor, and am unsatisfied. The references I look at start with
"this is called a projective module, this is ...
- 27.9k
1
vote
1
answer
484
views
Serre fibrations
I am trying to figure out some commutative diagrams and am having difficulty with this one. We have two fibrations of $B$, $E'$ and $E$ such that $E' \subset E$ and $E' \to E$ is a cofibration. $B$, $...
- 257
6
votes
0
answers
410
views
Fundamental group of non-Hausdorff surfaces & actions of discrete Heisenberg group
Let $G$ be a discrete group, acting on a space $X$ (by homeomorphisms). I will say that the action is properly discontinuous if for any $x, y \in X$, there are neighborhoods $U_x$ and $U_y$ such that ...
- 616
13
votes
0
answers
783
views
What's so difficult about $\pi_{15}(SO)$?
Regarding the table of $SO(n)$s-of-origin in Davis+Mahowald (if you can get MathSciNet), is there a good reason that it should take longer for $\pi_{15}(SO)$ to be representable than $\pi_{19}(SO)$, ...
- 1,987
3
votes
0
answers
151
views
Equivariant Poincare Series of Based Loop Group of SU(2)
Let $\Omega SU(2)$ denote the based loop group of $SU(2)$, and consider the action of $S^1$ on $\Omega SU(2)$ as a maximal torus of $SU(2)$. (This is not the "loop rotation" action.) Is there an ...
- 4,920
3
votes
0
answers
173
views
More questions about high-dimensional knot invariants
In a question yesterday I asked about the existence of algebraic invariants for embeddings of n-manifolds into n+2-spheres. The answers in the positive dimension all made certain assumptions about ...
- 1,025
1
vote
1
answer
595
views
When is a bijective map between bundles a homeomorphism?
Let $F \rightarrow E_i \rightarrow X_i$ be a bundle with fibre $F$ for i=1,2.
Let $f:E_1 \rightarrow E_2$ be a bijective continuous map and $h: X_1 \rightarrow X_2$ a homeomorphism.
Is f then also ...
- 165
3
votes
0
answers
129
views
Robust Invariants of a solution of systems of equations
Let $X$ be a $k$-dimensional triangulated manifold and $A:=\partial X$ be its boundary. Let $f:X\to R^d$ be a continuous function such that $g:=f|_{A\cup X^{(i-1)}}$ avoids zero and let $z_g\in Z^{i} \...
- 972
3
votes
2
answers
250
views
A model structure on the category of "dualizing maps"
Let $C$ be the category with objects being maps $h:M\to A$ where $A$ is a commutative graded $\mathbb{Q}$-algebra (cdga), $M$ is a differential graded (dg) $A$-module and $h$ is an $A$-dg-module map, ...
- 23.5k
1
vote
0
answers
187
views
Gysin sequence for $\mathbb S^3$ bundle
Let the group $G=\mathbb S^3$ act semi freely on a paracompact space $X$. Then exercise 12 of G.E. Breadon's book , Introduction to compact transformation groups pg 169 asks to derive the following ...
- 119
4
votes
1
answer
243
views
Topological Localization of (the simply-connected cover of) SO or Spin
This is essentially a (cw) reference request, because it seems like the sort of thing that should have been looked at already.
Setting aside, for now, how to think what the localization of a general ...