# Tagged Questions

Homotopy, stable homotopy, homology and cohomology, homotopical algebra.

**3**

votes

**0**answers

10 views

### E-infinity structure on singular cochains

Is there a transparent explanation of why the singular cochain complex of a topological space X is an $E_\infty$ algebra. There are combinatorial proofs using, say, the surjection operad, but is there ...

**4**

votes

**1**answer

269 views

### $\Omega X$-action on spectral $X$-bundles

I can generalize the notion of a space over $X$ (where $X$ is a based and connected space) to the notion of a spectrum over $X$ by considering functors of quasicategories $X\to Mod_\mathbb{S}$. In the ...

**-1**

votes

**0**answers

21 views

### What is the formalism for a function that returns the adjacent vertex positions of a given adjacency matrix?

How do I represent the math formalism for a function that returns the adjacent vertex positions of a given adjacency matrix?
Example:
V = undirected adjacency matrix
$$ V =
\left[
\begin{matrix}
...

**4**

votes

**0**answers

55 views

### Moore spectra are not E-infinity (oldest known proof)

Fix a prime $p$. Let $M_p(i)$, the $i$-th Moore spectrum at the prime $p$, be the cofiber of the map
$$ S^0 \overset{p^i}\longrightarrow S^0 $$
where $S^0$ be the sphere spectrum. In the Mathoverflow ...

**6**

votes

**1**answer

248 views

### Cellular model structures on continuous functors

The category of enriched functors from finite based CW complexes to based topological spaces
has a projective model structure. The fibrations
are the objectwise Serre fibrations and the weak ...

**0**

votes

**1**answer

59 views

### free action on product of two spaces [on hold]

Let $G$ be a compact Lie group acting freely on $X\times Y$ , product of two Hausdorff spaces. Is is true that $G$ must act freely on one of the factor spaces ($X$ or $Y$). For example the group ...

**5**

votes

**1**answer

143 views

### Left Properness of Simplicial Commutative Algebras

A bit of light googling turns up several sources asserting that the model structure on simplicial commutative algebras over a ring is left proper (for example, 2.9 in Charles Rezk's paper Every ...

**0**

votes

**0**answers

108 views

### Morse theory in zero dimensions? [on hold]

Are there any known results for Morse theory of a compact 0-dimenionsal manifold (i.e. set of points)? In particular, can one define the analogue of a gradient flow for a finite set of points and ...

**1**

vote

**1**answer

101 views

### Special Case of the Toral Rank/Halperin-Carlsson Conjecture

The Toral Rank conjecture in its original form runs something along the lines of: suppose we have an almost free action of the $n$-torus $T^n$ on a nice topological space $X$. (Say a closed CW ...

**5**

votes

**2**answers

295 views

### On combinatorial and cellular model categories and infinity categories

I am looking for a counterexample. Let me first give the set-up. When you work with model categories, it is extremely common to assume they are cofibrantly generated. For me, this means the definition ...

**5**

votes

**0**answers

126 views

+50

### How to get a polygon from a translation surface $(X,\omega)$

Let $S_g$ be a compact topological surface of genus $g$. I know there is the correspondence
$\{$Abelian differentials on compact Riemann surfaces of genus g$\}\leftrightarrow\{$ Translation surfaces ...

**7**

votes

**1**answer

321 views

+200

### Deformations of Ext rings

Let $k$ be a base ring and $k[x]$ the ring of polynomials in an indeterminate $x$ over $k$. Consider a (not necessarily commutative) algebra $A$ over $k[x]$ and two $A$-modules $M$ and $N$. Then for ...

**12**

votes

**0**answers

435 views

### “To operate the machine, it is not necessary to raise the bonnet.”

The quotation in the title is attributed to Frank Adams and appears in several places:
In the preface of [2002, Operads in algebra, topology and physics]: "to operate the machine, it is not ...

**0**

votes

**0**answers

175 views

### Whitehead group [closed]

Whitehead group (WG) is known for some groups (e.g. free abelian group, cyclic group,Braid group etc).
For example-1)The Whitehead group of the trivial group is trivial.
2)The Whitehead group of a ...

**16**

votes

**3**answers

2k views

### What do cohomology operations have to do with the non-existence of commutative cochains over $\mathbb{Z}$?

Let $X$ be a topological space. In elementary algebraic topology, the cup product $\phi \cup \psi$ of cochains $\phi \in H^p(X), \psi \in H^q(X)$ is defined on a chain $\sigma \in C_{p+q}(X)$ by ...

**2**

votes

**2**answers

419 views

### What is the $\mathbb Z/2$-cohomology of $\mathrm B^n(\mathbb Z/2)$?

I would like to know the cohomology groups $\mathrm H^\bullet(\mathrm B^n(\mathbb Z/2);\mathbb Z/2)$. I assume that this is a standard computation, but I'm not sure where to look up the answer (and, ...

**6**

votes

**3**answers

439 views

### Higher refinement of Seifert-van Kampen theorem on the language of hocolim

I like the following version of SvKT. If $\Pi_1$ is the functor of fundamental groupoid and $(X_i)_{i\in I}$ is a diagram of spaces then
$$\Pi_1({\sf hocolim}\: X_i)\simeq {\sf hocolim}\: ...

**6**

votes

**3**answers

673 views

### Ambidexterity and Quantization

Here the nlab says about Hopkins-Lurie's ambidexterity paper:
"The discussion in the article is apparently motivated as part of what it takes to make precise the discussion of quantization in ...

**0**

votes

**0**answers

181 views

### A question about the Leray-Serre spectral sequence

Suppose $F \to E \stackrel{p}{\to} B$ is a fibration with $B$ simply connected. The $E_2^{p,q}$ page of the Leray-Serre spectral sequence is given by $H^p(B;H^q(F))$. Suppose futhermore that $k$ is a ...

**6**

votes

**1**answer

427 views

### Lurie's Endomorphism Space vs. Endomorphisms

In Jacob Lurie's book Higher Algebra, for an object $M$ of a monoidal $\infty$-category $\mathcal{C}$, he constructs a category $\mathcal{C}[M]$ which can be thought of as "maps in $\mathcal{C}$ of ...

**2**

votes

**1**answer

202 views

### A homeomorphism between total spaces with same fiber and base spaces not homotopic

Is there a counterexample to the following assertion?:
Let $p_1:E_1\to B_1$ and $p_2:E_2\to B_2$ be fibrations with the same fiber $\mathbb S ^1$ such that $E_1$ and $E_2$ are homeomorphic (and both ...

**9**

votes

**2**answers

656 views

### Homology of localisations of spectra

Let $H^*$ and $K^*$ be two cohomology theories, and $X$ a reasonable spectrum. Here, I'm thinking that $H^*$ is singular cohomology (and for my purposes, rational cohomology will suffice), and $K$ is ...

**3**

votes

**1**answer

78 views

### Involution on the set of isomorphism classes of 2n-dimensional vector bundles induced by an involution of $O(2n)$

Denote by $\varphi$ the automorphism of $O(2n)$ given by $\varphi (A)=det(A)\cdot A$.This induces a self-map $B\varphi$ of $BO(n)$, so it induces a self-map (actually an involution)
$B\varphi ^*$ on ...

**3**

votes

**1**answer

162 views

### Can a homology $n-1$-sphere divide $\mathbb{S}^{n}$ into non-contractible components?

This is a follow-up to my earlier question.
Let $\Sigma\subset \mathbb{S}^{n}$ be a hypersurface -- here $\mathbb{S}^{n}$ is a smooth sphere (possibly exotic ... if this makes a difference). If ...

**6**

votes

**0**answers

269 views

### What's the detailed proof of “the composition of planar tangles is well-defined”?

In the planar algebra theory (see here or there section 2), a planar tangle is an isotopy class; then to define the composition of two tangles, we need to choose a representative in each classes. See ...

**3**

votes

**1**answer

148 views

### Topology of hypersurface of sphere fixed by homeomorphic involution

I'm not an topologist, so I apologize in advance if this is a silly question.
I have the following situation, let $\mathbb{S}^n$ (for $n\geq 3$) be the standard (smooth if it matters) $n$-sphere and ...

**5**

votes

**1**answer

230 views

### If a topological space has vanishing $n$th homology for every possible homology theory, does it have vanishing $n$th homotopy?

I don't have any strong preference as to whether or not the homology theories are required to be ordinary.
Also, if this does not hold in general, does it hold for some nice category of spaces, like ...

**1**

vote

**0**answers

127 views

### Homotopy type of a CW complex

The only dimension in which not every compact manifold is homeomorphic to a CW complex is $4$. Does every such manifold have the homotopy type of a CW complex?

**17**

votes

**4**answers

2k views

### Compact open topology on $\mathrm{Homeo}(X)$

Let $X$ and $Y$ be topological spaces. Define the compact open topology on the set $\mathrm{M}(X,Y)$ of continuous maps from $X$ to $Y$ via the subbase $[K,O]$ of all maps $f:X\rightarrow Y$ s.t. ...

**0**

votes

**0**answers

96 views

### Simple homotopy type of 2-dimensional simplicial complexes

Simple Homotopy type is a particular case of homotopy equivalence, of a combinatorial nature which is governed by the Whitehead group $wh(\pi)$ of the fundamental group ...

**10**

votes

**0**answers

301 views

### Combinatorial results by Poincaré duality

For the n-dimensional Torus, the k-th homology group (with integer coefficients) is isomorphic to the direct sum of $n \choose k$ copies of $\mathbb{Z}$.
Poincaré duality thus gives us a somewhat ...

**6**

votes

**1**answer

231 views

### Definitions of the module $R/(x_0^\infty,x_1^\infty,\ldots,x_{n-1}^\infty)$

There are several constructions of the Prüfer group $\mathbb{Z}/p^\infty$; here are two that are relevant for this question.
It can be constructed via the short exact sequence
$$
0 \to \mathbb{Z} ...

**3**

votes

**1**answer

167 views

### Does the CGWH-fication change the (weak) homotopy type?

Let $Top$, $CG$, $WH$, $CGWH$ be the categories of topological spaces, compactly generated spaces, weak Hausdorff spaces and compactly generated weak Hausdorff spaces.
There is the CG-ification ...

**-1**

votes

**0**answers

32 views

### Properties of Pushout [migrated]

suppose we have a pushout square in $\mathrm{Top}$:
\begin{align*}
\require{AMScd}
\begin{CD}
X_0 @>{\mu_1}>> X_1\\
@V{\mu_2}VV @VV{\alpha_1}V \\
X_2 @>>{\alpha_2}> X
...

**4**

votes

**2**answers

240 views

### Need M combinatorial for existence of injective model structure on $M^G$?

I'm doing some work with model categories and operads, and to check a certain hypothesis I've had to learn a bit of equivariant homotopy theory. Let $M$ be a model category and $G$ be a finite group. ...

**12**

votes

**1**answer

430 views

### Condition on a Hopf operad for tensor product in the base category to be a (categorical) coproduct for algebras

A Hopf operad will be an operad endowed with a coproduct $P(n) \longrightarrow P(n) \otimes P(n)$ which is compatible in the obvious sens with operad laws (no more structure is assumed a priori. ...

**7**

votes

**1**answer

368 views

### Waldhausen Additivity in a More General Context

The following arose when I was thinking about a talk at the Midwest Topology Seminar:
Background
I want to consider a generalization of a Waldhausen-like structure on a category $C$ with 0-object ...

**5**

votes

**1**answer

484 views

### How do small changes in a filtered complex affect the associated spectral sequence

I have recently been learning about spectral sequences, not in the context of any problem, but mostly out of curiosity. There are two questions that have occurred to me which I've been unable to ...

**50**

votes

**16**answers

5k views

### Solving algebraic problems with topology

Often, topologists reduce a problem which is - in some sense - of geometric nature, into an algebraic question that is then (partiallly) solved to give back some understanding of the original problem.
...

**8**

votes

**1**answer

197 views

### Tensor products over operads and bar constructions

Let $O$ be an operad in spaces, $A$ an $O$-algebra and $R$ an right $O$-module. One can define $R \otimes_O A$ as the coequalizer of the two maps $ROA$ to $RA$. One can also define $B(R,O,A)$ (as in ...

**6**

votes

**1**answer

171 views

### Closed orientable 4-manifold with $H^1(M;\Bbb Z_2)=\Bbb Z_2$ and non-zero cup product $H^1\times H^1\to H^2$

I am looking for an example of a closed orientable 4-manifold $M$ with $H^1(M;\Bbb Z_2)=\Bbb Z_2$ and non-zero cup product $H^1(M;\Bbb Z_2)\times H^1(M;\Bbb Z_2)\to H^2(M;\Bbb Z_2)$.
A non-orientable ...

**23**

votes

**2**answers

4k views

### The error in Petrovski and Landis' proof of the 16th Hilbert problem

Edit:For a recent progress on the Hilbert 16th problem see the following note which consider an infinite dimensional nature for this (apparently 2 dimensional) amazing problem . Best wishes for ...

**10**

votes

**5**answers

868 views

### Computing homology of very large posets

I'm studying the homology of a couple of very large posets (one has over 4 million vertices, though the dimension is only 3). I want to show the posets are spherical (homology vanishes except in top ...

**40**

votes

**6**answers

3k views

### Total spaces of $TS^2$ and $S^2 \times R^2$ not homeomorphic

Hello,
I'm looking for an invariant to distinguish the homeomorphism types of homotopy equivalent spaces. Specifically, how does one show that the total spaces of the tangent bundle to $S^2$ and the ...

**7**

votes

**1**answer

284 views

### Z/p action on finite contractible complex

Let $p$ be a prime and $X$ a finite contractible CW-complex. Assume $\mathbb Z/p$ acts on $X$. Then it is easy to see that there has to be a fixed point. (E.g. use Lefschetz's fixed point theorem or ...

**3**

votes

**1**answer

208 views

### Fixed point property for intersection of spaces which are homeomorphic to a disk

The following question is question 9.8 from Miller's paper ``Some interesting problems
'':
Question Suppose $D_n$ a subset of the plane is homeomorphic to a disk and for every
$n\in \omega, ...

**4**

votes

**1**answer

434 views

### Topology of algebraic varieties

Let $X$ be a projective variety (lets say normal and irreducible) with the topology coming from being a subspace of $\mathbb{P}^N$ (and not the Zariski topology). Surely one can then define the ...

**37**

votes

**11**answers

4k views

### Compelling evidence that two basepoints are better than one

This question is inspired by an answer of Tim Porter.
Ronnie Brown pioneered a framework for homotopy theory in which one may consider multiple basepoints. These ideas are accessibly presented in his ...

**10**

votes

**1**answer

542 views

### Non-unique splittings of homotopy idempotents

By a homotopy idempotent I mean a map $f:X\to X$, where $X$ is a space, equipped with a homotopy $f\circ f \sim f$. In contrast to the situation in stable homotopy theory (where $X$ would be a ...

**7**

votes

**2**answers

512 views

### Embedded (framed) cobordisms

[The title initially was "Actions of gauge groups on framed cobordisms. This has been changed.]
This question is a follow-up to my answer to When is a submanifold of $\mathbf R^n$ given by global ...