# Tagged Questions

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

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134 views

### Functors similar to $H^i(\cdot)$

Suppose $T$ is a contravariant functor from the category of pointed topological spaces to the category of abelian groups, then we have homomorphisms $\alpha\colon T(X)\times T(Y)\to T(X\times Y)$ and ...

**9**

votes

**1**answer

181 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 ...

**5**

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**0**answers

115 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 ...

**0**

votes

**1**answer

64 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 ...

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112 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 ...

**12**

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**0**answers

453 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 ...

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**0**answers

140 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 ...

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205 views

### Whitehead group [closed]

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

**7**

votes

**1**answer

343 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 ...

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183 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 ...

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**1**answer

429 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

204 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 ...

**2**

votes

**2**answers

424 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, ...

**3**

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**1**answer

79 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 ...

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**3**answers

677 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 ...

**3**

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**1**answer

163 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 ...

**3**

votes

**1**answer

149 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

**2**answers

299 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 ...

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**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?

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**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 ...

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vote

**1**answer

102 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 ...

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98 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 ...

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**3**answers

445 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}\: ...

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**0**answers

305 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 ...

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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 ...

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**1**answer

232 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} ...

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**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
...

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votes

**1**answer

174 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 ...

**4**

votes

**1**answer

287 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 ...

**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 ...

**8**

votes

**1**answer

199 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 ...

**4**

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**1**answer

436 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 ...

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votes

**2**answers

414 views

### Who is currently researching topics concerning applying algebraic topology and/or differential geometry to numerical methods? [closed]

I am interested in pursuing a PhD in mathematics from a top ranked university with a faculty member researching something akin to the following description: applications of algebraic topology and/or ...

**35**

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**0**answers

364 views

### Topological cobordisms between smooth manifolds

Wall has calculated enough about the cobordism ring of oriented smooth manifolds that we know that two oriented smooth manifolds are oriented cobordant if and only if they have the same ...

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**0**answers

92 views

### First to note the relation between Stasheff polytopes (associahedra) and compositional inversion?

In my answer to MO-Q: Enumerative geometry and nonlinear waves, I outline the relation between the refined face polynomials of the Stasheff polytopes (associahedra) and the partition polynomials for ...

**3**

votes

**1**answer

256 views

### localization and $E_{\infty}$-spaces

Let $\mathrm{Top}$ be the model category of topological spaces. Define a new model structure on $\mathrm{Top}$ where $f:X\rightarrow Y$ is a weak equivalence iff ...

**8**

votes

**1**answer

232 views

### Pull-back of a fibration along a homotopy equivalence and homotopy classes of sections

I previously asked this on Math.SE but didn't receive a satisfactory answer.
Let $p:E\rightarrow B$ be a fibration (i.e. have the homotopy lifting property with respect to all spaces), and $f: ...

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**0**answers

113 views

### Persistence barcodes and spectral sequences

Persistent homology is a well-developed tool which allows topological analysis of large data sets. From a topological perspective, the input is a filtered complex, and the output is a sequence of ...

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votes

**2**answers

403 views

### What are algebras for the little n-balls/n-cubes/n-something operads exactly?

As a non expert in the theory of topological operads, I find it pretty hard, to understand what algebras for little balls/cubes/something operads are.
For all the other famous operads I know (like ...

**4**

votes

**1**answer

293 views

### Cone structures on $\mathbb R^n$

We know that we can put two different structures on $\mathbb R^5$ in the topological category. First is $C(S^{4})$ and second is $C(\Sigma X^3)$, where $X^3$ is a homology sphere and $C(\cdot)$ stands ...

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**0**answers

85 views

### Reference: K-theory of real Grassmann manifolds

Was the complex K-theory of the unoriented real Grassmann manifold $G_{\mathbb{R}}(k,n)$ computed somewhere, at least for certain values of $k,n$?

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**1**answer

232 views

### Can we construct a Baas-Sullivan presentation of TMF?

Quick Review:
The Baas-Sullivan construction cones off generators $\alpha_1, ..., \alpha_n \in \pi_*(MU)$ from $MU$ to get a new spectrum $MU/(\alpha_1, ..., \alpha_n)$, which is isomorphic to some ...

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votes

**0**answers

97 views

### How can one “extend scalars” for (motivic) ring spectra and for modules over it?

Let $S$ be a (motivic symmetric) ring spectrum (more generally, one can possibly consider a ring object in a symmetric stable model category); let $R$ be a flat associative commutative unital algebra ...

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**0**answers

98 views

### group completion of topological monoid

Let $M$ be a topological monoid. The group completion of $M$ is defined as $\Omega BM$, where $BM$ is the classifying space of $M$.
In Lecture Notes in Math. 533, The homology of C n+1 spaces, n>=0, ...

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**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.
...

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**0**answers

182 views

### Can Z/2 x Z/2 act freely on an infinite dimensional sphere?

Using that all groups that act freely on some sphere $S^n$ have periodic cohomology, one can see that $\mathbb Z/2 \times \mathbb Z/2$ can not act freely on any $S^n$. But can it act freely on ...

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votes

**0**answers

95 views

### Nilpotency of BG^+

I have become interested in knowing if there are conditions, in the literature, under which the action of the fundamental group of BG^+ acts nilpotently on its higher homotopy groups, where the + ...

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votes

**0**answers

116 views

### There is no quasiregular diffeomorphism from punctured ball into ring (on the plane)

The idea is to use l2 cohomology as a quasiregular map invariant.
It is easy to see that there are closed 1-forms on the ring which are not exact, but it occures that every closed l2-form
$f_1(x,y)dx ...

**1**

vote

**1**answer

294 views

### Construction of Highly Structured Quotient Groups in Quasicategories

Suppose we have a map of $E_n$-spaces $X\to Y$. Then there is a highly structured action of $X$ on $Y$, $X\wedge Y\to Y\wedge Y\to Y$, using the multiplication of $Y$. As such, I believe that there ...

**4**

votes

**0**answers

131 views

### Contractibility of regular CW sphere minus open star

Let $S$ be any regular CW decomposition of (a space homeomorphic to) the $n$-sphere, and consider a cell $\sigma$ of dimension $d \in \{0,\ldots,n\}$. Let $S'$ be the regular CW complex which remains ...