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

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9
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1answer
137 views

Counter examples for strengthening Whitehead's theorem?

Let $f:X\to Y$ be a pointed map of pointed connected $n$-dimensional CW complexes. Whitehead's theorem says that if $f_*:\pi_qX\to \pi_qY$ is an isomorphism for $q\le n$ and a surjection for $q=n+1$, ...
2
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0answers
106 views

homology fibrations induced by actions of topological monoids

In the paper Homology fibrations and group completion theorem, McDuff-Segal, page 280, lines from bottom 3-6, Proposition 2: If $M$ is a topological monoid which acts on a space $X$, and for each ...
-1
votes
0answers
21 views

Freedom of speech in scientific discussions - An invitation to more tolerance in Scientific debates [migrated]

I hope this post enjoys some tolerance, and don't get closed or put on hold immediately. I believe that freedom of speech in scientific discussions is one of the key values which enriches the debates ...
1
vote
0answers
71 views

Orientation form on the blow up of a Kaehler manifold

Let $(X,\omega)$ be a complex Kaehler manifold of (complex) dimension $d$, and let $Y\subset X$ a complex submanifold of dimension $k$. Evidently $[\omega]^d\in H^{2d}(X,{\mathbb{R}})$ is always ...
3
votes
1answer
310 views

$\pi_8(S^5)=\pi_8(SO(6))=\mathbb{Z}/24$?

I am currently thinking about a physics model related to framed bordism $\Omega_3^{fr}=Z/24=\pi^s_3$, and the first stable example is $\pi_8(S^5)$, so I was curious about the generator, and happened ...
4
votes
1answer
130 views

Prove that the Log-Euclidean distance is negative-definite

Let $\Bbb{S}_{++}^n$ be the $\frac{n(n+1)}{2}$-dimensional Riemannian manifold of the symmetric positive definite (SPD) $n\times n$ real matrices. The Log-Euclidean distance between two points of ...
1
vote
0answers
143 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
1answer
221 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
votes
0answers
127 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
1answer
65 views

free action on product of two spaces [closed]

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 ...
0
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0answers
113 views

Morse theory in zero dimensions? [closed]

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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468 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 ...
5
votes
0answers
155 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 ...
0
votes
0answers
206 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
1answer
369 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 ...
0
votes
0answers
184 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
1answer
436 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
1answer
205 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
2answers
425 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
votes
1answer
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 ...
6
votes
3answers
682 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
votes
1answer
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
1answer
150 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
2answers
300 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 ...
1
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0answers
130 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?
5
votes
1answer
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
1answer
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 ...
0
votes
0answers
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 ...
6
votes
3answers
451 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}\: ...
10
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0answers
307 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 ...
3
votes
1answer
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 ...
6
votes
1answer
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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0answers
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 ...
6
votes
1answer
175 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
1answer
289 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
1answer
285 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
1answer
202 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
votes
1answer
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 ...
2
votes
2answers
415 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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0answers
365 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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0answers
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
1answer
257 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
1answer
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: ...
10
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0answers
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 ...
7
votes
2answers
406 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
1answer
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 ...
3
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0answers
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 $k,n$?
8
votes
1answer
233 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 ...
3
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0answers
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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0answers
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, ...