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

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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 ...
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1answer
145 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 ...
4
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1answer
399 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 ...
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157 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
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1answer
254 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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148 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 ...
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1answer
74 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 ...
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497 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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170 views

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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1answer
391 views

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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189 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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1answer
447 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 ...
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1answer
209 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
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2answers
443 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, ...
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1answer
88 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
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3answers
713 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 ...
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1answer
176 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 ...
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1answer
163 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 ...
6
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2answers
402 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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133 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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237 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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1answer
115 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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99 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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3answers
472 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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316 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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1answer
171 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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1answer
235 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} ...
6
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1answer
180 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 ...
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1answer
295 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 ...
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1answer
292 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 ...
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1answer
215 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 ...
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1answer
449 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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2answers
430 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 ...
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381 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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93 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 ...
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1answer
262 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 ...
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2answers
415 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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119 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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2answers
419 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
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1answer
294 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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87 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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1answer
257 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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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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100 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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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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185 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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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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117 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 ...
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1answer
296 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 ...
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132 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 ...