Questions tagged [at.algebraic-topology]

Homotopy theory, homological algebra, algebraic treatments of manifolds.

Filter by
Sorted by
Tagged with
7 votes
4 answers
630 views

Realizing complexes with bases as cellular complexes

This is a question a friend of mine asked me some time ago. I suspect the answer is "no" but can't prove it. Every free complex of abelian groups is isomorphic to the reduced cellular complex of some ...
algori's user avatar
  • 23.2k
7 votes
2 answers
637 views

Naive Z/2-spectrum structure on E smash E?

Let $E$ be a spectrum. Then $E \wedge E$ is a $\mathbb{Z}/2$-spectrum in the naivest possible sense, i.e., an object with $\mathbb{Z}/2$-action in the (∞,1)-category of spectra. Can I make it ...
Reid Barton's user avatar
  • 24.8k
7 votes
1 answer
240 views

Computing $\pi_1$ of the complement of a non-singular plane curve

The following is a well-known fact: Theorem. The fundamental group of the complement of a non-singular curve of degree $d$ in the complex projective plane is cyclic of order $d$. This was further ...
Marco Golla's user avatar
  • 10.4k
7 votes
1 answer
175 views

Are morphisms in a stable $\infty$-category generated by split injections?

I've seen it stated in the $\infty$-categorical literature (without proof or reference) that every object in the $\infty$-category $\operatorname{Fun}(\Delta^1, \mathcal{C})$ of morphisms in a stable $...
Ishai Dan-Cohen's user avatar
7 votes
1 answer
426 views

Why does the tangent classifier classify the tangent (micro)bundle?

Let $\mathcal{M}\mathrm{fld}_n$ denote the $\infty$-category of topological manifolds (without boundary) and embeddings; more precisely, it is the homotopy coherent nerve of the simplicial category ...
Ken's user avatar
  • 1,749
7 votes
1 answer
535 views

Long exact sequences for parametrized cohomology

I'm reading Michael Shulman's articles on cohomology in HoTT here and here, as well as Floris van Doorn's thesis here. Given $E: Z \to \mathsf{Spectrum}$ a family of spectra over a homotopy type $Z$, ...
ಠ_ಠ's user avatar
  • 5,933
7 votes
1 answer
287 views

A finitely presented group whose rational cohomology is not nilpotent

Does there exist a finitely presented (preferably $\text{FP}_{\infty}$) group $\Gamma$ and an element $\alpha \in \text{H}^{\ast>0}(B\Gamma;\mathbf{Q})$ that is not nilpotent? If non-discrete ...
Jens Reinhold's user avatar
7 votes
1 answer
364 views

Implications of Geometrization conjecture for fundamental group

Hempel proved that Haken manifolds have residually finite fundamental groups. With the Geometrization conjecture, this now holds for any compact and orientable 3-manifold. How exactly does the ...
aceituna's user avatar
  • 121
7 votes
2 answers
302 views

Homotopicity of two certain sections of frame bundle of $GL(n,\mathbb{R})$

Edit: According to comment of Prof. GoodWillie we revise the question. Put $M=GL(n,\mathbb{R})$. We identify $M_n(\mathbb{R})$ with $\mathbb{R}^{n^2}$: The identification is based on the ...
Ali Taghavi's user avatar
7 votes
1 answer
444 views

Twisted spin bordism invariants in 5 dimensions

[Note]: My question will be a bit long. So, first, thank you for your careful reading, generous comments, helps and answers, in advance! The spin $G$-bordism invariant can be twisted in the way that ...
wonderich's user avatar
  • 10.3k
7 votes
1 answer
361 views

Characteristic classes of the bundle of trace free, skew adjoint endomorphisms

In "Floer Homology groups in Yang-Mills theory", Donaldson says that if we take an $U(2)$-vector bundle $E$ and we construct the bundle $\mathfrak{g}_E$ of trace-free, skew adjoint automorphisms of $...
Overflowian's user avatar
  • 2,523
7 votes
1 answer
460 views

Commutativity up to homotopy implies strict commutativity, for lifting problems

Suppose we have a commutative diagram $\require{AMScd}$ \begin{CD} A @>>> X \\ @VVV & @VVV \\ W @>>> Y\\ \end{CD} where the map $A\rightarrow W$ is a cofibration and the ...
Diego95's user avatar
  • 511
7 votes
1 answer
1k views

What is the scope of validity of Kunneth formula for de Rham?

In books like Bott-Tu or all pdf texts I have found on internet, the Kunneth formula for manifolds $M$ and $N$ and their de Rham cohomology $$ H^{\bullet}_{dR}(M \times N) \simeq H^{\bullet}_{dR}(M) \...
ychemama's user avatar
  • 1,326
7 votes
1 answer
437 views

Two mixed Hodge structures on equivariant cohomology for actions by finite groups

The answer to the following question might be obvious but I haven’t found a full proof yet (neither by myself nor in the literature). So my apologies if it is trivial. Let $X$ be a (for simplicity ...
Florian's user avatar
  • 143
7 votes
1 answer
262 views

Does every equivalence of operads in the category of small categories have a weak inverse?

Call a map of operads $\mathcal{O}\rightarrow \mathcal{U}$ in the category of small categories an equivalence, if each functor $\mathcal{O}(n)\rightarrow \mathcal{U}(n)$ is an equivalence of ...
simon's user avatar
  • 71
7 votes
1 answer
635 views

Fundamental group of the space of maps into a classifying space

Let $P : E \to X$ be a principal $G$-bundle, where $G$ is a connected topological group. $P$ is classified by a map $f: X \to BG$. The group of gauge transformations $\mathcal{G}$ of $P$ is defined to ...
Bilateral's user avatar
  • 3,064
7 votes
1 answer
958 views

higher algebraic homotopy groups for schemes?

I think I understand how to define the algebraic fundamental group $\pi^{alg}_{1}(X)$ of a scheme and I think I understand the relation between $\pi^{alg}_{1}(X)$ and $\pi_{1}(X(\mathbb{C}))$, where $...
Ofra's user avatar
  • 1,603
7 votes
1 answer
482 views

Homotopy of orthogonal groups in the unstable range

We fix an integer $n$ and consider the stabilization map $O(n)\to O$. Using rational methods one can easily check that the map $\pi_{4i-1}(O(n))\to \pi_{4i-1}(O)\cong\mathbb{Z}$ vanishes for ...
Ulrich Bunke's user avatar
7 votes
1 answer
492 views

liftings of principal bundles

I would like to know what structure has the category of liftings of a principal bundle. Let me be more precise. Fix $k$ an algebraically closed field and $X$ a smooth projective variety over it (for ...
Dragos Fratila's user avatar
7 votes
1 answer
724 views

Thom isomorphism from the ABGHR perspective

In ABGHR Thom spectra are described in the following way: we start with a morphism of Kan complexes $X\to \mathbb{S}\text{-line}$, where $\mathbb{S}\text{-line}$ is an $\infty$-groupoid which is ...
Jonathan Beardsley's user avatar
7 votes
1 answer
313 views

When does simplicial localization commute with functor categories?

Let $(C,W)$ be a category with a class of weak equivalences, and $D$ a small category. Then I can form the diagram category $(C^D,W^D)$ with objectwise weak equivalences, and its simplicial ...
Mike Shulman's user avatar
7 votes
1 answer
414 views

How does the kernel of the map $\Omega^{\bullet}(X)\rightarrow \Omega^{\bullet}(G\times X)$ relate to equivariant cohomology?

This question may be trivial for experts. Consider a (compact, connected) smooth manifold $X$ and a (compact connected) Lie group $G$ act on $X$. Then we have the action map $$ \mu: G\times X\...
Zhaoting Wei's user avatar
  • 8,657
7 votes
1 answer
2k views

Finite homotopy limits commute with sequential homotopy colimits

I would like to know for what kind of model category finite homotopy limits commute with sequential homotopy colimits. Would cofibrantly generated and finitely locally presentable be enough? It seems ...
Emanuele Dotto's user avatar
7 votes
1 answer
936 views

Explicit 2-Cocycles of G=Z2×Z2xZ2 over U(1)

We know that group cohomology $H^2(G,U(1))$ consists of 2-cocycles $\beta(A,B)\in U(1)$ corresponding to elements in the group $H^2(G,U(1))$, where $A\in G,B \in G$. Note that $\beta(A,B)$ satisfies 2-...
wonderich's user avatar
  • 10.3k
7 votes
1 answer
452 views

Are constructible derived categories invariant up to weak homotopy equivalence?

Let $X$ and $Y$ be two topological spaces and $R$ be commutative ring. Let $D_c^b(X, R)$ and $D_c^b(Y,R)$ be their respective bounded derived categories of constructible sheaves of $R$-modules. I ...
deltmu's user avatar
  • 121
7 votes
1 answer
595 views

Image of J in the classical Adams Spectral Sequence

Hey all, I know that in some versions of the Adams Spectral Sequence you can easily identify the image of $J$, and I was wondering if there was a way to identify the image of $J$ in the $E_2$ page of ...
Joseph Victor's user avatar
7 votes
4 answers
1k views

Quotient rings of $C(X)$

Let $X$ be a Tychonoff topological space. Consider the ring $C(X)$ of all continuous real valued functions on $X$. For what conditions on an ideal $I$ of $C(X)$, we could deduce that the quotient ring ...
Ali Reza's user avatar
  • 1,778
7 votes
1 answer
903 views

Is there a cheap proof that (homotopy) endomorphisms are functorial?

This is, in some sense, the homotopy version of this question.) If $C$ is a category with $iC$ the subcategory of isomorphisms, there is a functor $X \mapsto End(X)$ from $iC$ to the category of ...
Tyler Lawson's user avatar
  • 51.1k
7 votes
1 answer
719 views

More on completion/compactification of open manifolds

This is a follow up to one of my previous questions (81714). Suppose that $M$ is an open manifold, say with a single end. Previously, I was concerned with realizing $M$ as the interior of a compact ...
Igor Khavkine's user avatar
7 votes
2 answers
691 views

Euler class of S^1-orbibundle

Given a topological space $M$ with a locally free $S^1$ action on it, assume the slice representation holds,(this is often the case, e.g. M is a smooth manifold) then this will make $M$ a principal $S^...
Shisen Luo's user avatar
7 votes
1 answer
547 views

When do we need the axiom of compact support for a homology theory to be uniquely defined?

Looking over the treatment of the Eilenberg-Steenrod axioms in a few of my favorite introductory algebraic topology texts, I see that some include an "axiom of compact support", while others do not. ...
Greg Friedman's user avatar
7 votes
1 answer
418 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 ...
7 votes
1 answer
389 views

When do covering spaces correspond to characteristic subgroups?

Given a covering space $p \colon X \to Y$, we get an injection $p^* \colon \pi_1(X) \to \pi_1(Y)$, and we know that the image $p^*(\pi_1(X))$ is normal in $\pi_1(Y)$ if an only if $p$ is regular, that ...
Anschel Schaffer-Cohen's user avatar
7 votes
1 answer
230 views

Relation between cohomology operations and the Adams spectral sequence

$\newcommand{\Z}{\mathbb Z} \DeclareMathOperator{\Ext}{Ext} \DeclareMathOperator{\Cone}{Cone}$ I'm trying to understand how higher order cohomology operations are related to the Adams spectral ...
Shivang's user avatar
  • 71
7 votes
2 answers
479 views

Simplicial nerve of a topological group

Given a group $G$, one way to construct the classifying (topological) space $BG$ is to regard $G$ as a category with one object and morphisms $G$, take its nerve, and then apply the geometric ...
Ken's user avatar
  • 1,749
7 votes
1 answer
642 views

Diagonal maps, Goodwillie calculus, and $T(n)$ local homotopy theory

Here is a collection of facts that all seem true, but together seem to give a nonsensical solution: After $T(n)$-localization, all natural transformations $F \sim G$ between homogenous functors $F,G:...
Connor Malin's user avatar
  • 5,191
7 votes
1 answer
555 views

What is the closure of the Eilenberg-MacLane spectra under limits? under colimits?

Every bounded spectrum is in the closure of the Eilenberg MacLane spectra under finite co/limits. Thus every bounded below (resp. above) spectrum is in the closure of the EM spectra under limits (resp....
Tim Campion's user avatar
  • 60.6k
7 votes
1 answer
257 views

Stallings' binding tie

I came to know that the statement below could be proved using Stallings' binding tie argument, though I have no reference article proving the statement by the binding tie argument. Can anyone help me ...
Random's user avatar
  • 927
7 votes
1 answer
817 views

Filtered homotopy colimits and singular homology

Suppose I have a functor $$ X_\bullet: I \to \text{Spaces} $$ where $I$ is a small filtered category. It seems to be a "folk theorem" that the homomorphism $$ \underset{\alpha\in I}{\text{...
John Klein's user avatar
  • 18.6k
7 votes
1 answer
234 views

On the comparison map $MU^\bullet(X)\otimes_{MU^\bullet(pt)}E^\bullet(pt)\to E^\bullet(X)$ for complex oriented multiplicative cohomology theories

Whatever complex oriented multiplicative cohomology theories are, they come with two basic properties (among many others): i) a complex oriented multiplicative cohomology theory is a contravariant ...
domenico fiorenza's user avatar
7 votes
1 answer
320 views

When do two topoi have the same cohomology of constant sheaves

Recently, I have some questions for some generalizations from algebraic topology. I learn some homotopy theory in algebraic topology. We know that, if two spaces are homotopy, then they have same ...
user106561's user avatar
7 votes
1 answer
288 views

Is the canonical model structure on strict $\infty$-Cat left proper?

Is the canonical (or Folk) model structure on the category of (strict) $\infty$-categories as constructed by Lafont, Métayer and Worytkiewicz in A folk model structure on omega-cat left proper ? All ...
Simon Henry's user avatar
  • 39.9k
7 votes
1 answer
273 views

Inducing a model structure using a cosimplicial object

In a recent paper, Hiroshi Kihara induced a model structure on the category of diffeological spaces. He generates the classes of fibrations, cofibrations, and weak equivalences by constructing a ...
Ben MacAdam's user avatar
  • 1,253
7 votes
1 answer
302 views

What is difference between working with small and large category of spaces?

The following construction have always bugged me. This is p328, Remark 5.1.6.1 in Lurie's Higher Topos Theory. Lurie begins with the following: Construction: Let $C$ be a simplicial set. $S$ denote ...
Bryan Shih's user avatar
7 votes
1 answer
235 views

Diffeomorphisms pushing forward vector field to any non-zero multiple

Is there a closed smooth manifold $M$ such that for each real $x\neq 0$ there is a nowhere vanishing vector field $v$ on $M$ and a diffeomorphism $\phi:M\to M$ such that $\phi_*v=xv$?
user avatar
7 votes
1 answer
446 views

Visualizing a Whitehead product: the attaching map $S^3\to S^2\vee S^2$

There are informative and easily accessible images and videos that illustrate the Hopf fibration $S^3\to S^2$ by describing what happens to the fibers in the unit cube $(0,1)^3\approx S^3\backslash \...
J.K.T.'s user avatar
  • 487
7 votes
1 answer
570 views

Maps into a Postnikov tower of the sphere

Suppose I have a CW complex $Y$ of dimension $n+2$ and let $X_{n+2}$ be the third non-trivial Postnikov stage of $S^n$ (i.e. there is a map $S^n \to X_{n+2}$ which is an $(n+2)$-equivalence). We ...
Panagiotis Konstantis's user avatar
7 votes
2 answers
447 views

Critical points and high homotopy groups

Is there any known or interesting relation between critical points (possibly degenerate, or maybe only nondegenerate) of a function on a manifold and generators/relations of high homotopy groups? I ...
Andrej Luzhin's user avatar
7 votes
1 answer
195 views

Homology of a limit of semidirect products

Suppose I have two families of groups $A_k$ and $B_k$ indexed by the natural numbers and suppose $B_k$ acts on $A_k$. Suppose there are groups homomorphisms $A_{k+1} \rtimes B_{k+1} \to A_k \rtimes ...
2223's user avatar
  • 111
7 votes
2 answers
587 views

Which topological spaces contain dense simply connected subspace?

And when can this subspace be chosen to be open? As the answer to this question indicates, any manifold contains an open dense subset, which is homeomorphic to $\mathbb{R}^{n}$, and so for manifolds ...
erz's user avatar
  • 5,385

1
64 65
66
67 68
165