Questions tagged [at.algebraic-topology]
Homotopy theory, homological algebra, algebraic treatments of manifolds.
8,227
questions
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 ...
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 ...
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 ...
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 $...
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 ...
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 $Z$, ...
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 ...
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 ...
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 ...
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 ...
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 $...
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 ...
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) \...
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 ...
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 ...
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 ...
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 $...
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 ...
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 ...
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 ...
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 ...
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\...
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 ...
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-...
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 ...
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 ...
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 ...
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 ...
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 ...
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^...
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. ...
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 ...
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 ...
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 ...
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:...
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....
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 ...
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{...
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 ...
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 ...
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 ...
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 ...
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 ...
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 $\phi_*v=xv$?
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 \...
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 ...
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 ...
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 ...
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 ...