Questions tagged [at.algebraic-topology]
Homotopy theory, homological algebra, algebraic treatments of manifolds.
8,255
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what is this simple topological space?
Take $M_p$ the mapping cylinder (MC) of the $p=3$-fold cover of $S^1$, $M_q$ the MC of the $q=2$-fold cover of $S^1$, where for both, the identification of the MC is done on the side $\{0\}$ of $[0,1]$...
3
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"Totally real" linear transformations
Identify $\mathbb{C}^n$ with $\mathbb{R}^{2n}$ via the equality $$(z_1, z_2, \ldots, z_n)=(x_1, \ldots, x_n, y_1, \ldots , y_n)$$
Where $z_j=x_j + iy_j$.
We call a linear invertible map $A: \mathbb{R}^...
user515519
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1
answer
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What topological principle is at work here?
[I'm cross-posting this from MSE. I initially asked there 10 days ago, and the question was well-received, but left unanswered.]
My question is inspired by a problem I discovered in Putnam and Beyond,...
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Homotopy group of maps into S^3 using its Lie group multiplication to define the group operation
The Bruschlinsky group of maps of a space X into S1 up to homotopy, using the multiplication on S1, is well-known to equal the first cohomology group of X (at least assuming X is a reasonably nice ...
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What is $\pi_{23}(S^2)$?
The first $22$ homotopy groups of the $2$-sphere were worked out by Toda in 1962, but I cannot find any results extending that to any higher homotopy groups of $S^2$.
Are any more of these groups ...
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Possible Euler characteristics of manifolds with tangential structures
Let $p:B\to BO$ be a fibration. We say that a manifold has a $B$-structure if its stable tangent bundle lifts to $B$. I am interested in the question of whether there exists, for a given even ...
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Are the spaces BG for compact connected groups G ind-projective or ind-Kähler?
Let $G$ be a compact connected group, or maybe better its complexification. By thinking about the simplicial Borel space, or using $n$-acyclic $G$-spaces for higher and higher $n$, it's "easy&...
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Recover the $C_k$-action of a cyclic object as from the $S^1$-action on Hochschild chain
$\newcommand{\Fun}{\operatorname{Fun}}$Let $X\in\Fun(\Lambda^\mathrm{op}, \mathcal{C})$ be a cyclic object in the ($\infty$-)symmetric monoidal category $\mathcal{C}$, where $\Lambda$ is the cyclic ...
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Extending topological vector bundles and obstruction theory
This is a question that has appeared in various forms on MathOverflow, see here and here, for example. But as opposed to these more algebraic questions, I am interested in the purely topological ...
- 1,735
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Weakly contractible $X$, but none of the maps $*\to X$ are cofibrations
Let $\mathrm{Top}$ be the category of all topological spaces and continuous maps. The Quillen model structure on $\mathrm{Top}$ has weak equvalences $W = \{ \text{weak homotopy equivalences} \}$, ...
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Categories on which one can determine all model structures?
Famously, there are exactly nine model structures on the category of sets, which are detailed here. In this case, one can exhaustively determine all six weak factorization systems and then see which ...
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Strøm model structures on the category of simplicial sets
Let $X,Y$ be simplicial sets. A simplicial homotopy is a simplicial map of the form $h:X\times\Delta^1\rightarrow Y$. There are two distinguished maps
$$
in_0:X\cong X\times\Delta^0\xrightarrow{1\...
- 5,016
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Existence of covering space with trivial pullback map on $H^1$
I have seen somewhere the following claim (but can't remember where): let $M$ be a connected orientable closed smooth manifold with $b_1(M)=1$, then there exists a connected covering space $p:\tilde{M}...
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Is every strongly causal spacetime purely electric?
Take a time 4-dimensional orinted Lorentzian manifold $(M,g)$.
A spacetime is called Strongly Causal at point $p$ if and only if for every neighbourhood $U$ of the point $p$ there exists a ...
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Linearity of topological periodic cyclic homology
Let $A$ be an $E_\infty$ ring spectrum, $B$ a ring spectrum. Then if I understand correctly, $TP(A)$ is a ring spectrum by the lax monoidal property of $TP$. Suppose there is a map of ringed spectra ...
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On the equivalence of two definitions of cohomological dimension for locally compact topological spaces
$\mathbf{The \ Problem \ is}:$ Let $X$ is a locally compact, separable metric space. Let $G$ be an abelian group. Now I came across two definitions of cohomological dimension of $X.$ One is the usual ...
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2
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Generalized cohomology on the one point space
I am reading Hatcher's algebraic topology for an assignment on generalized cohomology theories, and in section 4.E p. 447 he says the following
The wedge axiom implies that $h(\textit{point})$ is ...
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Understanding the maps of the long exact sequence of cohomology from a Koszul complex
Suppose $\mathcal E$ is a bundle over a rational homogeneous smooth variety $X\subset \mathbb P^n$ such that the zero-locus $Z_s$ of a generic global section $s$ is a reduced zero-dimensional scheme. ...
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For topological torus action, there is a subcircle whose fixed point is the same as the torus
Let $T=\mathbb{S}^{1}\times \mathbb{S}^{1}\times \cdots \times \mathbb{S}^{1}
$ ($n$ times) be an $n$-dimensional torus acting on any topological space $X$.
The group $G$ is said to act on a space $X$ ...
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How to calculate $\mathrm{TP}(\mathbb{F}_p[t])$?
$\DeclareMathOperator\TP{TP}$I am trying to learn about topological periodic cyclic homology following the notes:
https://www.uni-muenster.de/IVV5WS/WebHop/user/nikolaus/Papers/Lectures.pdf
https://...
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Comparing notions related to $(\infty,2)$-categories
I am trying to understand two related notions:
$(\infty,2)$-category as in Definition 5.5.1.3, Kerodon
weak $\infty$-bicategory as in Definition 4.1.1 in "$(\infty,2)$-Categories and the ...
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Poincaré-Hopf Theorem for domains with a point of vanishing curvature
Consider $\Omega \subset \mathbb{R}^2$ a convex planar domain having positive curvature on the boundary except for a point $p \in \partial \Omega$ where the curvature vanishes.
I would like to know ...
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When is $\smash{\check{H}}^{q}(X,A;R)\cong H_{c}^{q}(X-A;R)$ for a pair $(X,A)$?
I'm trying to understand the proof of Corollary 1.3 part b. in a paper by Bestvina and Mess titled 'The Boundary of negatively curved groups'. I do not understand why $\smash{\check{H}}^{q}(X,A;R)\...
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Does the symmetric algebra functor preserve inclusions?
Theorem: For any compact abelian group $G$, the homogeneous component $%
H^{2}\left( B_{G};%
%TCIMACRO{\U{2124} }%
%BeginExpansion
\mathbb{Z}
%EndExpansion
\right) $ of degree $2$ is naturally ...
- 1,161
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A question about fixed point set of the compact group actions
Let $G$ be an infinite compact Lie group acting on a compact space $X$.
Denote $F=F(G,X)=\{x\in X$ : $gx=x$ for all $g\in G\}$.
Show that if $H^*(B_{G_x};\mathbb{Q})=0$ for all $x \notin F$ and $T^1$ ...
- 1,161
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Homogeneous regular (= polynomial component) maps with odd degree and their being global homeomorphisms in dimensions higher than one?
Let $F:\mathbb{R}^m \to\mathbb{R}^m, F:=(F_1\dots F_m)$ be a regular map, i.e. with components $F_i$ that are polynomials.
Assume further that each $F_i$ is an odd degree (say $d$) homogenous ...
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Applications of Thom's first isotopy lemma
Thom's first isotopy lemma says that given a smooth map $f:M\to P$ between smooth manifolds, and a closed Whitney stratified subset $S$ of $M$, such that
$f|_S:S\to P$ is proper and $f|_X:X\to P$ is a ...
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Non-triviality of Whitehead products in wedges of CW-complexes
Suppose $X$ and $Y$ are finite, simply connected, based CW-complexes and $m,n\geq 2$. If $a\in \pi_m(X)$ and $b\in \pi_n(Y)$, then one can regard these as elements of the homotopy groups of $X\vee Y$. ...
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If homotopy groups of spaces are identical, then stable ones are also identical?
Is it true that if pointed spaces $X, Y$ have the same homotopy groups $\pi_n(X) \cong \pi_n(Y)$, then they have the same stable homotopy groups $\pi^S_n(X) \cong \pi ^S_n(Y)$?
In particular, is this ...
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Triviality of map $(\Sigma \theta)^*$
We know that there is a cofibration sequence
$$S^{4n+1}\xrightarrow{\theta}\Sigma^{4m-1} Q_{n-m} \rightarrow \Sigma^{4m-1} Q_{n-m+1} \rightarrow S^{4n+2}\xrightarrow{\Sigma\theta}\Sigma^{4m} Q_{n-m}.$$...
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Homotopy groups of finite CW complex finitely generated as Lie algebra
This is probably a well-known question, but I haven't found the answer on MO or MSE.
It is well-known that the homotopy groups of a finite CW complex $X$ need not be finitely presented, even as $\...
- 27.3k
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Structure of second homotopy group of a compact CW complex
I am interested in the second (and higher as well) homotopy groups of compact CW complexes. I know these groups don't need to be finitely generated (e.g. for $S^1 \vee S^2$ they are not), but I'd like ...
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Cohomology ring of $\mathbb{P}(\mathcal{O}(-1)\oplus \mathcal{O})$
Let $\mathcal{O}(-1)$ be the Hopf bundle over $\mathbb{C}\mathbb{P}^\infty$. Let $\mathcal{O}$ be the trivial rank one bundle. Consider the projectivization of the rank two bundle $\mathcal{O}(-1)\...
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Contractible orbit space of action of compact Lie group on Euclidean space
R. Oliver proved that the following in https://www.jstor.org/stable/1970955
Theorem: Any action of a compact Lie group on a Euclidean space has contractible orbit space.
My question is that this ...
- 1,161
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1
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Two spectral sequences arising from a simplicial spectrum
Let $X_\bullet$ be a simplicial spectrum, and let $X = |X_\bullet|$ be the geometric realization.
Let's assume each $X_n$ is connective.
From this situation, we can form two filtrations on $X$: the ...
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do all two manifolds admit a three-colorable triangulation?
A triangulation of a two-manifold $M$ is three-colorable if all vertices of the triangulation can be colored red, green, or blue without any two adjacent vertices having the same color.
My question: ...
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Complex cobordism and integrable systems
In Jack Morava's paper On the complex cobordism ring as a Fock representation, it was remarked right at the beginning that complex cobordism may play a role in the theory of integrable systems. In ...
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The algebras and coalgebras of the homology functor
My question is very simple, but I suspect far from the intuition with which singular homology is introduced.
Consider singular homology as a functor
$$H_n : {\sf Top}\times{\sf Ab} \to \sf Ab$$
This ...
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Are there known minimal models for the cohomology of semisimple Lie algebras?
My student and I recently found a cute construction of a minimal model for the cohomology of a Lie algebra $\mathfrak{g}$. This is a "minimal model" in the sense that it is a minimal chain-...
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When does homology preserve inverse limits of Eilenberg-MacLane spaces?
Let $... \to G_3 \to G_2 \to G_1$ be an inverse system of abelian groups and $G$ the limit of the system. By a theorem of Goerss the integral homology of the Eilenberg-MacLane space $K(G,n)$ for $n &...
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On the existence, for $\langle X,R\rangle$ a finite presentation of a group $G$, of an exact sequence of $\mathbb{Z}G$ modules
From this Q&A -- for $\langle X,R\rangle$ a finite presentation of a group $G$, there is an exact sequence of $\mathbb{Z}G$ modules
$$0\rightarrow\pi_{2}(Z)\rightarrow \mathbb{Z}G^{\oplus R}\...
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For a map $x: S^0 \to X$, $J(X,x) \otimes Y = 0$ iff $x \otimes 1_Y$ is nilpotent?
Let $X$ be a spectrum, and let $x : S^0 \to X$ be a map. Let the stable James construction $J(X,x)$ denote the free $E_1$ ring on the $E_0$-ring $(X,x)$. It is computed as the colimit of the $\Delta^{...
- 61.5k
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The James and Morse filtrations of homotopy groups
Denote by $JX$ the James construction on a path connected, well-pointed space $X$. This space is filtered by subspaces $X=J_1X\subseteq\dots\subseteq J_nX\subseteq\dots\subseteq JX$ and is the domain ...
- 5,016
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Homotopy classes of homeomorphism vs. Homotopy classes of a biholomorphism
This is a more detailed question about my first question Representation theory and topology of Teichmüller space, I asked there how to understand:
$$T_{g}\hookrightarrow Hom(\pi_{1}({S}),PSL_{2}(\...
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On the initiality of the inclusion from the simplex category to the paracycle category
Thm B.3 of Nikolaus and Scholze shows that the natural inclusion $\Delta \to \Lambda_\infty$, from the simplex category to the paracycle category, is an initial functor, i.e. satisfies the hypotheses ...
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Proposition 4.3.8 Qing Liu about flat morphisms of schemes
I have a problem with a detail of Qing Liu's proof of Proposition 4.3.8 (pag. 137 of "Algebraic Geometry and Arithmetic Curves").
The statement is:
Let $Y$ be a scheme having only a finite ...
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Is this $\mathbb C$-fibration over compact Riemann surface trivial?
I have a question about a complex manifold $M$ and a holomorphic submersion $p : M \to S$ to a compact complex curve satisfying the following conditions:
$p^{-1}(x)$ is biholomorphic to $\mathbb{C}$ ...
- 619
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0
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167
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Geometric realization of crossed square
Given a crossed square of groups, you can "totalize" it and get a 2 crossed module in the sense of Conduché "Modules croisés généralisés de longueur 2", then you can apply his ...
2
votes
1
answer
284
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Hypercover and hyper descent
I am trying to understand the descent condition using hypercovers. The condition says that a hyper cover of a scheme $X$ is a simplicial set $Y_{\bullet}$ that satisfies the condition $Y_n\rightarrow ...
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Relation between trivial tangent bundle $\Leftrightarrow$ certain characteristic classes of tangent bundle vanish [closed]
We know that
framing structure means the trivialization of tangent bundle of manifold $M$.
string structure means the trivialization of Stiefel-Whitney class $w_1$, $w_2$ and half of the first ...
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