Questions tagged [at.algebraic-topology]
Homotopy theory, homological algebra, algebraic treatments of manifolds.
8,228
questions
-2
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1
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214
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Action of $\mathbb{Z}/3\mathbb{Z}$ on $P^{1}$ [closed]
I am reading from the book Topics in Galois theory by Serre.
I have the following question ,
take $G=\mathbb{Z}/3\mathbb{Z}$. The group $G$ acts on $P^1$ by
$$\sigma x\;=\;1/(1-x)$$
where $\sigma$ ...
1
vote
1
answer
541
views
Vector bundles and equivariant vector spaces
It seems commonly accepted that most of the results of equivariant geometry for vector spaces yield analog result for vector bundles.
In so far as I understand it, the reason for that is the ...
2
votes
0
answers
160
views
Chern character (form) of a Gauss-Manin connection
Consider the trivial fibration $\mathbb{T}^2\to\mathbb{S}^1$, where $\mathbb{T}^2$ is the two-torus. Denote by $\mathbb{C}\to\mathbb{T}^2$ the trivial line bundle over $\mathbb{T}^2$, and equip it ...
3
votes
0
answers
287
views
Manifolds and CW-complexes
Let us consider a category $C$ formed by topological spaces and continuous functions (or by smooth manifolds and smooth functions). We consider the morphism category $C_{2}$. An object of $C_{2}$ is a ...
8
votes
1
answer
456
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A model structure on semi-simplicial algebraic Kan complexes?
By an algebraic semi-simplicial kan complex I mean a semi-simplicial set (i.e. a presheaf on the category of finite ordered sets and injective order preserving maps), which is a Kan complex (in the ...
4
votes
0
answers
284
views
Galois categories and the connected components functor
In stacks 0BMQ, a Galois category is defined to be a functor $F:\mathsf C\longrightarrow \mathsf{FinSet}$ such that $\mathsf C$ is finitely bicomplete, every object ...
4
votes
2
answers
367
views
cohomology of configuration space of punctured variety
Given a smooth projective variety $X$ of dimension $l$, we denote with $F(X,n)$ the configuration space of points
$$
F(X,n):=\{(x_{1}, \dots, x_{n})\in X^{n}\: : \: x_{i}\neq x_{j}\text{ for each }i,j ...
1
vote
1
answer
258
views
cohomology ring of homogenous manifold
Let $[d_1^{t_1}, \dotsc, d_s^{t_s}]$ be a partition of a positive integer $n$, i.e., $\sum d_r t_r = n$. I want to know the de Rahm cohomology ring of the following types of homogenous spaces :
$$
G/H ...
3
votes
0
answers
182
views
Quaternionic projective bundle in complex Grassmann bundle
"What is the fundamental class of the projective bundle of lines of a quaternionic bundle in the Grassmann bundle of 2-planes of the underlying complex bundle?"
In Quaternionic projective space in ...
5
votes
1
answer
448
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Quaternionic projective space in complex Grassmannian
I would like to consider the quaternionic projective space $\mathbb{PH}^{n-1}\subset\mathbb{G}_2(\mathbb{C}^{2n})$ as a subvariety of the Grassmannian of complex 2-planes.
For a real vector $e\in\...
2
votes
1
answer
268
views
Intersection of two curves is not Cohen Macaulay
Let be $R=\mathbb{C} \lbrace x,y,z \rbrace$ the formal series ring and let $f_{1},f_{2},f_{3} \in R$ be nonzero elements of $R$.
(a) Consider the varieties $M:=V(f_{1},f_{2})$ and $N:=V(f_{2},f_{3})$ ...
1
vote
0
answers
81
views
Homotopy invariant deletions of open faces of simplicial complexes
Given a finite simplicial complex (as a topological space) $\Delta$ and a face $\tau$, suppose we delete the interior of $\tau$ (a point if $\tau$ is a vertex, otherwise homeomorphic to an open ball ...
0
votes
0
answers
50
views
Intersection of ideals corresponding to simplicial complexes at different points?
Suppose I have two simplicial complexies $\triangle_1$ and $\triangle_2$. Consider their Stanley-Reisner ideals $I(\triangle_1)$ and $I(\triangle_2)$. I want to get their intersections when they meet ...
7
votes
3
answers
964
views
Is $S^1\vee S^1$ an Eilenberg-Mac Lane Space to a Homotopy Purist?
I believe that $S^1\vee S^1$ is the Eilenberg-Mac Lane space $K(\mathbb{Z}\ast\mathbb{Z},1)$. One can prove this by constructing its universal cover and observing that it is contractible.
My question ...
10
votes
2
answers
625
views
Seeking very regular $\mathbb Q$-acyclic complexes
This question was raised from a project with Nati Linial and Yuval Peled
We are seeking a $3$-dimensional simplicial complex $K$ on $12$ vertices with the following properties
a) $K$ has a complete $...
2
votes
0
answers
104
views
Multiplicativity of the analytic index (or of kernel bundle)
What I want to ask is the multiplicativity of the analytic index of a family of Dirac operators.
In the single operator case the analytic index of elliptic operator is multiplicative. This is proved ...
4
votes
0
answers
331
views
Baum Connes Conjecture [closed]
I have recently decided on a topic for my master thesis. I want to compare the Baum Connes conjecture as it is formulated in topology to the conjecture as it is formulated in functional analysis. I ...
7
votes
3
answers
887
views
Lifting symmetries to the universal cover
If $X$ is a connected topological space with universal cover $p: \tilde{X} \to X$, I believe any homeomorphism $f : X \to X$ can be 'lifted' to a homeomorphism $\tilde{f} : \tilde{X} \to \tilde{X}$. ...
3
votes
1
answer
267
views
$\mathbb Z_2$-homotopy type of a $k$-connected, $(k+1)$-dimensional simplicial complex with a free involution
If $K$ is a finite, $k$-connected, $(k+1)$-dimensional simplicial complex then, by the theorems of Hurewicz and Whitehead, $|K|$ is homotopy equivalent to a point or to a wedge of $(k+1)$-dimensional ...
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 ...
3
votes
0
answers
251
views
Can ring spectra be thought of as some sort of operad in $Top$?
It is a result of May's work on operads that the homotopy category (or $\infty$-category, if you prefer) of connective spectra is equivalent to a full subcategory of the category of representations of ...
3
votes
1
answer
464
views
Equivalent definition of a homotopy of functions
It is well known that given $X,Y$ arbitrarily topological spaces, $I$ the unit interval, and continuous functions $f, g : X \rightarrow Y,$ a homotopy between the functions is a continuous function $H ...
2
votes
1
answer
603
views
Homotopy type of an oriented, closed, simply connected manifold
It is well known that every closed, oriented, simply-connected four-manifold $M$ is homotopy equivalent to a CW-complex consisting on a 0-cell, a wedge of two spheres and a 4-cell.
I was wondering ...
9
votes
1
answer
806
views
explicitly embedding a simplicial $d$-complex into $\mathbb{R}^{2d+1}$, or algorithms for doing so
A classical result in topology for which I can't find a reference for is that a simplicial complex $K$ of dimension $d$ with $n$ vertices can be linearly embedded into $\mathbb{R}^{k}$ when $k=2d+1$. ...
4
votes
1
answer
569
views
Manifolds whose diffeomorphism group has the homotopy type of a manifold itself
I have a very stupid question.
Let $M$ be a closed smooth manifold. In particular cases the homotopy type of the diffeomorphism group $Diff(M)$ can be very pathological. For example, in the case of $M=...
7
votes
4
answers
727
views
Topological invariance of Stiefel-Whitney classes for open smooth manifolds
It is well known that Stiefel-Whitney classes are homotopy invariant for closed smooth manifolds. But in the case of open manifolds even $w_1$ is not a homotopy invariant (take just open cylinder and ...
6
votes
1
answer
1k
views
Kunneth formula for de Rham cohomology twisted by flat vector bundle
It is well known that for two manifolds $M$ and $N$ (let's say they are compact), the Kunneth formula says that $H(M\times N)=H(M)\otimes H(N)$, where $H$ denotes de Rham cohomology with complex ...
9
votes
1
answer
528
views
Infinite families in stable homotopy groups
I will be very grateful for any advise or reference on the following.
1- How much is known about infinite families in ${_2\pi_*^s}$, the $2$-component of the stable homotopy ring?
2- How much is ...
4
votes
0
answers
179
views
'Noether normalization' for finite group schemes
Throughout let $p$ be a prime, and let $k$ be a field of characteristic $p$.
Let $G$ be a compact Lie group. Such a $G$ can always be embedded as a closed subgroup of $SU(n)$ for some $n$. This ...
11
votes
1
answer
878
views
Computational complexity of computing simplicial homology
Is there any literature regarding the fastest known algorithm to compute the homology groups of a simplicial complex (on n vertices)? What about computing the fundamental group? The context is to tell ...
0
votes
1
answer
223
views
Inverse Galois Problem ; Galois group of some branched cover of $P^{1}$ defined over $\mathbb{Q}$
I was trying to read a paper on Inverse Galois problem . I understands what the inverse Galois problem is. It asks if every finite group is the Galois group of some extension of the rationals.
The ...
8
votes
0
answers
247
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What is known about maps between loop spaces of Spheres? - Reference request
What is know in general about the maps $\Omega^rS^n\rightarrow\Omega^sS^m$ between loop spaces of Spheres, or, perhaps to phrase it better, the groups $[\Omega^rS^n,\Omega^sS^m]$ for various values ...
11
votes
1
answer
830
views
Is the natural isomorphism $|FX_\bullet| \cong F|X_\bullet|$ lax symmetric monoidal?
Let $\mathcal{V_1}$ and $\mathcal{V_2}$ be cocomplete symmetric monoidal categories, each endowed with a cosimplicial object $\Delta^\bullet=\Delta^\bullet_{\mathcal{V}_i}:\Delta \to \mathcal{V}_i$. ...
8
votes
0
answers
687
views
Homology of inverse limits over inverse systems more complicated than towers
Most textbooks discussions of homology of inverse limits of chain complexes consider only “towers,” i.e. inverse systems indexed by the natural numbers. I’d like to find a reference that explains what ...
1
vote
1
answer
274
views
$G_1 \rtimes G_2 \cong G_3 \implies BG_1 \times BG_2 \simeq BG_3$? $G_1 \times G_2 \cong G_3 \implies BG_1 \times BG_2 \simeq BG_3$? [closed]
As summarized in the title, suppose there is an isomorphism between $G_1 \times G_2$ and $G_3$, is it always true that $BG_1 \times BG_2$ is homotopy equivalent to $BG_3$? If it is not always true ...
6
votes
0
answers
77
views
Examples of maps with nontrivial Hopf invariant but Lusternik-Schnirelmann category of the cofiber doesn't increase?
Let $A$ be a suspension and $X$ be a space with Lusternik-Schnirelmann category $n$ and let $\alpha: A\to X$. It is easy to see that the cofiber $C_\alpha$ has $\mathrm{cat}(C_\alpha) \leq n+1$. One ...
6
votes
2
answers
794
views
Can a Morse function define a unique structure on a closed manifold?
I was thinking about the doubt that if $M$ and $N$ are closed manifold and if there exists two Morse function $f$ and $g$ respectively on $M$ and $N$ with the following property that they both have ...
2
votes
0
answers
101
views
Does attach-one-cell have a stable homotopy transfer?
Specifically, I am thinking of attaching one ordinary cell to an ordinary space in your favourite convenient category of spaces; so, given a cofiber sequence
$$ \mathbb{S}^k \to_c X \to_p X', $$
on ...
15
votes
0
answers
350
views
Existence of flat connections via characteristic classes, for nice groups
I have two questions about what I write below (which honestly seems pretty elementary).
Is it true (more or less)?
Is there a clean reference that I can cite.
Let $G$ be a compact Lie group, $M$ a ...
6
votes
1
answer
579
views
Criterion for a equalizer to be a homotopy equalizer in spaces
Let $f,g\colon X\rightarrow Y$ be maps between spaces.
I am looking for criteria for the equalizer of $f$ and $g$ to be a homotopy equalizer and I am happy to get answers for whatever model category ...
10
votes
1
answer
341
views
Cohomology operations on unoriented cobordism
In unoriented cobordism there exist stable cohomology operations looking similar to Steenrod squares (they were used by Quillen to compute the unoriented cobordism ring with its formal group law ...
7
votes
1
answer
468
views
Can the Hochschild cochain complex be given the structure of a "homotopy BV algebra"?
In a 1993 letter, Deligne posed the following (paraphrased from a paper of Gerstenhaber and Voronov's):
Conjecture (Deligne). The Hochschild cochain complex $CC^*(A)$ of an associative algebra ...
2
votes
0
answers
119
views
Can relative homotopy groups be represented as relative homology groups of some Moore complex?
Daniel Kan defined a combinatorial version of the homotopy group $\pi_n(X)$ of a simplicial set $X$ as the $(n-1)$st homology of the (non-commutative) Moore complex $\tilde{G}(X)$, where $G_iX$ is ...
6
votes
4
answers
2k
views
List of invariants that distinguish homotopy equivalent non-homeomorphic spaces
It is written on wikipedia article (https://en.wikipedia.org/wiki/Analytic_torsion) that the Reidemeister torsion is the first invariant that could distinguish between spaces which are homotopy ...
5
votes
1
answer
676
views
Under what condition is a fiber bundle cobordant to the trivial bundle?
Let $E$ be the total space of a fiber bundle with base $B$ and fiber $F$, where $B$ and $F$ are smooth manifolds.
Under what condition is $E$ unoriented cobordant to $B\times F$?
And what happens ...
3
votes
0
answers
231
views
How far can one reconstruct the boundary of a manifold M given its interior $int M$? [duplicate]
Suppose I keep in my pocket a manifold with boundary $M$ , and I provide you access to $int M := M \setminus \partial M$ up to homeomorphism/diffeomorphism. What can you deduce about $\partial M$? can ...
4
votes
0
answers
284
views
Various definitions of the odd Chern character form
I am asking this question from my possibly defected memory, so the things below may not be accurate.
I want to know how many different definitions of the odd Chern character form using differential ...
4
votes
0
answers
422
views
Topology on $\mathcal{C}(X,Y)$ to work with homotopy
We know that the compact open topology on $\mathcal{C}(X,Y)$ is a good choice for topology on the set of continuous maps, but this seems really efficient, both naively and with respect to existence of ...
7
votes
1
answer
222
views
Computing homology of subvarieties of Euclidean spaces by persistent homology
Let $M$ be a submanifold of the Euclidean space $\mathbb{R}^n$. Let $G$ be a finite group acting on $M$ freely. I want to compute the homology (or even the cohomology ring) of $M/G$.
Suppose the ...
2
votes
2
answers
413
views
For a universal covering morphism $p:E\rightarrow B$, how to prove $E$ connected implies $B$ connected?
Definition. An arrow $\alpha:A\rightarrow B$ in $\mathsf C=\mathsf{Fam}(\mathsf A)$ is said to be a covering morphism if there exists an effective descent morphism $p:E\rightarrow B$ that splits it, i....