Questions tagged [at.algebraic-topology]

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

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14
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2answers
710 views

When is a homotopy pushout contractible?

Let $B \leftarrow A \to C$ be a span of spaces, and consider the homotopy pushout $B \cup_A C$. Question: When is $B \cup_A C$ contractible? This is a pretty open-ended question. I'm interested in ...
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0answers
88 views

About the Moore composition of paths

I work with weak Hausdorff $k$-spaces (so all spaces are $T_1$). The internal hom is denoted by $\mathbf{TOP}(-,-)$. Let $\mathcal{G}$ be the topological group of nondecreasing homeomorphisms from $[0,...
3
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0answers
46 views

Computations of Bredon homology of $S(1+\sigma)$ with Universal Coefficient S.S

What I am trying to do is to compute $\mathbb{Z}$-graded Bredon homology of $S(1+\sigma)$ over $Q\times\Sigma_2$, where $Q$ is a cyclic group of order 2 $\sigma$ is its real sign representation $\...
4
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1answer
154 views

pullback and fiber sequence

Let $A\rightarrow D\leftarrow C$ a diagram of connected pointed toplogical space where $A\rightarrow D$ is a fibration. Denote $P=A\times_{D}C$. We obtain a homotopy fiber sequence $$ \Omega D\...
8
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0answers
107 views

Computations using “Stover's spectral sequence”

In this article from 1990, Stover describes a specral sequence which converges to the higher homotopy groups of the homotopy colimit of a diagram $\underline{X}$ of topological spaces. The second ...
4
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0answers
161 views

Reference request: Weil II for derived schemes over a finite field

https://arxiv.org/abs/1401.1044 gives a survey of derived algebraic geometry, and we can define etale site for derived schemes. I am really a beginner. Is there any reference about Weil II for ...
8
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0answers
137 views

Dualizing module for $\operatorname{Aut}(F_n)$

In The complex of free factors of a free group (pdf at Hatcher's page), Hatcher and Vogtmann defined a simplicial complex $FC_n$ called the ``complex of free factors'' of the free group $F_n$. They ...
8
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1answer
268 views

Topological mapping class groups of 4-manifolds

It is a classical result of Quinn that for a simply-connected closed $4$-manifold $X$ the isometries of its intersection form are in one-to-one correspondence with $\pi_0 \text{Homeo}(X)$. (Isotopy ...
34
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2answers
991 views

Incorrect information in an old article about the Kervaire invariant

In the Soviet times there was a famous Encyclopedia of Mathematics. I think it is still familiar to every Russian mathematician maybe except very young ones, and yours truly is in possession ...
4
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0answers
119 views

Extended double 2-cocycle conditions: Mathematical structure behind?

Note: For experts, to save your time, you can just read the highlighted texts and Eqs directly. The ordinary group 2-cocycle condition: Let us remind the usual so-called homogeneous group 2-cocycle $...
5
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1answer
314 views

Sign in May’s General algebraic approach to Steenrod operations

In the first section of J. P. May’s General algebraic approach to Steenrod operations, May defines for $\pi\subseteq\Sigma_r$ an integer $q\in\mathbb{Z}$ and a commutative ring $\Lambda$, the $\Lambda\...
3
votes
2answers
184 views

K-theory of free G-sets and the classifying space, and generalization [reference-request]

Let $G$ be a finite group, $\mathcal{G}^0$ be the category of finite free $G$-sets and isomorphisms between them. Then $\mathcal{G}^0$ is a symmetric monoidal category with respect to the disjoint ...
4
votes
1answer
281 views

Postnikov tower for $S^2$

I am learning Postnikov towers from this lecture. Here is the first part of the proof that I am studying Why is true the marked statement? For example, let be $X = S^2$. To build $Y_1$ (i.e, with ...
6
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0answers
202 views

Self diffeomorphism of $S^2\times S^2$

The main question is motivated from the answer of this question https://math.stackexchange.com/questions/2481200/finite-groups-gs-which-acts-freely-on-s2-times-s2 Is it true that every self ...
8
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1answer
222 views

On the classification of $\mathrm{SU}(mn)/\mathbb{Z}_n$ principal bundles over 4-complexes

In The Classification of Principal PU(n)-bundles Over a 4-complex, J. London Math. Soc. 2nd ser. 25 (1982) 513–524, doi:10.1112/jlms/s2-25.3.513 Woodward proposed a classification of $\mathrm{PU}...
8
votes
1answer
192 views

Different definitions of formality for groups

Let $X$ be a space with fundamental group $G$. Recall that the de Rham fundamental group of $X$ is the inverse limit of the Malcev completions of the nilpotent truncations of $G$. This has a Lie ...
1
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1answer
166 views

CW complexes obtained by attaching cells not with increasing dimension

CW-complexes are defined by attaching cell with increasing dimension: you start with a set of points, then attach 1-cells, then 2-cells and so on. Why are defined so? My question is: why is it ...
6
votes
2answers
438 views

moving from sphere spectrum to finite spectrum

I am reading Hatcher's treatment of the Adam's spectral sequence. http://pi.math.cornell.edu/~hatcher/SSAT/SSch2.pdf On page 20, he states "Thus for each $i$ the groups $\pi_i(Z^k)$ are zero for all ...
7
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0answers
190 views

Does Farjoun's “fiberwise localization” have a universal property?

Let $\mathcal S_L$ be any accessible reflective subcategory of the $\infty$-category of spaces. In his book, Farjoun discusses "fiberwise $L$-localization" of a map of spaces $E \to B$, i.e. a ...
6
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1answer
279 views

Could a motivic spectrum have a “zeta function”?

I'm currently learning about zeta functions, so I apologize in advance if this is riddled with nonsense. Suppose you have a sequence $E=(E_0,E_1,...)$ of motivic spaces along with structure maps $s_i:\...
12
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3answers
1k views

How does one find vanishing algebraic cycles?

I have a question, related to what I asked before. Let's consider a smooth hyperplane section $X$ of a smooth projective variety $Y$ over $\mathbb C$. According to Weak Lefschetz theorem, cohomology ...
6
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0answers
92 views

Complete invariant of filtered chain complexes under chain homotopy equivalence

Betti numbers are a complete invariant of chain complexes of vector spaces modulo chain homotopy equivalence. Can we similarly find complete invariants for (say, finite dimensional) filtered chain ...
24
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6answers
2k views

Why are we interested in permutahedra, associahedra, cyclohedra, …?

The following families of polytopes have received a lot of attention: permutahedra, associahedra, cyclohedra, ... My question is simple: Why? As I understand, at least the latter two were ...
7
votes
2answers
583 views

How do small changes in a filtered complex affect the associated spectral sequence

I have recently been learning about spectral sequences, not in the context of any problem, but mostly out of curiosity. There are two questions that have occurred to me which I've been unable to ...
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0answers
93 views

1-connected infinity groupoids, groupoids and 1-connected spaces

I am exploring a bit the world of groupoids. What I have in mind is that infinity groupoids correspond to spaces. So my first question is the following: Consider the model category $\infty-Grpd$ of ...
1
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0answers
75 views

Nilpotency of topological groups

A group $G$ is said to be nilpotent if $G$ has a central series of finite length, that is, a series of normal subgroups $$ \{1\} = G_0 \triangleleft G_1 \triangleleft \cdots \triangleleft G_n = G $$ ...
14
votes
2answers
596 views

Counter-example to the existence of left Bousfield localization of combinatorial model category

Is there any known example of a combinatorial model category $C$ together with a set of map $S$ such that the left Bousefield localization of $C$ at $S$ does not exists ? It is well known to exists ...
2
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0answers
83 views

Surjectivity of colimit maps for topological spaces

From this post and to (co)completness of the category Top of topological spaces and continuous functions we know that for any diagram $B_i$ and an object $A$ in Top, there are natural maps of sets:$\...
13
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2answers
931 views

Is there any relationship between the topologies of the clique complex and the independence complex?

Let $G$ be a simple graph on a finite vertex set. The clique complex $X(G)$ is the simplicial complex whose faces are complete subgraphs of $G$, and the independence complex $I(G)$ is the simplicial ...
18
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4answers
800 views

Cayley graph of $A_5$ with generators $(1,2,3,4,5),(1,4,3,2,5)$

The Cayley graph of $A_5$ with two generators of order 5 seems rather complicated. What is its graph genus (orientable or non-orientable)? The best I could get by trial and error is an embedding ...
7
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0answers
215 views

Generalization of familiar theorem about singular homology to general model category

I have two questions, the first one is just wether the following statement is true or not? Is there a reference for this? The second question is maybe related, I don't know. But anyway, given $U:\...
12
votes
2answers
612 views

Is there a version of algebraic de Rham cohomology that can be used to calculate torsion classes?

Much work has gone into the construction of cohomology theories which are defined on algebraic varieties (étale, crystalline, etc.) and comparison isomorphisms between them. Say $X$ is an algebraic ...
2
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0answers
40 views

Superlevel sets of a parametrized quadratic forms

Let $N$ be an odd integer, $n\in\mathbb{N}$, and $-\frac{2T}{NR^2}\leq a\leq0$ for given $R,T\in\mathbb{R}$ with $\frac{T}{NR^2}\leq\frac{\pi}{2}$. Now consider the quadratic form $\Omega(a)=\sum_{l\...
11
votes
1answer
188 views

W H Lin's thesis and Hopf subalgebras of the Steenrod algebra

If $B$ is a subalgebra of $A$, you can ask whether the $B$-module structure on $B$ can be extended to give an $A$-module structure on $B$. W H Lin, in his 1973 PhD thesis at Northwestern, showed that ...
3
votes
1answer
146 views

Bott periodicity homeomorphisms for spaces of Clifford extensions

I am trying to prove the following statement of real Bott periodicity, on the level of actual spaces of Clifford module extensions (i.e., not equivalence classes of modules). Let $W = \mathbb{R}^{\...
12
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0answers
157 views

Hauptvermutung for non-manifolds

The Hauptvermutung proposes the following: if two finite simplicial complexes are homeomorphic then they are PL-homeomorphic, meaning that they have a common refinement. People are mostly interested ...
11
votes
3answers
480 views

A binary operation on vector bundles that adds Chern classes?

Let $E$ and $F$ be two complex vector bundles over a space $X$. There's a fairly well-known binary operation called the direct sum, written $E\oplus F$, which has the property that its first Chern ...
5
votes
1answer
333 views

Künneth theorem for G-space

Let $X$ and $Y$ be a right $G$-space and a left $G$ space, respectively, where $G=H \rtimes K$, $H$ a finite group and $K$ a compact Lie group. Moreover, suppose that the $G$-action on $X$ is free. ...
10
votes
3answers
619 views

Classifying space for fibrations with Eilenberg-MacLane space fibers and nontrivial fundamental group actions

Let $A$ be an abelian group and let $n \geq 2$. For any connected CW complex $X$, it is standard that a fibration $f\colon E \rightarrow X$ whose fibers are homotopy equivalent to a $K(A,n)$ is ...
4
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1answer
153 views

Representing spaces of $\infty$-stacks

In The stable moduli space of Riemannsurfaces: Mumford’s conjecture, Madsen and Weiss introduce the representing space $|\mathcal{F}|$ of a sheaf of sets $\mathcal{F}$ on the site $\mathscr{X}$ of ...
0
votes
1answer
65 views

finding Morse index for the following functional

not sure if this meets the standards here in this forum. I was dealing with the following functional $I(u)=\frac{1}{p}\int_{\Omega}|\nabla u|^{p}dx-\frac{1}{q}\lambda\int_{\Omega}|u|^qdx$ for $p \geq ...
7
votes
2answers
277 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
1answer
257 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 ...
5
votes
1answer
214 views

Equivariant cohomology algebra of toric variety

Let $X$ be a complex projective and smooth toric variety of complex dimension $n$. It is acted by the real torus $T=(S^1)^n$. Is it true that the $T$-equivariant cohomology $H^*_T(X,\mathbb{Z})$ ...
10
votes
4answers
841 views

Reading list for Equivariant Cohomology

I was applying equivaraint cohomology, in particular in the symplectic setting, for some time, but I feel like I am missing some nice books/course notes/articles. Could you advise me some literature, ...
15
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0answers
258 views

Can the intermediate Chern classes be expressed as Euler classes?

General question: We know that the top Chern class $c_n(\xi)$ of an $n$-dimensional complex vector bundle $\xi$ is its Euler class, while the first Chern class, $c_1(\xi)$, is the Euler class of its ...
9
votes
1answer
303 views

When is Thom isomorphism a ring map?

Let $R$ be an $E_{\infty}$-ring spectrum and $B$ be an $E_\infty$-space. Suppose we have an $E_\infty$-map $$ f: B \to BGL_1(S^0)$$ such that the composite $$f_R: B \to BGL_1(S^0) \to BGL_1(R) $$ is ...
7
votes
0answers
189 views

Positive instances of the Eilenberg-Ganea conjecture with families

The original Eilenberg-Ganea conjecture, which remains unsettled, can be formulated as: any (discrete) group $G$ of cohomological dimension $\operatorname{cd}(G)=2$ has geometric dimension $\...
4
votes
1answer
151 views

Defining chain complexes for cellular spaces with local coefficients

Let $X$ be a nice finite cellular complex (a regular CW complex or a simplicial one), equipped with a local system $\mathcal{F}$ of free rank 1 modules over some Noetherian commutative ring $R$. What ...
13
votes
1answer
532 views

Calculation of $H^{10}(K(\mathbb{Z}, 3); \mathbb{Z})$

I was trying to calculate $H^q(K(\mathbb{Z}, 3); \mathbb{Z})$ for some $q$ with the Serre spectral sequence associated to the fibration $K(\mathbb{Z}, 2) \to PK(\mathbb{Z}, 3) \simeq * \to K(\mathbb{Z}...