# Questions tagged [at.algebraic-topology]

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

6,177
questions

**11**

votes

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232 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 ...

**8**

votes

**0**answers

161 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 ...

**4**

votes

**0**answers

123 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 $...

**7**

votes

**0**answers

246 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 ...

**4**

votes

**1**answer

171 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\...

**4**

votes

**1**answer

303 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 ...

**3**

votes

**2**answers

191 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 ...

**8**

votes

**1**answer

200 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 ...

**6**

votes

**2**answers

460 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 ...

**8**

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205 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**

votes

**1**answer

294 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:\...

**8**

votes

**1**answer

302 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 ...

**1**

vote

**1**answer

169 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

**0**answers

95 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 ...

**1**

vote

**0**answers

76 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
$$
\{1\} = G_0 \triangleleft G_1 \triangleleft \cdots \triangleleft G_n = G
$$
...

**1**

vote

**0**answers

96 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 ...

**2**

votes

**0**answers

90 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:$\...

**2**

votes

**1**answer

51 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

**1**answer

201 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 ...

**35**

votes

**3**answers

1k 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 ...

**12**

votes

**2**answers

671 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 ...

**14**

votes

**0**answers

170 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 ...

**25**

votes

**6**answers

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 ...

**5**

votes

**1**answer

334 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.
...

**7**

votes

**0**answers

219 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:\...

**4**

votes

**1**answer

158 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 ...

**10**

votes

**3**answers

643 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 ...

**11**

votes

**3**answers

503 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

**1**answer

340 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\...

**15**

votes

**0**answers

266 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 ...

**7**

votes

**1**answer

263 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 ...

**9**

votes

**1**answer

309 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 ...

**10**

votes

**1**answer

285 views

### Homotopy extension of $E_{\infty}$-spaces

Suppose that $X$ is a connected $E_{\infty}$-space, naturally $\Omega X$ is also an $E_{\infty}$-space. Can we classify all $E_{\infty}$-extensions of $X$ by $\Omega X$ (up to homotopy). I mean the ...

**13**

votes

**1**answer

548 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}...

**7**

votes

**2**answers

282 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

**0**answers

191 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

**0**answers

126 views

### Group homomorphisms between Eilenberg-MacLane spaces

It is well-known that for two (discrete) abelian groups $G$ and $H$, the set $[K(G,n),K(H,n)]_*$ of based homotopy classes of maps between the corresponding Eilenberg-MacLane spaces is in canonical ...

**4**

votes

**1**answer

156 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 ...

**5**

votes

**1**answer

277 views

### Relation between “triangulated bordism”, MO, and $H\mathbb{F}_2$

The unoriented bordism theory $MO$ has a map to $H\mathbb{F}_2$ which is easily described for a space $X$ by pushing forward the fundamental class of a singular manifold to $H_*(X)$. Since $MO$ and $H\...

**3**

votes

**1**answer

228 views

### Homotopy class of maps into Stiefel manifolds

Motivation
Hopf theorem, asserts that $C^0$-maps $f:M^n\to \mathbb{S}^n$ from an orientable, closed n-manifold into an n-sphere are classified up to homotopy by their degree $deg(f)$.
The theorem ...

**3**

votes

**1**answer

391 views

### Homotopy of functors

Recently I have read two different proposals for a notion of homotopy between functors, and I am curious which contexts each best lend themselves to. The first comes from Ming-Jung Lee's 1972 paper ...

**20**

votes

**0**answers

450 views

### Is the mapping class group of $\Bbb{CP}^n$ known?

In his paper "Concordance spaces, higher simple homotopy theory, and applications", Hatcher calcuates the smooth, PL, and topological mapping class groups of the $n$-torus $T^n$. This requires an ...

**3**

votes

**0**answers

135 views

### Integral cohomology of compact Lie groups and their classifying spaces

Let $G$ be a compact Lie group and let $BG$ be its classifying space. Let $\gamma\colon \Sigma G \to BG$ be the adjoint map for a natural weak equivalence $G \xrightarrow{\sim} \Omega BG$. Taking ...

**10**

votes

**4**answers

867 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, ...

**2**

votes

**0**answers

56 views

### Characterization of degeneracy of spectral sequence of a fiber bundle at the second term

Let $f\colon E\to B$ be a fiber bundle of compact manifolds with fiber $F$. Assume that the push-forward $Rf_*(\underline{\mathbb{F}})$ in the derived category of the constant sheaf with coefficients ...

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99 views

### Identifying the two points of a subspace homeomorphic to a Sierpinski space

Let $X$ be a $\Delta$-generated space having a subset $A=\{a,b\}$ such that the relative topology is the Sierpinski topology with for example $\{a\}$ closed and $\{b\}$ open (the Sierpinsky space is a ...

**5**

votes

**1**answer

217 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 $H^*_T(X,\mathbb{Z})$ ...

**3**

votes

**1**answer

120 views

### Cohomology algebra of a fibration whose spectral sequence degenerates in the second term

Let $f\colon E\to B$ locally trivial bundle of 'nice' topological spaces (say finite CW-complexes) with fiber $F$. Assume also that the base $B$ is simply connected.
Assume that either the cohomology ...

**1**

vote

**1**answer

102 views

### Relative version of the cohomology product

Let $f\colon X\to Y$ be a continuous map of 'nice' topological spaces (e.g. $f$ is a smooth map of smooth manifolds; $f$ might be assumed to be proper although I am not sure it is relevant). Let $\...

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**0**answers

50 views

### Singular chain complex of balanced products

Let $\pi\subseteq\Sigma_r$ and $V$ be a right $\pi$-space. We may assume that $V$ is free, if necessary. Consider the morphism of singular chain complexes (over a fixed commutative ring)
$$f:C_*(V) \...