# Questions tagged [at.algebraic-topology]

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

6,033
questions

**7**

votes

**1**answer

198 views

### Where should I search for computations of group cohomology rings of not-too-complicated finite groups?

A computation I'm trying to make uses as input the cohomology rings of not-too-complicated finite groups in low
degrees, and I'd like to determine where to search for preexisting computations.
...

**1**

vote

**0**answers

63 views

### 3rd Cohomology of a fibration with Flag varieties as fibers

Let $X$ be a smooth projective rational variety over $\mathbb{C}$, let $Y$ be another smooth projective variety, both of dimension bigger than 2, and let $\pi : Y \rightarrow X$ be a locally trivial ...

**4**

votes

**1**answer

110 views

### Reference request for a calculation of a Toda-bracket of spheres

For example a calculation of $S^5\overset{\eta}{\to} S^4\overset{2}{\to} S^4\overset{\eta}{\to} S^3$.
I know this exists in Toda's book.
However, I'm looking for a fairly elementary proof that is ...

**9**

votes

**1**answer

131 views

### Fundamental group under Gelfand duality

Gelfand duality states that the functor of continuous functions $C(-)$ from compact Hausdorff topological to commutative $C^*$-algebras is an equivalence of categories. In other words, all topological ...

**5**

votes

**2**answers

358 views

### Example of a space X exhibiting the Landweber non-exactness of the additive formal group over the integers?

Landweber exactness gives a criterion for when a complex oriented cohomology theory $E$ can be recovered from the formal group law over $E_{*}$ determined by the complex orientation. That is it gives ...

**2**

votes

**2**answers

162 views

### A question on the nature of the vortex number

In the Yang-Mills-Higgs (also called magnetic Ginzburg-Landau) model in the plane the energy has the form
$$
E(A,\phi)=\int_{\mathbb{R}^2}\left(|(d-iA)\phi|^2+\frac{1}{2}F_{jk}F_{jk}+\frac{1}{4}(1-|\...

**18**

votes

**1**answer

550 views

### Which cohomology classes are detected by tori?

Given a space $X$, I am looking for a characterization of classes $\alpha \in H^n(X;\bf Q)$ such that there is a map $f\colon T^n \to X$ so that $f^{\ast} \alpha$ pairs non-trivially against the ...

**4**

votes

**0**answers

202 views

### Homotopy type of a $4$-manifold with finite fundamental group (paper by S. Bauer)

I'm studying Stefan Bauer's paper
The homotopy type of a 4-manifold with finite fundamental group. In: tom Dieck T. (eds) Algebraic Topology and Transformation Groups. Lecture Notes in Mathematics,...

**7**

votes

**0**answers

141 views

### Duality of Hopf algebras and duality of spectra

Let $S$ be the sphere spectrum, and for $X$ a topological space, let $S(X)$ be the mapping spectrum from the free loop spectrum on $X$ to the sphere spectrum. This is an $E_\infty$ ring spectrum (also ...

**12**

votes

**1**answer

330 views

### Homotopic classification of maps $M \to \mathbb{RP}^n$ where $M$ is a compact orientable $n$-dimensional manifold

It is well known that if $M$ is a compact orientable $n$-dimensional manifold, then $[M, \mathbb{S}^n] \cong \mathbb{Z}$, i.e the maps are classified by their degree.
What is known about $[M, \mathbb{...

**0**

votes

**0**answers

78 views

### Question on the proof of locally contractibility of CW complex

I'm very new to algebraic topology. I'm currently reading Hatcher, and get stuck at proposition A.4. which states that CW complexes are locally contractible.
Suppose $x \in X^m-X^{m-1}$, I ...

**3**

votes

**1**answer

230 views

### Stacks as local quotients or via atlases

If one looks up the definition of a Deligne--Mumford stack or an Artin stack, one usually finds something like:
A DM (resp. Artin) stack is a stack $X$ satisfying [insert condition in the diagonal ...

**-1**

votes

**0**answers

51 views

### Intersection of two trajectories lying interior a sector [on hold]

In what publication (textbook or task book) the following statement is proved?
Let $e_1$, $e_2$ form a basis of $R^2$, $T_1>0$, $T_2>0$ and
$x_1 : [0,T_1]\rightarrow R^2$,
$x_2 : [0,T_2]\...

**2**

votes

**1**answer

133 views

### Quotient of normalizers is the fixed points of a homogeneous space

Let $G$ be a finite group, with subgroups $A \leqslant H$. Is there an isomorphism of $N_G A$-sets (or just sets)
$$ N_G A / N_H A \cong (G/H)^A ?$$
This dropped out of some calculations of Mackey ...

**7**

votes

**1**answer

164 views

### Formal group law for oriented bordism

From this answer I learned that the coefficient ring $MSO^{*}[1/2]$ of oriented bordism with 2 inverted supports an odd formal group law and is infact the universal such ring. Is there a reference/...

**2**

votes

**1**answer

136 views

### Model structure for fiberwise Bousfield localization

I think the following should be in the literature but couldn't find it.
Recall that around the 1970's Bousfield described the $R$-localization $EX$ of any space $X$, for $R$ a fixed ring. The ...

**1**

vote

**1**answer

92 views

### The currents homology of closed orientable surfaces and Birkhoff Ergodic theorem?

I just know very little about currents but I need vexedly. Thanks for your help.
Let $M$ be a closed orientable surface and $I=(f_t)_{t\in[0,1]}$ be an isotopies from identity to $f$. Suppose that $\...

**6**

votes

**0**answers

267 views

### Example of a commutative, cocommutative, $p$-torsion Hopf algebra which is dualizable but not self-dual?

Let $C$ be a symmetric monoidal category with split idempotents, and let $H$ be a Hopf algebra object in $C$. If $H$ is dualizable as an object of $C$, then $H^\vee = L \otimes H$ for some $\otimes$-...

**1**

vote

**0**answers

95 views

### Quasi-isomorphism and homotopy equivalence between diagram of chain complex

Given two diagram of chain complex $C_*,D_*$(with field coefficient) and a map $f$ between them, assume there are filtration $F_1^*,F_2^*$ on this two diagram and $f$ respect filtration. Let $Gr(C_*),...

**2**

votes

**1**answer

160 views

### Multiplicity of indecomposable stable summands of $BG^{\wedge}_p$

I am reading the article Homotopy stable classification of $BG^{\wedge}_p$ by Martino-Priddy. Let $P_u$, $P_v$ be $p$-subgroups of a finite group $G$, such that $P_u\leq x^{-1}P_v x$ for some $x\in G$,...

**22**

votes

**5**answers

1k views

### Does anyone know a basepoint-free construction of universal covers?

Let $X$ be a real manifold (for simplicity). The standard construction of the universal cover $\varphi: \widetilde{X} \longrightarrow X$ involves fixing a basepoint $p \in X$ and considering homotopy ...

**7**

votes

**1**answer

434 views

### Geometric intuition behind this chain homotopy

My question has to do with the chain homotopy that appears in Lee's Introduction to Topological Manifols and Rotman's Introduction to Algebraic Topology proofs that the inclusion
$$C_\bullet^\mathcal{...

**3**

votes

**1**answer

123 views

### Two paths to the boundary with no holes in between

Let $X\subset \mathbb{R}^2$ be open connected (and let's say bounded), let $x\in X$ and $y\in\partial X$ be such that there is a Jordan curve $\gamma:[0,1]\to X\cup\{y\}$ such that $\gamma(0)=x$ and $\...

**4**

votes

**2**answers

593 views

### Canonical reference for Chern characteristic classes

I'm a little uncertain about the definitions for
1) Chern roots
2) Chern classes
3) Chern characters
From perusing several discussions, I gather that if one correlates the nomenclature with that ...

**11**

votes

**0**answers

178 views

### Is there a model category describing shape theory?

Is there a nice model structure on some category of topological spaces compatible with shape theory? In particular, weak equivalences should induce isomorphisms on sheaf cohomology.
As an example, ...

**7**

votes

**1**answer

198 views

### A Thom spectrum from “doubled” tautological bundles?

Let us consider real vector bundles, and denote by $V_k$ the tautological bundle $V_k\to BO(k)$. From
$$
Thom(V\oplus 1_{\mathbb{R}}\to X)=\Sigma Thom(V\to X)
$$
and from $j^*V_{k+1}=V_k\oplus1_{\...

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64 views

### Poset of degree zero bundles

Let’s assume we are working on a smooth projective curve $X$. For any vector bundle $E$ on $X$, the poset of non-trivial proper sub-bundles of $E$ is in bijection with the poset of non-zero proper sub-...

**3**

votes

**1**answer

185 views

### Does the projectivization of a vector bundle have sections?

Let $E \to X$ be a homomorphic vector bundle over a projective variety $X$. Does $\mathbb{P}(E)$ always have holomorphic sections? If not what is the obstruction?

**6**

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81 views

### Is the deformation along flow lines a simple homotopy equivalence?

Let $(M,g)$ be a compact, smooth $n$-manifold with boundary $\partial M$ and let $f: M \to [a,b]$ be a Morse function, whose critical points are interior and which satisfies $f^{-1}(b) = \partial M$.
...

**5**

votes

**1**answer

130 views

### Integral (co)homology of $SU/SO$

I would like to know the integral cohomology of $SU(\infty)/SO(\infty)$ (to degree 5 or 6, say.)
Mimura-Toda says $H^*(SU/SO,\mathbb{Z}/2\mathbb{Z})=\wedge[w_2,w_3,\ldots]$ where $w_i$ is a pullback ...

**3**

votes

**0**answers

83 views

### Existence of a homotopy cofibration with particular cofibre

Let $X$ be a finite, pointed, CW-complex. For $n \geq 1$, define the $n$-th fat wedge:
$$FW^n(X) := \bigcup_{i=1}^n X \times X \times... \times * \times ... \times X \subseteq \prod_{i=1}^nX$$
where ...

**2**

votes

**0**answers

192 views

### What does one call a Morse function whose nondegenerate condition is relaxed?

In robotics, navigation functions are of utmost interest to plan a path from an initial location $q_0$ to a target location $q^t$. A function $f: \mathbb{R}^2 \rightarrow \mathbb{R}$ is a navigation ...

**6**

votes

**1**answer

391 views

### Classification of fibrations $\Bbb S^k\longrightarrow\Bbb S^d\longrightarrow B$

Does there exist a complete classification of all fiber bundles $\Bbb S^k\longrightarrow\Bbb S^d\longrightarrow B$, that is, fibrations of $\smash{\Bbb S^d}$ with each fiber homeomorphic to $\smash{\...

**9**

votes

**1**answer

189 views

### Topological Spin manifolds in dimension 4

In his ICM Adress at Nice (Proceedings of the International Congress of Mathematicians Nice, September, 1970, Gauthier-Villars, editeur, Paris 6 e ,1971, Volume 2, pp. 133-163.),
Robion Kirby ...

**4**

votes

**1**answer

226 views

### Geometric intuition for Mather's cube theorem

Mather's cube theorem for the category of topological spaces says that given a homotopy-commutative cube:
If one pair of opposite faces are homotopy pushouts and the two
remaining faces adjecent ...

**7**

votes

**1**answer

183 views

### Delooping the quotient space $SU/SU(n)$

Let $SU$ denote the infinite unitary group. Does the quotient space $SU/SU(n)$ admit a delooping $X$? One could also ask that this space $X$ sit in a fiber sequence $BSU(n)\to BSU\to X$, but this is ...

**3**

votes

**0**answers

65 views

### When are there continuous families of pull-backs of a discrete cohomology class of a compact Lie group?

Let $\mathcal{G}$ be a compact Lie group. Then define $H^n(\mathcal{G},\mathrm{U}(1))$ to be the cohomology of measurable cochains $\mathcal{G}^{\times n} \to \mathrm{U}(1)$ with the usual coboundary ...

**2**

votes

**1**answer

229 views

### Tables for stable homotopy groups of $\mathbb{R}P^{\infty} \wedge \mathbb{R}P^{\infty}$

I am wondering if there is any computation of stable homotopy groups of $\mathbb{R}P^{\infty}\wedge \mathbb{R}P^{\infty}$ in low dimensions? I would be very grateful for any reference.

**2**

votes

**0**answers

134 views

### Factoring a topological universal cover

Let $X$ be a compact, connected, locally path-connected, and semilocally simply connected topological group with $\pi_1(X) \cong \mathbb{Z}$.
Let $u\colon \widetilde{X}\to X$ be its topological ...

**5**

votes

**1**answer

180 views

### Group of parallelizations of $M^3$ finitely generated?

Let $M^3$ be a compact orientable 3-manifold. Then $TM$ is trivial and let's go ahead and fix a trivialization $\tau : M \times \mathbb{R}^3 \to TM$. Then given a map $g : (M, \partial M) \to (SO(3),...

**3**

votes

**0**answers

177 views

### Fundamental group of boundary divisor of $\overline{\mathcal{M}}_3$

Let $\overline{\mathcal{M}}_3$ be the moduli stack of stable genus $3$ curves defined over $\mathbb{C}$. The stack $\overline{\mathcal{M}}_3$ has
(1) A locally closed substack $D_0$, parameterizing ...

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**0**answers

178 views

### Zero in colimit of sheaves category

This question is motivated by showing that the category $\mathbf{Sheaves} (X)$ from the open subset excluding the empty set category to the category of abelian group $\mathbf{Ab}$ has enough injective ...

**3**

votes

**1**answer

170 views

### Homology modules and symmetry

Let $B$ be a cellular (simplicial, semi-simplicial etc) complex having $\mathbb{Z}^n$-symmetry (that is, there is a free action of $\mathbb{Z}^n$ on $B$, commuting with the boundary operators) and let ...

**6**

votes

**1**answer

217 views

### Zero differential in Serre spectral sequence for configuration spaces

I moved this question from Math StackExchange.
I am trying to compute homology of $Conf(n, \mathbb{R}^2)$ - ordered configurations of $n$ points on the plane - using Serre spectral sequence. I know ...

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**0**answers

162 views

### Status of the problem: find $X \subset \mathbb{R}^3$ such that $\pi_1(X)$ is not torsion-free [duplicate]

Which is the status of the following problem?
Does it exist (path-connected) $X \subseteq \mathbb{R}^3$ such that $\pi_1(X)$ is not torsion-free?
Specifically, my questions are:
Is it still an ...

**5**

votes

**1**answer

277 views

### Delooping a fibration sequence with loopspace fiber and finite CW complexes

The following question is somewhat similar to a previous one on MathOverflow, except that my application does not directly involve Eilenberg-MacLane spaces $K(G,n)$, and so I don't see the immediate ...

**15**

votes

**2**answers

335 views

### Which elements of $H^2(M,\mathbb{Z}/2)$ are the $w_2(E)$ for a real bundle $E$?

Any element of $H^1(M,\mathbb{Z}/2)$ is the $w_1(E)$ of a real line bundle $E$ over $M$.
I wonder how to characterize (probably using the Steenrod squares) which elements of $H^2(M,\mathbb{Z}/2)$ are ...

**1**

vote

**0**answers

108 views

### Modules over quasiisomorphic DG algebras

Suppose there is a quasiisomorphism $q: A \to B$ between DG algebras. Is there some reasonable description of induced functor $q^*: B-mod \to A-mod$? Can we say something better if it was a $B$-...

**13**

votes

**1**answer

469 views

### Explicit isomorphism $\pi_{n+1}(\mathbb{RP}^n) \cong \pi_1(\mathbb{RP}^{n-1})$

From covering space theory we know that $\pi_{n+1}(\mathbb{RP}^n) \cong \pi_{n+1}(\mathbb{S}^n)$.
From wikipedia I can notice that $\pi_{n+1}(\mathbb{S}^n) \cong \pi_1(\mathbb{RP}^{n-1})$.*
My ...

**13**

votes

**2**answers

512 views

### Does injectivity of $\pi_1(\partial U) \to \pi_1(M)$ imply injectivity of $\pi_1(U) \to \pi_1(M)$?

Let $M$ be a smooth compact manifold of dimension $n$, and let $U$ be a smooth compact manifold with boundary, of the same dimension $n$, embedded in $M$.
The embedding induces maps on $\pi_1$.
If $...