Questions tagged [at.algebraic-topology]

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

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7
votes
1answer
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
0answers
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
1answer
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
1answer
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
2answers
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
2answers
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
1answer
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
0answers
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
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0answers
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
1answer
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
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0answers
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
1answer
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
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0answers
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
1answer
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
1answer
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
1answer
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
1answer
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
0answers
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
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0answers
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
1answer
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
5answers
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
1answer
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
1answer
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
2answers
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
0answers
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
1answer
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_{\...
1
vote
0answers
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
1answer
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
votes
0answers
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
1answer
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
0answers
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
0answers
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
1answer
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
1answer
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
1answer
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
1answer
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
0answers
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
1answer
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
0answers
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
1answer
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
0answers
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 ...
1
vote
0answers
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
1answer
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
1answer
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 ...
1
vote
0answers
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
1answer
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
2answers
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
0answers
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
1answer
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
2answers
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 $...