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Questions tagged [at.algebraic-topology]

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

3
votes
3answers
345 views

Maps from 2-Torus to SO(3)

Could someone please point me to a reference for topologically nontrivial maps from 2-Torus to SO(3), and how they are classified? [I'm a physicist, so a simple explanation would be useful]
2
votes
2answers
241 views

Hairy ball theorem for odd-dimensional spheres

Let $\mathbb S^n$ be the $n$-sphere: $$\mathbb S^n=\left\{x \in \mathbb R^{n+1}: \left\|x\right\|=1\right\}.$$The hairy ball theorem can be formulated as follows: If $n$ is even and $f\,\colon\, \...
5
votes
2answers
188 views

Zero surgery on a Seifert fiber space

I have a problem with understanding what is a neighbourhood of a singular fiber in a Seifert fibered space coming from the zero surgery. For me a 3-manifold $Y$ is a SFS if it has a decomposition into ...
3
votes
0answers
62 views

Eilenberg-Moore spectral Sequence calculation

I want to use the cohomology Eilenberg-Moore spectral sequence to calculate the cohomology of the fibre of the map $$ S^{n} \to \Omega S^{n+1}. $$ Question 1: Is anyone aware of any references for ...
5
votes
1answer
500 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
0answers
141 views

Fundamental group of a topological group

It is well know that the fundamental group of a path connected topological group is abelian. Suppose that $G$ is a connected topological group and let $Ab(G)$ the abelianization of the topological ...
8
votes
2answers
285 views

Iterated free infinite loop spaces

Let $Q$ denote $\Omega^\infty\circ \Sigma^\infty$ the free infinite loop space functor. Given some space $X$, we see that $QX$ carries all the stable homotopy information about $X$. Naturally I wanted ...
18
votes
2answers
450 views

Vanishing of characteristic numbers vs vanishing of characteristic classes

A famous result by Thom states that Oriented Bordism classes are determined by characteristic numbers; specifically, two closed manifolds are orientedly bordant if and only if they have the same ...
3
votes
0answers
64 views

Chirality and Anti-Chirality of links in 3 and in 5 dimensions

We know there is a chiral knot which is a knot that is not equivalent to its mirror image. It is well known in the mathematical field of knot theory: https://en.wikipedia.org/wiki/Chiral_knot My ...
1
vote
1answer
88 views

The table reduction morphism of operads from Barratt-Eccles to Surjection

The Barratt-Eccles operad $E$ in the category of simplicial sets is obtained by applying the nerve functor to the canonical operad $\{\Sigma_n\}_{n>0}$ in groups. Berger-Fresse defined here an ...
8
votes
1answer
211 views

Use of Steenrod's higher cup product and the graded-commutativity

In Steenrod's ``Products of Cocycles and Extensions of Mappings (1947),'' which derives [Theorem 5.1] $$ \delta(u\cup_{i} v)=(-1)^{p+q-i}u\cup_{i-1}v+(-1)^{pq+p+q}v\cup_{i-1}u+\delta u\cup_{i}v+(...
9
votes
3answers
671 views

Who discovered this definition of Stiefel-Whitney classes?

I would define Stiefel-Whitney classes as the pullbacks of generators of $H^*(BO, \mathbb{Z}/2)$ under a classifying map, and I gather this is a pretty common definition. However, the book "...
21
votes
2answers
525 views

Does Spin cobordism vanish in dimension $4k-1$?

For the purposes of a remark in a paper in preparation, I would like to know if anyone can confirm that $\Omega^{spin}_{4k-1} = 0$. In the Atiyah-Patodi-Singer paper, Spectral asymmetry and ...
3
votes
1answer
252 views

Non-abelian cohomologies

Let A be a non-commutative algebra and let X be some geometric space (such as a topological space or an algebraic variety or scheme). Is there a notion of cohomology ring of X with coefficients in A? ...
16
votes
1answer
777 views

What was a “cusp” to Hurwitz in 1892?

Let $d\in\mathbb{N}$ be squarefree. Let $\mathcal{O}_d$ be the ring of integers of $\mathbb{Q}(\sqrt{-d})$. Let $\Gamma_d=\mathrm{PSL}_2(\mathcal{O}_d)$. Let $\mathcal{H}^3$ be the upper half-space ...
12
votes
0answers
38 views

How many upper sets in this decomposition of finite posets

Let $X$ be a finite poset. If $$X = X_1 \cup X_2$$ where $X_1$ and $X_2$ are strict upper sets, then a lot of properties of $X$ can be inferred from the smaller posets $X_1, X_2$ and $X_1\cap X_2$ (...
2
votes
0answers
66 views

Why is a homotopy limit of a cosimplicial space not the ordinary limit?

I've been trying to compute a homotopy limit of a cosimplicial object $X: \Delta \to \mathscr{M}$, where $\mathscr{M}$ is some simplicial model category, we may take it to be spaces. I'm wondering ...
7
votes
2answers
362 views

What are the advantages of simplicial model categories over non-simplicial ones?

Of course, there are general results allowing one to replace a model category with a simplicial one. But suppose I want to stay in my original non-simplicial model category (say for some reason I'm a ...
1
vote
0answers
97 views

Fibre transfer of $\mathbb{S}^1$-bundles

Let $p:E\to X$ and $p':E'\to X'$ be two orientable $\mathbb{S}^1$-bundles. Denote their homological transfers by $p_!:H_*(X)\to H_{*+1}(E)$ resp. $p'_!$. Now let $(u,f)$ be a bundle morphism ($u:E\to ...
6
votes
1answer
100 views

Two models for the classifying space of a subgroup via the geometric bar construction

Let $H$ be a topological group which is a subgroup of two other topological groups $G$ and $G'$. It follows (from Rmk 8.9 in May - Classifying spaces and fibrations (MSN, free)) that there exist weak ...
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vote
0answers
29 views

$\omega$-nilpotent cover of a recurrent surface

Theorem. Any $\omega$-nilpotent cover of a recurrent Riemannian manifold is Liouville. $\omega$-nilpotent ($\Gamma=\bigcup_{i=1}^{\infty}Z_{i}$, $Z_{i}$ normal in $\Gamma$, where $Z_{n+1}$ maps to ...
3
votes
0answers
39 views

Tensor product of an L-infinity algebra with the cochains on the 1-simplex

I would like to understand the $L_\infty$ structure on the tensor product of an $L_\infty$ algebra (over $\mathbb{R}$) $L$ with the normalized cochains on the one-simplex $N^*(\Delta^1)$. This latter ...
8
votes
1answer
282 views

Left Bousfield localization without properness, what is known?

I'm interested in the existence of several example of left Bousfield localization of model categories that are not left proper (nor simplicial). I'm relatively convince that I can construct all those ...
8
votes
0answers
165 views

$\Gamma$-sets vs $\Gamma$-spaces

I know that every $\Gamma$-space is stably equivalent to a discrete $\Gamma$-space, i.e. a $\Gamma$-set. For example, Pirashvili proves, as theorem 1.2 of Dold-Kan Type Theorem for $\Gamma$-Groups, ...
3
votes
1answer
107 views

Locally trivial fibration over a suspension

For $X$ a paracompact space, I am trying to classify all locally trivial fibration with base the suspension $SX = X \times [-1,1]\, /\, (X \times \{-1\} \cup X \times \{1\})$, and fiber-type a space $...
3
votes
1answer
204 views

Divisibility of a divisor

Let $X$ be a smooth complex projective curve and $f \colon X \to Y$ an étale Galois cover, whose Galois group $G$ is finite and of order $r$. For any $g \in G$, define $$\Delta_g = \{(x, \, g \cdot x) ...
4
votes
0answers
109 views

References on $HZR$ theory

Are there references available on $HZR$ theory ? I found on ncatlab that this is "genuinely $\mathbb Z/ 2\mathbb{Z}$-equivariant cohomology version of ordinary cohomology" Found nothing on wikipedia,...
3
votes
0answers
56 views

Globalising fibrations by schedules

In a paper in Fund Math 130.2 (1988): 125-136. http://eudml.org/doc/211719 Dyer and Eilenberg give an account of the local-global theorem for fibrations by proving a "Schedule Theorem" that, given a ...
2
votes
0answers
246 views

Topological data of $K3\times T^{2}$

I need some help in order to clarify some topological data of a $K3\times T^{2}$ Calabi Yau manifold in which $K3$ part has been obtained as a resolution of a $T^{4}/ \mathbb{Z_{2}}$ orbifold . EDIT:...
10
votes
1answer
714 views

Homology of the fiber

Let $f:X\rightarrow Y $ be a fibration (with fiber $F$) between simply connected spaces such that $H_{\ast}(f):H_{\ast}(X,\mathbb{Z})\rightarrow H_{\ast}(Y,\mathbb{Z})$ is an isomorphism for $\ast\...
8
votes
1answer
235 views

Different definitions of the linking number

Assume that $$ \iota_1:\mathbb{S}^k\to\mathbb{R}^n, \quad \iota_2:\mathbb{S}^\ell\to\mathbb{R}^n, \quad k+\ell=n-1, $$ are two embeddings of spheres with disjoint images $\iota_1(\mathbb{S}^k)\cap\...
3
votes
1answer
52 views

Concerning the Spanier group relative to an open cover

Let $\mathcal{U} = \{ U_i \; |\; i\in I \}$ be an open covering of $X$‎. Spanier defined $\pi (\mathcal{U}‎, ‎x)$ to be the subgroup of $\pi_1 (X‎, ‎x)$ which contains all homotopy classes having ...
2
votes
1answer
137 views

$X \rtimes Y \simeq X \vee (X \wedge Y)$ for $X$ a co-H-Space

I have asked the below question on MathSE (with a 200 point bounty) but have yet to receive an answer there, and so am trying here. I am happy to remove it if it is nevertheless decided that this ...
15
votes
3answers
290 views

Vietoris-Rips complex homology of a higher degree than the ambient dimension

Assuming we have a set of points $X=\{x_1,..,x_n\}$, all in $\mathbb{R}^d$, and construct the Vietoris-Rips-Complex $V_\epsilon (X)$ for some distance parameter $\epsilon > 0$. Is it possible to ...
8
votes
2answers
222 views

What is the total square on the dual Steenrod algebra?

The dual Steenrod algebra ($p=2$) has generators $\xi_n$ and these have conjugates that are often labeled $\zeta_n$. I am curious about the left and right actions of the Steenrod algebra on its dual, ...
7
votes
4answers
679 views

Characteristic classes detecting nontrivial fiberwise homotopy of sphere bundles

I am looking for characteristic classes of vector bundles (either real or complex) with values in generalized multiplicative cohomology theories such that: i) they vanish if the bundle of unit ...
12
votes
2answers
671 views

Examples of odd-dimensional manifolds that do not admit contact structure

I'm having an hard time trying to figuring out a concrete example of an odd-dimensional closed manifold that do not admit any contact structure. Can someone provide me with some examples?
5
votes
1answer
197 views

Homotopy classes of maps between special unitary Lie groups

I am sorry to mislead the notations: $SU(n)$ should be replaced by $PSU(n)$. I will reformulate it now. We consider the special unitary Lie group $SU(n)$. Then its center is $\mathbb{Z}_n$ and we ...
12
votes
0answers
316 views

When is a map of topological spaces homotopy equivalent to an algebraic map?

My question is simple, but I don't expect there are any simple answers. Let $X$ and $Y$ be a pair of schemes, and let $X(\mathbb{C})$ and $Y(\mathbb{C})$ denote their respective spaces complex points....
7
votes
1answer
271 views

The connective $k$-theory cohomology of Eilenberg-MacLane spectra

Consider the connective $K$-theory spectrum $ku$. Let $H\mathbb{Z}$ be the Eilenberg-MacLane spectrum and $H\mathbb{F}_p$ be the mod-$p$ Eilenberg-MacLane spectrum. Is it known what $ku^{*}(H\...
7
votes
0answers
176 views

Higher homotopy groups of Calabi-Yaus

Is something known about the higher homotopy groups of Calabi-Yau threefolds? For example, one of the easiest CYs is the quintic, defined as the anticanonical divisor in $\mathbb{CP}_4$. What are its ...
1
vote
0answers
63 views

Homotopy classes of maps between special unitary Lie group. Correction [duplicate]

An hour ago I asked a question (under the same title) but I used a wrong notation. Here is the improved version. We consider a special unitary Lie group $SU(n)$. Then its center is $\mathbb{Z}_n$ and ...
1
vote
0answers
79 views

Action of the symmetric group on connected sums of manifolds (minus a disk)

Let $M$ be a connected compact topological $n$-dimensional manifold without a boundary and with a CW-structure $M= \bigcup M^i$. We have that $$ (\#^g M)\smallsetminus D^n \simeq \bigvee_{i=1}^gM^{n-...
1
vote
1answer
182 views

Homotopy of group actions

Let $G$ be a topological group and $X$ be a topological space. Let $\alpha$, $\beta:G\times X\to X$ be two group actions. We say that these two actions are homotopic actions if there is a continuous ...
12
votes
1answer
333 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
votes
0answers
49 views

Generators of the fundamental group of the solid torus [migrated]

I have a solid torus $T$ and a curve (or knot) $C$ that winds 2-times around the torus (parallel to the longitude). Can I say that $[C] = 2 \in \mathbb{Z} \cong \pi_1(T)$, that is the curve represents ...
6
votes
1answer
220 views

Is the Thomason model structure the optimal realization of Grothendieck's vision?

In Pursuing Stacks, Grothendieck uses the category $Cat$ of small categories to model spaces. A recurring theme is the question of whether there is a Quillen model structure supporting this homotopy ...
18
votes
2answers
748 views

What is the status of the 4-dimensional Smale Conjecture?

4-dimensional Smale conjecture claims the following: The inclusion $SO(5)$ → $SDiff(S^4)$ is a homotopy equivalence. or Does $Diff(S^4)$ have the homotopy-type of $O(5)$ ?. The inclusion $SO(n + 1$)...
1
vote
0answers
84 views

Classifying map of a simple circle bundle

Let $\mathbb{K}_0 \subset \mathbb{K}$ be two tori (subtori of $(S^1)^n$). We suppose that $\mathbb{K}_0$ is obtained from $\mathbb{K}$ by the following procedure: consider, on the lie algebra $\text{...
3
votes
0answers
180 views

Do smooth manifolds admit unique cubical structures?

It seems to me that a smooth manifold should admit the structure of a cubical complex by Morse theory, since handle attachments seem to be perfectly cubical maps. Is this cubical structure "...