# Questions tagged [at.algebraic-topology]

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

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votes

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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

**2**answers

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

**2**answers

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

**0**answers

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

**1**answer

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

**0**answers

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

**2**answers

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**

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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 ...

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**0**answers

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

**1**answer

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 ...

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votes

**1**answer

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

**3**answers

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

**2**answers

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

**1**answer

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? ...

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votes

**1**answer

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

**0**answers

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 $X_1\cap X_2$ (...

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votes

**0**answers

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

**2**answers

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 ...

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vote

**0**answers

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

**1**answer

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

**0**answers

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 ...

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votes

**0**answers

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

**1**answer

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

**0**answers

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

**1**answer

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

**1**answer

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) ...

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votes

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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,...

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votes

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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 ...

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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:...

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votes

**1**answer

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\...

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votes

**1**answer

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

**1**answer

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 ...

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votes

**1**answer

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 ...

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votes

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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 ...

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votes

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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, ...

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votes

**4**answers

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 ...

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votes

**2**answers

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?

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votes

**1**answer

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 ...

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votes

**0**answers

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....

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votes

**1**answer

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\...

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votes

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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 ...

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vote

**0**answers

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 ...

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vote

**0**answers

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-...

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vote

**1**answer

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 ...

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votes

**1**answer

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 ...

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votes

**0**answers

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 ...

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votes

**1**answer

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

**2**answers

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 $SO(n + 1$)...

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vote

**0**answers

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{...

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votes

**0**answers

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 "...