# Questions tagged [at.algebraic-topology]

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

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### Transgressive elements in Hopf algebra spectral sequence

Let $E^r$ be a commutative Hopf algebra homology spectral sequence over $\mathbb{Z}/p$, i.e. such that every sheet is a commutative Hopf algebra.
I am struggling with proving (probably simple) fact -...

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115 views

### Duality in Hopf algebras and Milnor-Moore paper

I am going through Milnor and Moore - On the structure of Hopf algebras (MSN) (I have already posted one question on that, another one is coming).
My question is about Proposition 4.9, more ...

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### On the classification of $\mathrm{SU}(mn)/\mathbb{Z}_n$ principal bundles over 4-complexes

In
The Classification of Principal PU(n)-bundles Over a 4-complex, J. London Math. Soc. 2nd ser. 25 (1982) 513–524, doi:10.1112/jlms/s2-25.3.513
Woodward proposed a classification of $\mathrm{PU}...

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**1**answer

127 views

### Homology of universal abelian cover of a manifold

If one define the universal abelian covering $M_0$ of a manifold $M$ as the abelian covering (i.e. normal covering with abelian group of deck transformations) that covers any other abelian covering, ...

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224 views

### When localisation preserves isomorphy of homotopy groups

Let $E$ be a generalized cohomology theory. Let's agree that $E$ satisfies property $(*)$ if for any two finite CW complexes with isomorphic homotopy groups their $E$-localizations also have ...

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160 views

### A user guide to the theory on Corks

I am trying to digest the meanings of the corks from the both:
algebraic topology
and
geometry topology
perspectives.
Studying corks is important for understanding the exotic phenomenon of 4-...

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148 views

### Compatible algebraic Spanier-Whitehead dual

Let me first ask an intuitive version of the question:
Let $Sp$ be the homotopy category of spectra. Let $E$ be a ring spectrum. Let $$D:Sp \to Sp$$ be the Spanier-Whitehead dual functor (maybe we ...

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303 views

### Reference on complex cobordism

I am trying to study a little of algebraic cobordism and I lack background from the classic setting. Hence, I am looking for a textbook or expository writing covering the basics of complex cobordism.
...

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365 views

### “Free” Hopf algebra

I am reading through the Dyer and Lashof's paper "Homology of Iterated Loop Spaces". They are quoting the following theorem:
If $A$ is a connected, free, associative, commutative, primitively ...

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79 views

### Explicit map with Hopf invariant two in any even dimension

It is known that the Hopf invariant for maps from $\mathbb{S}^{4n - 1} \to \mathbb{S}^{2n}$ is nontrivial (and captures the rational homotopy of the spheres). For $n = 1$, the Hopf fibration provides ...

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137 views

### Blocksum induces a unital H-space structure on the space of Fredholm operators

Fix a complex separable infinite-dimensional Hilbert space $H$. It is well known that the space of (bounded) Fredholm operators $Fred(H)$ with the norm topology is a classifying space for the ...

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### Cohomology of the space of generic immersion maps of surface into 3-space

In "Local Invariants of Mappings of Oriented Surfaces Into 3-Space", V.Goryunov classified singular maps of surface into 3-space and considered their resolution and local invariants.
It is natural ...

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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\...

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86 views

### When are cohomology operations determined by their action on coefficients?

It is well-known that K-theory operations are determined by the action on coefficients, but I don't know the right way to prove this fact, nor a reference for the same. On the other hand, clearly this ...

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### Madsen-Tillmann spectrum $MTE$ of the group $E$ which is defined in Freed-Hopkins's paper

In Freed-Hopkins's paper, the group $E(d)$ is defined to be the subgroup of $O(d)\times\mathbb{Z}_4$ consisting of the pairs $(A,j)$ such that $\det A=j^2$, where $\mathbb{Z}_4=\{\pm1,\pm\sqrt{-1}\}$ ...

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### homotopy MC element

A homotopy MC element is the Linfty analog of a Maurer-Cartan
element for a Lie algebra. Where is anything written about
homotopy MC elements as perturbations of strict MC elements?

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**1**answer

256 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\, \...

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### Modern Algebraic Geometry and Analytic Number Theory

I am currently discovering the algebraic geometry of Grothendieck. I have the impression that this theory, which leads to categories, schemas, topos etc. alone can encompass all modern mathematics (...

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### Singular homology: Lifting simplices gives map in homology

Let $X$ be a space, $k=k_1+\dotsb+k_r$ and let $G:=\mathfrak{S}_{k_1}\times\dotsb\times \mathfrak{S}_{k_r}$ act freely on the right on $X$. Fix a commutative ring $R$ and another space $Y$.
Then the ...

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**1**answer

153 views

### Explicit chain homotopy for the Alexander-Whitney, Eilenberg-Zilber pair

Let $A$ and $B$ be simplicial abelian groups, and let $N_\ast(-)$ denote the normalized chain complex functor. Let
$$AW_{A,B}\colon N_\ast(A\otimes B)
\longrightarrow N_\ast(A)\otimes N_\ast(B)$$
and
...

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**1**answer

146 views

### Homotopy groups of Diffeomorphisms of punctured d-dim ball

Let $\mathbb{D}_n^d$ be the $d$-dimensional unit ball with $n$ punctures. I am interested in the groups of orientation-preserving diffeomorphisms $Diff(\mathbb{D}_n^d)$ that fix the punctures. In ...

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### References and resources for (learning) chromatic homotopy theory and related areas

What references and resources (e.g. video recorded lectures) are available for learning chromatic homotopy theory and related areas (such as formal geometry)?

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130 views

### “Left Brace Module”

Let $A$ be an algebra over the brace tree operad and $M$ a module over some base ring.
Is there a good notion of a "left brace module" over a brace algebra?
I do not think the definition of a module ...

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141 views

### Is the tensor product of pretriangulated dg-categories a pretriangulated dg-category?

In "Grothendieck ring of pretriangulated categories", Bondal, Larsen and Lunts define a product of perfect (pretriangulated with Karoubian homotopy category) dg-categories as $A\bullet B:=Perf(A\...

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111 views

### Is the natural geometric realization of symmetric simplicial sets homotopically correct?

Recall that the category of $\sigma Set$ of symmetric simplicial sets is the category of presheaves on $\Sigma$, the category of finite nonempy sets and all functions. The inclusion $v: \Delta \to \...

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146 views

### Ring spectra whose homotopy is the ring of ordinary differential operators

Consider $R=\mathbb{C}[[x]][\frac{d}{dx}]$ as a graded ring (assign $x$ zero weight, and $\frac{d}{dx}$ some non-zero weight). Are there some $A_n$-ring spectra naturally arising in homotopy theory ...

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### Contiguity for simplicial maps between simplicial sets

I begin by recalling the definition of contiguous simplicial maps between abstract simplicial complexes:
Definition. Two simplicial maps $\varphi,\psi\colon K \to L$ are said to be contiguous if for ...

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### Is every orientable $I$-bundle over an orientable surface trivial?

Is every orientable $I$-bundle over an orientable surface $F$ trivial, $I \times F$? Is this also the case for vector all bundles?
Similarly, is every orientable $I$-bundle over an nonorientable ...

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236 views

### Does the cubical nerve preserve weak equivalences of simplicial sets?

The finite cartesian powers of $\Delta[1]$ form a cubical object in simplicial sets, inducing a "cubical nerve" functor $N_\Box: sSet \to Set^{\Box^{op}}$.
$N_\Box$ is a right Quillen equivalence, ...

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360 views

### Comparing real topological K-theory and algebraic K-theory

Let $R$ be a commutative unital ring and let $i$ be a non-negative integer such that $K^i_{alg}(R)$ is finitely generated abelian group. Is it possible that there does not exist weak homotopy type of ...

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157 views

### Generalizing the formula between Wu class and the Steenrod square

I know that on the tangent bundle of $M^d$, the corresponding Wu class and the Steenrod square satisfy
$$
Sq^{d-j}(x_j)=u_{d-j} x_j, \text{ for any } x_j \in H^j(M^d;\mathbb Z_2) .
\tag{eq.1}$$
...

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### $n$-point map $S$ from $k$-manifold $X$ to $\mathbb{R}^k$

It is known that for every continuous map $M:S^1\to \mathbb{R}$ there are infinitely many double points. Also by the Borsuk-Ulam theorem, this is true for each continuous map $N:S^n\to \mathbb{R}^n$, $...

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311 views

### A question about Poincare duality

Let $k$ be a field. Let $C$ be a small category and assume that for every $i\geq 0$ we have a functor $H^i:C\rightarrow FinDimVect_k$. Assume that there is a function $dim:Obj(C)\rightarrow \mathbb{Z}...

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126 views

### Leray-Serre spectral sequence for projective bundles

Let $\mathcal{E} \rightarrow X$ be a complex vector bundle of rank $r+1$ and let $F=\mathbb{P}^r \rightarrow E = \mathbb{P}\mathcal{E}\rightarrow X$ be the associated projective bundle. We know that ...

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308 views

### Are framed manifolds cubulatable?

Let's say an $n$-manifold is cubulated if it is glued out of cubes $[0,1]^n$ in a way that looks locally like the standard cubulation of $\mathbb R^n$. For instance, the face $[0,1]^{k-1} \times \{1\} ...

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103 views

### Twisted spin-bordism invariant and a possible Postnikov square from $d=2$ to $d=5$

This is a follow up more advanced question following Twisted spin bordism invariants in 5 dimensions. We follow the definitions in the earlier post.
I had discussed my computation of
$$
\Omega_5^{...

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**1**answer

155 views

### Cofibrations of functors

Let $\cal M$ and $\cal N$ be model categories, $S,T:\cal M\to N$ functors, and $\alpha:S\to T$ a natural transformation. Say that $\alpha$ is a <blank> cofibration if for any cofibration $i:A\to B$...

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### Twisted spin bordism invariants in 5 dimensions

[Note]: My question will be a bit long. So, first, thank you for your careful reading, generous comments, helps and answers, in advance!
The spin $G$-bordism invariant can be twisted in the way that ...

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60 views

### Alexander-Whitney for cyclic objects

What is known about the extension of the AW map from simplicial to cyclic Abelian groups? Homological perturbation theory implies there is an A infinity-like sequence of maps, but is it known ...

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### Example of a finite group $G$ with low dimensional cohomology not generated by Stiefel-Whitney classes of flat vector bundles over $BG$

In Stiefel-Whitney classes of real representations of finite groups, J. Algebra 126 (1989), no. 2, 327–347, Gunawardena, Kahn and Thomas dealt with the question, whether the cohomology ring $H^...

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### Is there a natural homotopy inverse to the map $∣Sing(X)∣\rightarrow X$

Let $Sing:C \rightarrow SSet$ be the functor sending a CW complex to its singular complex (a simplicial set).
Let $∣-∣: SSet \rightarrow C$ be the geometric realization functor.
For every $X$ we ...

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536 views

### How do topological automorphic forms fit into homotopy theory and what makes them interesting?

Topological automorphic forms (TAF) were introduced by Mark Behrens and Tyler Lawson in 2007, being to Shimura varieties what topological modular forms (TMF) is to the moduli stack of elliptic curves.
...

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208 views

### Does the Hopf construction work for $S^0$?

Given a $0$-connected $H$-space $X$, we can prove that left and right multiplication are (weak) homotopy equivalences. This allows us to construct the Hopf fibrations for $S^1$, $S^3$ and $S^7$. I ...

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### homotopy classes of maps from $S^3/\Gamma$ to $X$

Given a finite subgroup $\Gamma <SO(4)$ and a topological space $X$ whose $i$th homotopy group $\pi_i(X)$ are known for any $i$, is there any way to compute $[S^3/\Gamma,X]$?
Here $[S^3/\Gamma,X]$...

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### Why not a Stacks project for Homotopy Theory?

The lack of resources bridging the gap between what one finds in Hatcher's algebraic topology text and modern research on homotopy theory has been brought several times before on MathOverflow [1, 2, 3]...

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205 views

### Classifying space of semidirect product of groups

Assume that $G$ and $H$ are two groups and $G\rtimes _\phi H$ is their semidirect product. My question is, how does the classifying space $B(G\rtimes_\phi H)$ of $G\rtimes _\phi H$ relate to $BG$ and $...

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527 views

### Does the Grothendieck construction satisfy Fubini's thorem

Suppose we are given a functor
$F:(A\times B)^{\operatorname{op}}\to \operatorname{Set}$.
It's well-known that the Grothendieck construction in this case evaluates as
$\int_{A\times B}F = (A\...

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272 views

### Model structure on the category of topological groups

Consider the category $TopGr$ of topological groups. I want to know that this is a model category (can one understand its model structure by understanding a model structure on the category of enriched ...

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### Approximation of homotopy avoiding a point in $\mathbb{R}^3$

For a proof that $\mathbb{R}^3\setminus \mathbb{Q}^3$ is simply connected using Baire category theorem I need to approximate an homotopy $H : [0,1]\times \mathbb{S}^1 \to \mathbb{R}^3$ from a loop $\...

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85 views

### Torus bundle over spheres

I was wondering what is the classification of all torus bundles over spheres? That is, to classify the fibration
$$
T^m \hookrightarrow M \to S^n.
$$
It is well known that if $n=1$, all fibrations ...