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Questions tagged [at.algebraic-topology]

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

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47 views

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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0answers
86 views

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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1answer
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, ...
4
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1answer
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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0answers
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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0answers
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 ...
10
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1answer
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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2answers
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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0answers
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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0answers
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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61 views

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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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\...
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0answers
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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0answers
151 views

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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0answers
106 views

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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1answer
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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4answers
1k views

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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0answers
82 views

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 ...
6
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1answer
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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1answer
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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2answers
682 views

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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1answer
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 ...
6
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1answer
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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0answers
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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0answers
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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0answers
36 views

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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0answers
102 views

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 ...
7
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1answer
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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0answers
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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0answers
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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0answers
60 views

$n$-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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1answer
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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1answer
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 ...
5
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1answer
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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1answer
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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1answer
201 views

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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0answers
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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1answer
223 views

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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0answers
169 views

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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0answers
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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1answer
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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85 views

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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2answers
2k views

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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1answer
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 $...
12
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1answer
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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1answer
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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1answer
136 views

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 $\...
3
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0answers
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 ...