# Questions tagged [at.algebraic-topology]

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

1,406
questions with no upvoted or accepted answers

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### Homotopy pullbacks of simplicial sets; Joyal vs Kan-Quillen model structures

I am interested in comparison of homotopy pullback squares in the category of simplicial sets with respect to Joyal' model structure and Quillen's one.
Suppose we are given a (strict) pullback square
...

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261 views

### Is there a citeable source for generators and relations of simplicial sets?

Simplicial sets can be specified using generators and relations,
in complete analogy with groups, rings, etc.
More precisely, a system of generators of relations for a simplicial set
consists of a ...

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145 views

### Generalize $\mathbb Z/p$-space for irrational $\alpha$

A free $\mathbb Z/p$-space is a topological space $X$ with an action $\varphi$ such that $\forall x\in X$ $\varphi^p(x)=x$ but $\varphi(x)\ne x$.
I would like to generalize this notion from $\frac 1p$ ...

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243 views

### Simplicial model for $\Sigma^{\infty}_+ \Omega^{\infty}X$?

Let $Sp^{\Sigma}$ be the category of symmetric spectra, and let $Sp^{\Sigma}-alg$ be the associated category of augmented commutative algebras. These are simplicial categories.
A question for experts:...

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337 views

### (Torsion in) homology of free nilpotent groups

It is known that for free $k$-step nilpotent group on $r$ generators $N(r, k)$ its integral homology is torsion-free in degrees $\leq 3$ (obvious for 1 and 2, Igusa&Orr computations for 3). ...

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327 views

### Triangulation of manifolds with corners

Let's begin with some definitions:
A (smooth) manifold with corners is a Hausdroff (and second countable if you want) space that can be covered by open sets homeomorphic to $\mathbb R^{n-m} \times \...

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137 views

### Countability assumption for good covers in Bott-Tu

In chapter II of their text Differential Forms in Algebraic Topology, Bott and Tu construct the Čech-De Rham complex with regards to an open covering indexed by some ordered and countable indexing set....

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165 views

### Eilenberg-Moore spectral sequence in etale cohomology?

Let $X,Y \rightarrow S$ be schemes over an algebraically closed field $k$. (Actually I'm interested in the case where they are stacks, but I'll ignore that for now.) The vague form of my question is: ...

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152 views

### How many components are there in the space of “generic” planar N-gons? (Mnev's revenge)

Call an ordered $N$-tuple of points in the Euclidean plane ${\mathbb R} ^2$ "in general position" if no three points of the points in the set are collinear. As a function of $N$ how many components ...

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397 views

### In terms of sheaf cohomology, what does Bott & Tu's relative de Rham cohomology $H^\bullet(f)$ compute for $f: S \to M$ a smooth map?

Given a map $f: S \to M$ of smooth manifolds, Bott & Tu define on page 78 a complex by $\Omega^q(f)=\Omega^q(M) \oplus \Omega^{q-1}(S)$ and $d(\omega, \theta)=(d\omega, f^*\omega - d\theta)$ where ...

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201 views

### Reference request: $H^* X$-module structure on the Mayer–Vietoris coboundary

Is the following presumed folklore fact written anywhere?
Let $E^*$ a multiplicative cohomology theory. Then the coboundary map in the Mayer–Vietoris sequence of an excisive triad $(X;U,V)$ preserves ...

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194 views

### Almost Poincaré duality

Let $M^n$ be a connected, closed manifold. It has Poincaré duality with $\mathbb{Z}/2$ coefficients $H^k(M;\mathbb{Z}/2)\cong H_{n-k}(M;\mathbb{Z}/2)$, induced by cap product with the fundamental ...

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279 views

### Equivariant obstruction theory done wrong

Let $G$ be a finite group, and let $p:E\to B$ be a $G$-fibration with fibre $F$. The correct framework for studying equivariant sections of $p$ is Bredon cohomology. With the right definitions, most ...

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393 views

### Motivic Galois theory and Betti realizations?

Why Motivic Galois groups are defined with Betti realizations? (In fact Absolute Galois groups can be defined in this way (with Betti realizations), why they are so related?).

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226 views

### Do symmetric monoidal groupoids model all connective spectra?

Thomason showed that symmetric monoidal categories model all connective spectra. But can it be done with groupoids? In order for this to be the case, the group completion prices process must be very ...

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174 views

### The Steenrod Algebra of the Dihedral Group $D_{2n}$, $n=0 \pmod{4}$

As the tile suggests, I'm interested in computing the action of the Steenrod Algebra on $H^*(D_{2n};\mathbb{Z}_2)$, for $n=0 \pmod{4}$. Let us start with some definitions/facts:
$$D_{2n} = \langle x,y ...

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229 views

### Isomorphisms between minimal $A_\infty$-algebras having identical $k$-truncations

Let $A_m =(A,0,m_2,m_3,\dots)$ and $A_n=(A,0,n_2,n_3,\dots)$ be two $A_\infty$-structures on a vector space $A$. Assume that
i) $A_m$ and $A_n$ are isomorphic, and
ii) $A_m$ and $A_n$ have the same ...

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333 views

### Explicit description of a subgroup of the braid group $\mathsf{B}_2(C_2)$

This is related to my previous MathOverflow question Fundamental group of $\mathrm{Sym}^2(C_g)$ minus the diagonal.
Let $C_2$ be a smooth curve of genus $2$ and $X:=\mathrm{Sym}^2(C_2)$ its second ...

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186 views

### Chern-Simons form and Rarita-Schwinger operator

The Rarita-Schwinger (RS) operator naturally generalizes the Dirac operator and in Physics it describes particles with spin-3/2.
I was wondering if there exists any reference concerning the ...

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563 views

### When does algebraic K theory behave like a cohomology theory

Let $\mathbb{F}$ be a field. Let $K(\mathbb{F})$ be its algebraic (Quillen) $K$-theory spectrum. Let $X$ be a (nice, finite CW) topological space and let $\text{Rep}\Omega(X)$ be the DG category of (...

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232 views

### Which nice subcategories of $\mathsf{Top}$ are locally cartesian closed?

For a class $\mathcal{J}$ of topological spaces, let $\mathsf{Top}_\mathcal{J}$ denote the category of $\mathcal{J}$-generated spaces, i.e. those spaces $X$ such that $U\subseteq X$ is open iff $f^{-1}...

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320 views

### When do contributions of $\pi_*(L_{K(n)}S^0)$ to $\pi_*(S^0)$ stabilize?

I apologize if this question is obvious as I am just beginning to learn this field. I am also abusing the dependance on $p$ for notational ease. The Chromatic Convergence theorem establishes that $S=\...

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254 views

### Is there a definition of an $\infty$-groupoid in HoTT whose terms are $n$-manifolds and whose higher morphisms are diffeomorphisms/isotopies/etc?

Suppose you want to work with TQFTs in homotopy type theory (HoTT). Working with $(\infty,n)$-categories, or even $(\infty,1)$-categories, is something that I gather is too difficult for HoTT at the ...

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548 views

### Standard model structures on $Top$

Call a model structure on $Top$ (the category of topological spaces) standard, if the weak equivalences are the weak homotopy equivalences. In this nLab page, two standard model structures on $Top$ ...

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147 views

### Why should we regard $PL(M)$ as a simplicial group?

Let $M$ be a manifold. If $M$ is smooth, it is clear what $\text{Diff}(M)$ should be, namely it should be the set of diffeomorphisms of $M$ equipped with the topology in which a sequence of ...

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363 views

### Combinatorial results by Poincaré duality

For the n-dimensional Torus, the k-th homology group (with integer coefficients) is isomorphic to the direct sum of $n \choose k$ copies of $\mathbb{Z}$.
Poincaré duality thus gives us a somewhat ...

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236 views

### Unicity of Johnson-Wilson Theories

Let $E$ be a ring spectrum with a $p$-typical complex orientation. Then we call $E$ a form of $BP\langle n\rangle$ or a generalized $BP\langle n\rangle$ if the induced map
$$\mathbb{Z}_{(p)}[v_1,\...

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468 views

### Twisted equivariant modular forms?

I'd like to know where I can find information about a class of objects which I think deserve to be called twisted equivariant modular forms. Let me guess a definition, indicate how it can be made more ...

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582 views

### On functoriality of the Leray spectral sequence

The Leray spectral sequence is functorial in the following sense: given a commutative square of spaces,
$$\begin{matrix} A & \to & B \\ \downarrow & & \downarrow \\ C & \to & D ...

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231 views

### Does simplicial localization with a 3-arrow calculus commute with functor categories?

Let $(C,W)$ be a category with a class of weak equivalences, and $D$ a small category. Then I can form the diagram category $(C^D,W^D)$ with objectwise weak equivalences, and its simplicial ...

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339 views

### extension problem for the Atiyah-Hirzebruch spectral sequence

For a generalized homology theory $h$ and a Serre fibration $F\rightarrow E\rightarrow B$, we can define an Atiyah-Hirzebruch spectral sequence\begin{equation}E^2_{p,q}=H_p(B,h_q(F))\Rightarrow h_{p+q}...

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170 views

### Homotopy theory of suplattices

In Quillen's monograph Homotopical algebra where he introduced the notion of model category, he showed that if $C$ is a bicomplete category with enough regular-projectives in which either (*) every ...

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958 views

### Eilenberg-Steenrod axioms of sheaf cohomology

Cohomology of a space is often defined axiomatically: a cohomology theory is a functor
from pairs of spaces to abelian groups
satisfying the Eilenberg-Steenrod axioms. Is there a similar ...

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221 views

### H-spaces without rational homology

Does there exist a simply connected, non-contractible manifold $M$, which is an $H$-space,
and whose rational homology groups vanish in positive degrees?
My space $M$ is in fact homotopy equivalent ...

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386 views

### Why does the index of the Dirac operator on a manifold with boundary live inside the Pfaffian line of the boundary Dirac operator?

I am trying to understand a statement from page 72 of What is an elliptic object? by Stolz and Teichner.
They have a spin Riemannian 2-manifold $\Sigma$, with boundary 1-manifold $Y$. The Dirac ...

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442 views

### Two constructions for BU×Z

Consider the following two ways of getting the zeroth space in the $K$-theory spectrum $BU \times \mathbb{Z}$:
1) Take the groupoid of finite dimensional complex inner product spaces with isometries ...

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422 views

### is a group $G$, that admits finite $k(G, 1)$ and has no Baumslag-Solitar subgroups, necessarily hyperbolic?

This is the first question asked in Bestvina's article "Questions in Geometric Group Theory". Does anyone know if there has been any progress made on this problem? Is the question answered if $G$ is ...

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443 views

### Applications of sheaf theory to the computation of invariants of LS-category type

I would like to know if sheaf theory can be applied to a particular class of questions in topology.
The Schwarz genus (also known as sectional category) of a continuous map $p\colon\thinspace E\to ...

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### Weight filtration over the integers

This is a follow up question to Weight filtration for smooth analytic manifolds
As mentioned in that question, the integral cohomology of some smooth complex analytic manifolds is equipped with a ...

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671 views

### Is a functor which is a sheaf for open covers and finite closed covers automatically a sheaf for covers by simplices?

Suppose $F:Top^{op}\rightarrow Set$ is a functor which is a sheaf for open coverings as well as for finite closed coverings, i.e. such that, apart from the usual sheaf property we also have that for ...

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186 views

### When does a locally symmetric space have no odd degree Betti numbers?

Let $G$ be a semisimple real lie group, $K$ be a maximal compact subgroup of $G$, $\Gamma$ be a torsion-free cocompact discrete subgroup. The Betti number the locally symmetric space $X_{\Gamma}:=\...

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209 views

### Are the braid groups good in the sense of Toën?

In this question it is asked whether the mapping class groups are 'good' in relation to pro-finite completion. Helpfully, one of the answers gives a link to a proof that the braid groups are good ...

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112 views

### Non-additivity of intersection forms

Given two oriented $4k$-manifolds $X_1$ and $X_2$, Novikov additivity tells us that
$$
\sigma(X_1 \sharp X_2) = \sigma(X_1) + \sigma(X_2).$$
More generally, if we glue the boundaries of two such ...

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255 views

### Lusternik-Schnirelmann Category of 4-Manifolds

Whilst reading 'the book' I stumbled across the following elegant little theorem, due to Gómez-Larrañaga and González-Acuña.
Let $M$ be a closed $3$-dimensional manifold. Then its Lusternik-...

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342 views

### Homology of a quotient space defined by an equivalence relation

Let $(X,x)$ be a pointed connected CW-complex. Let $f:X\rightarrow X$ be a (pointed) homeomorphism. Denote $Y=X\vee X$ and $Y^{ n}=Y\times\cdots \times Y$ $n$-times. Lets define a new homeomorphism $h:...

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129 views

### Example of a finitely-dominated CW complex which is not finite?

It's easy to find sources discussing Wall's finiteness obstruction, but harder to find discussion of examples. What's an example of a finitely-dominated CW complex which is not homotopy equivalent to ...

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202 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, ...

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236 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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195 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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126 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?