# Questions tagged [at.algebraic-topology]

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

6,177
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### finding Morse index for the following functional

not sure if this meets the standards here in this forum. I was dealing with the following functional $I(u)=\frac{1}{p}\int_{\Omega}|\nabla u|^{p}dx-\frac{1}{q}\lambda\int_{\Omega}|u|^qdx$ for $p \geq ...

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538 views

### Elementary topology of surfaces

Let $S$ be a compact connected orientable bordered surface of genus $g$ with $n$ holes (a hole is a component of the border homeomorphic to a circle). Consider a cell decomposition (the closure of ...

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190 views

### Free linear group actions on spheres with “strong” angle preservation

Suppose $G$ is a finite group and that $\rho: G\rightarrow O(d)$ is a faithful orthogonal representation, with action on $\mathbb{R}^d$ denoted $\cdot$. Let's say that $\rho$ is "strongly" angle ...

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175 views

### Test Categories which are not categories squelettiques?

Are there any examples of test categories which are not categories squelettiques in the sense of Cisinski?
Categories squelettiques are a type of generalized Reedy category, discussed for example on ...

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207 views

### Riemann-Hurwitz for real maps

Let $S$ be a (compact, connected) Riemann surface of genus $g$ and $f: S\to \mathbb CP^1$ be a degree $d$ meromorphic function. Then the Riemann-Hurwitz formula tells us that the number of ...

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177 views

### Is the tensor product of compactly generated Hausdorff abelian groups again Hausdorff?

Consider the tensor product $G \otimes_{\mathbb{Z}} H$ of two abelian groups $G$ and $H$. If $G$ and $H$ are topological groups, we can give $G \otimes_{\mathbb{Z}} H$ a topology as follows. For any $...

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433 views

### Do there exist acyclic simple groups of arbitrarily large cardinality?

Recall that a group $G$ is acyclic if its group homology vanishes: $H_\ast(G; \mathbb Z) = 0$. Equivalently, $G$ is acyclic iff the space $BG$ is acyclic, i.e. $\tilde H_\ast(BG;\mathbb Z) = 0$.
In ...

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128 views

### geometrical or physical interpretation of second Chern classes of Calabi-Yau threefold

It's my first post.
Consider Calabi-Yau threefold $M$ and its tangent bundle $TM$. I know $c_1(TM)=0$ means metric on $M$ is a solution of vacuum Einstein equation. Then my question is "are there any ...

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782 views

### Approximation of homeomorphism by diffeomorphism

Let $M$ be a smooth closed manifold. Let $f\colon M\to M$ be a homeomorphism.
Does there exist a sequence of diffeomorphisms $f_i\colon M\to M$ which conveges to $f$ uniformly, i.e. in $C^0$-...

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289 views

### About a generalization of the Borsuk-Ulam theorem

I've come upon this MO post, and I was wondering if there is a way to generalize what is said in the answer at the very bottom. Specifically, we have that given continuous maps $f: \mathbb{S}^n \to \...

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78 views

### Shapes of cores of symmetric monoidal $(\infty,n)$-categories (with duals)

According to the cobordism hypothesis, if $\mathcal{C}$ is a symmetric monoidal $(\infty,n)$-categories with duals, then framed fully extended TQFTs with target $\mathcal{C}$ are an $\infty$-groupoid, ...

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515 views

### Fundamental group of punctured simply connected subset of $\mathbb{R}^2$

(This question is originally from Math.SE where it was suggested that I ask the question here)
Let $S$ be a simply connected subset of $\mathbb{R}^2$ and let $x$ be an interior point of $S$, meaning ...

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210 views

### Chromatic Homotopy Theory and Physics

Chromatic homotopy theory is a subfield of stable homotopy theory that studies complex-oriented cohomology theories from the "chromatic" point of view, which is based on Quillen's work relating ...

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219 views

### Do spaces admit a weak cogenerating set?

Let $\mathcal C$ be a category. Say that a class of objects $\mathcal S \subseteq \mathcal C$ is weakly cogenerating if the functors $Hom_{\mathcal C}(-,S)$ are jointly conservative, for $S \in \...

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148 views

### Closed simple curves in $\mathbb{R}\mathbb{P}^2$

EDIT: The well known Jordan curve theorem says: let $C\subset S^2$ be a closed simple curve on the 2-sphere. Then its complement $S^2\backslash C$ consists of two connected components, both ...

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128 views

### A generalization of Jordan-Schoenflies theorem on simple plane curves

The well known Jordan-Schoenflies theorem says: let $C\subset \mathbb{R}^2$ be a closed simple curve. Then there exists a homeomorphism $f\colon\mathbb{R}^2\to \mathbb{R}^2$ such that $f(C)$ is the ...

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339 views

### (Stable) homotopy groups of Quillen plus construction

[Edit: in response to prof. Sullivan's answer, let me try to expand the question a bit.]
Let $G$ be a discrete group which is perfect (trivial abelianization) and let $BG \to BG^+$ denote Quillen's ...

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216 views

### Why does the Steenrod algebra act faithfully on $H^\ast(BC_p)$?

Define the Steenrod algebra $A^\ast$ to be the algebra of all stable mod $p$ cohomology operations. Without actually computing $A^\ast$, is it possible to see that $A^\ast$ acts faithfully on $H^\ast(...

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285 views

### Bringing cohomology recipes from algebra to topology?

In algebra, cohomology theories are often defined very roughly like this. Start with a scheme $X$ you want to study. Form a category-with-extra-structure (a site) $\mathcal{C}$ whose objects are, ...

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120 views

### action of étale fundamental group on the cover

I have a Galois cover $f \colon Y \rightarrow X$, i.e $f$ is finite étale and $deg(f) = Aut_X(Y)$. The étale fundamental group $\pi_1(X,\bar{x})$ acts on the geometric fiber $Y_{\bar{x}}$, but is that ...

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104 views

### Cofibrations and mapping spaces in compactly generated weak Hausdorff spaces

Assume that $X$ and $Y$ are compactly generated weak Hausdorff spaces (CGWH spaces for short). Assume that they are also well-pointed (so the inclusions of the base points are Hurewicz cofibrations).
...

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480 views

### Boardman's thesis or mimeographed notes

I would like to know if there is some online source where Boardman's 1964 thesis is available or his Warwick mimeographed notes. This is because by what I've heard Boardman's construction has a more ...

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233 views

### Computing K-theory for cellular vector bundles

One of the most computationally convenient properties of singular cohomology $X \mapsto H^\bullet(X;\mathbb{Z})$ is the fact that one can extract it from a good cover $\{U_i\}$ of $X$ via Cech ...

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256 views

### Contraction of a family of loops simultaneously

Let $LX$ denote the free loop space on $X$. We have an evaluation map $ev\colon LX\to X$ and we have an inclusion $X\hookrightarrow LX$ (where $x\in X$ is mapped to the constant loop at $x$).
Suppose ...

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303 views

### Cohomology theory with only one Adams operation?

Let $E$ be a multiplicative cohomology theory. Fix a prime p. Call a ring map $\psi^{p}:E\rightarrow E$ an Adams operation if it lifts the Frobenius map $E/p\rightarrow E/p$.
It is of course well-...

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199 views

### pullback of a local system

I have a smooth projective $k$-scheme $X$ with a local system $F$ (locally constant sheaf) of finite dimensional $k$-vector spaces (on étale topology). My question is whether there exists a finite ...

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320 views

### Compact manifold $X$ having fixed-point property but $X\times X$ does not

A manifold $X$ has the fixed-point property if for every continuous map $f:X→X$ there is $x∈X$ with $f(x)=x$. Examples of such spaces are disks and the real projective plane $\mathbb{RP}^2$.
Question:...

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248 views

### Do infinite products commute with trivial cofibrations, for simplicial sets?

I'm reading Voevodsky and Morel's book '$\mathbb{A}^1$-homotopy theory of schemes'. In Remark 3.1.15, it says that for any simplicial fibrant sheaf $F$ and open sets $U\subseteq V$, $F(V)\to F(U)$ is ...

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420 views

### Why is $\mathbb{S}^1$ a cogroup object in $\mathbf{Top.}$?

$\require{AMScd}$
This is basic level question, but this kind of questions usually find no answer on stackexchange.
I am trying to introduce my self to categories theory and advanced algebraic ...

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708 views

### How to compute the cohomology of a local system?

Suppose we have a reasonable topological space $X$ (i.e. a complex algebraic variety or a manifold) whose integral singular cohomology and fundamental group we understand well.
Suppose that we are ...

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201 views

### Splitting principle for real vector bundles

I'm reading the Book of John Roe, Elliptic Operators, Topology and Asymptotic Methods and got stuck at Lemma 2.27.
i) How does this lemma show that a real vector bundle can be given by a pullback of ...

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107 views

### Why is this equivariant cohomology a free module?

I try to understand the first part of the proof of the following lemma
($M$ is a variety with compact torus action $T$, $M^T$ is a set of fixed points, $H^*_T(M)$ is an equivariant cohomology of $M$,...

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147 views

### Bott periodicity homeomorphisms for spaces of Clifford extensions

I am trying to prove the following statement of real Bott periodicity, on the level of actual spaces of Clifford module extensions (i.e., not equivalence classes of modules).
Let $W = \mathbb{R}^{\...

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116 views

### Compactly supported cohomology of a topological DM stack

Consider a separated topological Deligne-Mumford stack $\mathfrak X$, i.e., a topological stack which is presentable by a proper etale topological groupoid (equivalently, $\mathfrak X$ is locally ...

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50 views

### Cohomological dimension of closed $G$-invariant subspaces on homology manifolds with a group action

Suppose $G$ is a compact topological group acting on an $m$-homology manifold $M$ over some ring $R$ by homeomorphisms.
Assume that the action of $G$ is effectively finite on a closed $...

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92 views

### Classifying space of an algebraic torus

I know that classifying space of a compact torus of n dimension is equal to n copies of infinite dimensional complex projective space. But how does a classifying space of an algebraic torus look like?

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130 views

### H spaces and Quillen's rational homotopy theorem

Quillen's theorem gives an equivalence
$$\text{Spaces}/\text{rational homotopy equivalance} \ \stackrel{\sim}{\longleftrightarrow} \ \text{dgla's}/\text{quasi-isomorphism}$$
such that if space $X$ ...

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563 views

### Is there ever a Kunneth isomorphism just for powers?

It's pretty rare for a multiplicative cohomology theory $E$ to have a Kunneth isomorphism $E^\ast(X \times Y) \cong E^\ast(X) \otimes_{E^\ast(pt)} E^\ast(Y)$ for all spaces $X,Y$. Are there any ...

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344 views

### Can we define fundamental groups functorially for non-pointed path connected topological spaces?

Let $\text{ppTop}$ denote the category of pointed and path connected topological spaces with morphisms base-preserve continuous maps. The fundamental group gives a functor $FG: \text{ppTop}\to \text{...

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130 views

### Is the meaning of “irreducible manifold”, “not reducible to other manifold”?

This is a cross post of MSE.
Q1: What does "irreducible manifold" mean (not definition)?
My understanding of "irreducible manifold" is "is not reducible (homotopic or deformation or homeomorph or ...

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100 views

### On the transgression in the Serre spectral sequence

This is probably very well known, something obvious to expect, and written somewhere. But, I do not recall any reference.
Let $F\to X\stackrel{p}{\to} B$ be a fibration where $F$ and $B$ are $(f-1)$-...

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407 views

### Parallelizability of 3-manifolds

Robert Bryant's answer here ( https://mathoverflow.net/a/149496/85500 ) states that any orientable 3-manifold is parallelizable.
Previously I was under the impression that only closed (compact & ...

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64 views

### Is open subspace of manifold with collared boundary a manifold with collared boundary?

An $n$-manifold (for this question) is a locally upper-$n$-Euclidean Hausdorff space. Hence, a manifold possibly has a boundary and possibly is not paracompact. An $n$-precell (for this question) is a ...

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98 views

### When are automorphisms of the cohomology ring realized by isometries?

Let $(M,g)$ be a closed smooth Riemannian manifold, and denote by $G$ a closed subgroup of its isometry group. By considering the maps $g^*$ induced by elements $g\in G$ in the (de Rham) cohomology $H^...

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98 views

### Asymptotics of the Steenrod algebra / $s$-partitions?

Recall that an $s$-partition is a partition of a natural number $n$ such that each part is of the form $2^r-1$. By a fundamental theorem of Milnor, the number $p_s(n)$ of $s$-partitions of $n$ counts ...

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191 views

### Are all pointed maps of classifying spaces of abelian groups homotopy equivalent to homomorphisms?

Let $G$ be a commutative topological group (e.g. $S^1$), and let $BG$ be its classifying space. Since $G$ is commutative, the space $BG$ is a group up to homotopy. It is well-known that we have a ...

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631 views

### Divisibility in the homotopy groups of spheres?

Consider the $k$-sphere (I'm particularly interested in $k=3$). For each positive integer $n$ let $P_{n}$ be the set of primes that divide the order of $\pi_{i}S^{k}$ for some $i\leq n$.
Does the ...

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1k views

### Unstable manifolds of a Morse function give a CW complex

A coauthor of mine and I want to use the following innocent looking statement in a forthcoming paper:
Statement. Let $M^{2n}$ be a compact manifold and let $f$ be a Morse function with critical ...

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105 views

### Does a homeomorphism of open cones restrict to a quotient map of the bases?

Write $CX$ for the (pointed, or reduced) cone on $X$, and $C^\circ X$ for the open cone inside of it.
Let's say a cone map is a map $g:CX\to CY$ such that $g(C^\circ X) \subseteq C^\circ Y$ and $g(X) ...

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150 views

### Pushforward in Compactly Supported Cohomology

Suppose $X,Y$ are locally compact Hausdorff spaces and $f:X\to Y$ is a topological submersion of relative dimension $n$. By this we mean that for all points $x\in X$, there exists an open neighborhood ...