# Questions tagged [at.algebraic-topology]

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

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### 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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### 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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### 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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### 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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### 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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### (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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### 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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### 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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### 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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### 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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### 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 ...
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 $H^... 0answers 98 views ### Asymptotics of the Steenrod algebra /$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 ... 1answer 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 ... 1answer 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 ... 3answers 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 ... 1answer 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) ...
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 ...