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1 vote

Alexander polynomials for a certain family of closed braids

The closure of the braid $\sigma_\kappa$ is a connected sum of torus links $T(2,k_i)$ (which are closures of 2-braids). Since the Alexander polynomial is multiplicative with respect to connected sums, ...
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4 votes
Accepted

If $\pi_\ast A$ is graded-commutative, then is $A_\ast$ a lax monoidal functor?

As pointed out in the comments, the functor $A_*$ cannot in general be lax symmetric monoidal without making some alterations. Here is an incomplete discussion of when $A_*$ can be lax monoidal. The ...
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3 votes
Accepted

Maps between unitary little disks operads and non-unitary little disks operads

A positive answer is the main theorem of my paper with Krannich and Horel, Two remarks on spaces of maps between operads of little cubes. The proof uses a result of Haugseng and Kock to reduce it to a ...
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1 vote

(Lower) homotopy groups from triangulations

Being a manifold or the dimension restriction $k\leq n$ doesn't matter, the following applies to finite simplicial complexes in general: As others have explained, if the fundamental group is not ...
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6 votes

Is there a flat manifold with trivial first homology?

Here is an idea for making examples. Let $F$ be a free group and let $N$ be a normal subgroup of $F$. Then $F/[N,N]$ is torsion-free. To see this, suppose $w\in F$ has finite order modulo $[N,N]$, and ...
4 votes
Accepted

Homotopy groups of cubical sets

I think a reference for this would be Theorem 3.24 of Homotopy groups of cubical sets, Daniel Carranza, Chris Kapulkin, 2022. arXiv:2202.03511, https://doi.org/10.48550/arxiv.2202.03511
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6 votes

Explicit generators from Serre spectral sequence

You have not specified your coefficients, but it sounds like you are working over the integers. In that case, if the $E^2$ term is not a free abelian group, then you only know that $H_*(E)$ has a ...
4 votes

The center of $\mathbf{hTop}$

Here's a partial answer constraining $\alpha_{S^1}$. First of all, I think my comment shows that (at least if we take $\operatorname{hTop}$ to be the homotopy category of CW complexes), it is possible ...
  • 6,439
12 votes

Is there a flat manifold with trivial first homology?

Andrzej Szczepański pointed me to Proposition 2.3.13 in the book [Perfect Groups, Derek F. Holt and Wilhelm Plesken, 1989], which gives an answer to my question. Namely, in a slightly different ...
2 votes

Proving that a countable group is not finitely generated

One possible way of proving that a countable group $G$ is not finitely generated is finding an infinite set $S$ and a mapping $\varphi$ from $G$ to the power set of $S$ such that the following hold: $...
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16 votes

Topology of the space of embedded genus $g$ surfaces in $S^3$

The first observation is that $\mathcal{E}_g$ is not connected when $g > 0$. This is due to the existence of "knotting". For example, in genus one, let $K$ and $K'$ be smooth knots. ...
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4 votes

Closed good cover of a triangulable space

Claim: Let $Z$ be a simplicial complex. For each simplex $\sigma\in Z$, let $N_2 (\sigma, Z)$ denote the simplicial neighborhood of $\sigma$ (or really, the second barycentric subdivision of $\sigma$) ...
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5 votes
Accepted

Injectivity of the cohomology map induced by some projection map

Ok, I will follow Fernando's advice and post an answer. I learned the computation below from the beginning of Pin(2)-equivariant Seiberg--Witten Floer homology and the triangulation conjecture. The ...
  • 8,728
4 votes

Injectivity of the cohomology map induced by some projection map

OK, just noticed mme's comment, which considers the very same counterexample below with a clean one-line argument. I'm leaving this here in case someone finds something of any value, but I think mme's ...
13 votes
Accepted

Stable torus that is not a torus

Suppose $M\times S^1$ is homeomorphic to $T^{n+1}$. Then $\pi_1(M\times S^1) \cong \pi_1(T^{n+1})$, so $\pi_1(M)\oplus\mathbb{Z} \cong \mathbb{Z}^{n+1}$, and hence $\pi_1(M) \cong \mathbb{Z}^n$. ...
0 votes

Equivariant K-theory for products of groups?

It is exactly the content of a paper of Minami (At least for compact Lie groups) A Künneth formula for equivariant $K$-theory. Haruo Minami. Osaka J. Math. 6(1): 143-146 (1969). Following ideas of ...
6 votes
Accepted

How many pairwise non-homeomorphic non-empty closed subsets of the Cantor set are there?

There are $2^{\aleph_0}$ different subsets of the Cantor set up to homeomorphism. There can't be more than $2^{\aleph_0}$ of them because any subset of the Cantor set is separable. To construct $2^{\...
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4 votes
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Explicit examples of Classical, Flat $U(2)$-connections on a torus link complement with non-trivial holonomy

For torus knots, all of the representations into $SU(2)$ were rather explicitly worked out by Eric Klassen (Representations of knot groups in $SU(2)$. Trans. Amer. Math. Soc. 326 (1991), no. 2, 795–...
12 votes
Accepted

Applications of equivariant homotopy theory to representation theory

There are decades and decades of algebraic results that use techniques from equivariant homotopy theory. Some examples ... (1) Quillen's work on ring theoretic aspects of the cohomology of finite ...
2 votes
Accepted

Spaces satisfying a strong Cartan-Hadamard theorem

Note that Hilbert spaces (of all dimensions finite or infinite) are the only geodesic spaces with extendable geodesics which are flat in the sense of Alexandrov. Therefore $X$ has to have extendable ...
9 votes
Accepted

When are filtered colimits of (trivial) cofibrations still (trivial) cofibrations?

If both cofibrations and weak equivalences are stable under filtered colimits, then so are trivial cofibrations. This happens for instance if $\mathcal{M}$ is a presheaf category on an elegant Reedy ...
4 votes

Homology of braid groups and loop spaces

Looping the fiber sequence $S^1 \to S^3 \to S^2$ gives $\Omega^2 S^2 \simeq \mathbb{Z} \times \Omega^2 S^3$. This is the group completion $\mathbb{Z} \times BB_\infty^+$ of $\coprod_{n\ge0} BB_n$, so ...
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1 vote

How much smaller is the Čech complex than the Vietoris-Rips complex?

I'm going to offer an answer mainly to get an idea off my brain and maybe someone will point out why this is incorrect. However, in my view, a lot of discussions about Čech Vs Vietoris-Rips seem to ...
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10 votes

Why the sphere spectrum is more correct than $\mathbb{Z}$?

An elementary answer to the first part of your question: Finite sets are more fundamental than their cardinalities. Consider the category of finite sets and bijective functions. Its geometric ...
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8 votes

"Singular homology = simplicial homology" relative to a fibration

My question is: Is the homology of $(C_*,\partial)$ isomorphic to the singular homology of E? Yes. Observe that the singular complex functor sends Serre fibrations to Kan fibrations. Thus, the map $...
5 votes
Accepted

Definition of S-reducibility and reducibility of a space

An $n$-dimensional CW complex with a single $n$-cell is reducible if the projection $X \to X/X^{(n-1)} = S^n$ onto the top cell admits a section up to homotopy. It is stably reducible, or S-reducible,...
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16 votes

Why the sphere spectrum is more correct than $\mathbb{Z}$?

For this to work, it is best to identify connective spectrum with spaces equipped with a group-like $E_\infty$-algebra structure (these are equivalent). From this point of view: $\mathbb{Z}$ is the ...
  • 35.1k
10 votes
Accepted

Is every retraction homotopic to a smooth retraction?

Using a collar of the boundary we resort to the case when $r\in C^0(M, \partial M)$ is given by the projection $\partial M\times I \to \partial M$ over the collar . Since $r$ is smooth on an open ...
3 votes
Accepted

Are there strictly connective smooth proper algebras over $\mathbb{F}_p$?

Here's an example: consider the $\infty$-category $Fun(\Delta^1,Perf(\mathbb F_p))$. It has two canonical generators $A= \mathbb F_p\to \mathbb F_p$ and $B=\mathbb F_p\to 0$; and I claim that $R= End(...
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