# Tag Info

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, ...
• 9,134
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 ...
• 48.4k
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 ...
• 7,658
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 ...
• 6,439

### 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 ...
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
• 783

### 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 ...
• 50.8k

### 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

### 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 ...
• 27.3k

• 5,172
Accepted

### 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–...
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 ...
• 9,794
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 ...
• 39.4k
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 ...
• 12.5k

### 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 ...
• 7,714
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 ...
• 1,241