43
votes
Thomason's "open letter" to the mathematical community
A copy of Thomason's open letter to the mathematical community can be found here.
37
votes
Why study finite topological spaces?
Long comment:
Note that a finite topological space is nothing but a finite pre-order (order, iff the topology is $T_0$), given by $x\leq y$ iff $x\in \overline{\{y\}}$. An equivalence relation on a ...
34
votes
Thomason's "open letter" to the mathematical community
Because a link to the original letter has been posted, I have posted here, for historical completeness, a link to the handwritten followup letter that Thomason sent out to the same mailing list a few ...
30
votes
Accepted
If homotopy groups of spaces are identical, then stable ones are also identical?
No, a counterexample is the rational sphere $S^{2n}_\mathbb{Q}$ and $K(\mathbb{Q},2n) \times K(\mathbb{Q},4n-1)$. By the work of Serre these have the same homotopy groups, though it is easy to see ...
27
votes
Why study finite topological spaces?
As Uri Bader's answer notes, finite $T_0$ topological spaces are equivalent to finite partially ordered sets (posets). Now, combinatorialists who are interested in the topology of finite posets most ...
26
votes
What are some toy models for the stable homotopy groups of spheres?
My favorite warmup example to the stable homotopy groups of spheres is the following differential graded algebra.
Let $A$ have the underlying ring
$$
\Bbb Z[y] \otimes \Lambda[x],
$$
a ring with a ...
25
votes
Can a simply connected manifold satisfy $𝑀\simeq 𝑀\times 𝑀$?
Lemma. If $A$ is an abelian group satisfying $A\otimes A=0$ and $\mathop{\rm Tor}(A,A)=0$ then $A=0$.
Proof. Since $\mathop{\rm Tor}$ is left exact on abelian groups, an inclusion of a finite cyclic ...
24
votes
do all two manifolds admit a three-colorable triangulation?
Sure. Start with an arbitrary triangulation $T$, and then takes its barycentric subdivision $T'$. That means that the vertices of $T'$ come in three types: (1) vertices of $T$, (2) midpoints of edges ...
23
votes
Accepted
Who wrote `if only I could understand the equation $d^2=0$'?
Here’s a synthesis of answers and comments by მამუკა ჯიბლაძე and Georges Elencwajg fleshed out by your obedient servant.
On June 25, 1980, Henri Cartan was admitted to the honorary degree of Doctor of ...
Community wiki
22
votes
Accepted
Why study finite topological spaces?
Prompted by the same excerpt, I asked Bill Thurston in 2011:
Can you point me somewhere
where I can read about some of your mental models and structures as
they relate to finite topological spaces?
...
19
votes
Accepted
What is known about the homotopy type of the classifier of subobjects of simplicial sets?
It’s not hard to check that the subobject classifier $\Omega$ is trivially fibrant, and so its homotopy type is trivial. Orthogonality of $\Omega \to 1$ against a map $i : A \to B$ corresponds to the ...
19
votes
Why study finite topological spaces?
Any finite simplicial complex is weakly homotopy equivalent to a finite topological space and vice versa:
Are finite spaces a model for finite CW-complexes?
So, when it comes to computing homotopy ...
19
votes
Accepted
What topological principle is at work here?
EDIT: I've added an argument at the end that proves a more general statement and does so more simply.
I'll show that if $X$ is a finite simplicial complex having even Euler characteristic then there ...
17
votes
Accepted
Are all homotopy equivalences realized by fibrations over [0,1]?
There is an old result due to Patricia Tulley which claims that this is possible.
P. Tulley, A strong homotopy equivalence and extensions for Hurewicz fibrations, Duke Math. J. 36(3): 609-619 (...
16
votes
Accepted
Is there a ring stacky approach to $\ell$-adic or rigid cohomology?
This is an interesting question.
First, I think the [PS] reference does not give the "correct" Betti stack. In my notes on 6 functors, I define a different stack $X_B$ such that $D_{\mathrm{...
15
votes
What are some toy models for the stable homotopy groups of spheres?
You could say that I've made a living out of looking at the stable module category of a finite group (or rather its slight enlargement, the homotopy category of complexes of injective modules, $\...
15
votes
Structure of second homotopy group of a compact CW complex
Let $X=S^1 \vee S^2$ have $\pi_1(X) = \mathbb{Z} = \langle t \rangle$, and so $\pi_2(X) = \mathbb{Z}[t, t^{-1}]$, generated by the inclusion $\iota : S^2 \to X$. Attach a 3-cell to $X$ along $t\iota - ...
15
votes
Accepted
what is this simple topological space?
These spaces $M_{p,q}=M_p\cup M_q$ are discussed in Examples 1.24 and 1.35 of my algebraic topology book. I don't know that they have a standard name, apart from $M_{2,2}$ which is the Klein bottle. ...
14
votes
For which spaces $S^n$ ($n\geq 2$) is a universal covering space?
The answer is quite complicated. To begin with, the universal cover of your space $X$ is a sphere $S^n$ with a free action of a finite group $G=\pi_1(X)$. The group $G$ has to have periodic cohomology....
14
votes
4-manifold $M$ with intersection form of Leech lattice
If you're assuming $M$ is simply-connected, then it would be spin (since the Leech lattice is even). So a smooth manifold would violate Rokhlin's theorem. In the topological case (still assuming ...
14
votes
Why study finite topological spaces?
And what would be some instances where 'standard circumlocutions' used to avoid them?
For example, one can define contractability and compactness (for metrizable spaces) in terms of maps of finite ...
14
votes
Connеcted components of irreducible algebraic varieties
For a (projective) smooth real plane curve $C \subset \mathbb{RP}^2$ the answer is known.
Such a curve is a compact smooth one-dimensional manifold without a boundary, so its connected components are ...
14
votes
Who proved the motivic 6-functor formalism?
My understanding is that Scholze was citing Gallauer for the universal property of the 6FF of motivic spectra (an unpublished result as far as I know), not for the existence of the 6FF. This result ...
13
votes
Accepted
Homotopy groups of finite CW complex finitely generated as Lie algebra
I think Ian Leary's answer to this question gives a counterexample. His construction shows that for every $k\ge 2$ there exists a group $G_k$, and a finite $k$-dimensional CW complex $X$ such that $\...
13
votes
Categories on which one can determine all model structures?
If I remember correctly, about ten years ago I calculated the following: if $A$ is an Artinian commutative ring in which the square of each maximal ideal is zero, then the category of $A$-modules has ...
13
votes
Accepted
Plus construction on Simplicial Sets?
The answer is yes. This is spelled out in the book The local structure of algebraic K-theory by Bjørn Ian Dundas, Thomas G. Goodwillie and Randy McCarthy. Check out Section 1.6.1 on page 26, where ...
12
votes
For which spaces $S^n$ ($n\geq 2$) is a universal covering space?
A nice and quick survey on the groups acting freely on the sphere is given in chapter 3 of https://webusers.imj-prg.fr/~bernhard.keller/ictp2006/lecturenotes/skowronski.pdf .
Theorem 3.26 gives a nice ...
12
votes
The image of the J-homomorphism of the tangent bundle of the sphere
There is probably more than one good way to look at this. I'll point out that $\tau_{S^n}$ is the boundary of a generator of
$$
\pi_n(SO(n+1),SO(n))\cong \pi_n S^n.
$$
The map
$$
\pi_jS^n\cong \pi_j(...
12
votes
Integral homology classes that can be represented by immersed submanifolds but not embedded submanifolds
This is a great question, and I don't have an answer but this is too long for a comment.
Working mod $2$, a codimension $k$ homology class $z\in H_{m-k}(M;\mathbb{Z}/2)$ is realizable by an embedding ...
12
votes
Categories on which one can determine all model structures?
Yes, this has been done in other settings. For example, Scott Balchin, Kyle Ormsby, Angélica M. Osorno, and Constanze Roitzheim wrote a paper, Model structures on finite total orders, that enumerates ...
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