All Questions
9,056 questions
31
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Can the fundamental group of any manifold be realized as the fund grp of a finite space?
Recently, I was asked to calculate the fundamental group of the space $X= \{a,b,c,d\}$ with open sets generated by $\{ a, c, abc, acd \}$.
Turns out, $\pi_1(X)\cong \mathbb Z$ and in fact, $X$ is the ...
- 899
31
votes
3
answers
3k
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Are the higher homotopy groups of the Hawaiian earring trivial?
The fundamental group of the Hawaiian earring is very complicated, but since it's "1-dimensional" one might guess that the higher homotopy groups vanish. Do they? Since the Hawaiian earring does not ...
- 28.1k
31
votes
4
answers
4k
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Fibrations and Cofibrations of spectra are "the same"
My question refers to a folklore statement that I have now seen a couple of times, but never really precise. One avatar is:
"For spectra every cofibration is equivalent to a fibration" (e.g. in the ...
- 1,558
31
votes
6
answers
5k
views
Book recommendation for cobordism theory
I am planning to organize a seminar on cobordism theory and I'm looking for a reference. Such a reference is preferably a book, but I'm open to other ideas.
The audience is familiar with ...
- 7,583
31
votes
4
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8k
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What is 'formal' ?
The key step in Kontsevich's proof of deformation quantization of Poisson manifolds is the so-called formality theorem where 'a formal complex' means that it admits a certain condition. I wonder why ...
- 571
31
votes
3
answers
2k
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Is the counit of geometric realization a Serre fibration?
Recall that a Serre fibration between topological spaces is a map which has the homotopy lifting property (HLP) for all CW complexes (equivalently for all disks $D^k$). The Serre fibrations are the ...
- 27.5k
31
votes
2
answers
2k
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Why is the motivic category defined over the site of smooth schemes only?
Fix a base scheme $S$. Stable and unstable motivic categories over $S$ are defined as certain categories of higher stacks on the Nisnevich site $Sm_S$ of smooth schemes over $S$. Why smooth?
As a ...
- 63.9k
31
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3
answers
3k
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Why aren't there more classifying spaces in number theory?
Much of modern algebraic number theory can be phrased in the framework of group cohomology. (Okay, this is a bit of a stretch -- much of the part of algebraic number theory that I'm interested in...)....
- 8,467
31
votes
2
answers
3k
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A natural construction of real numbers?
Summary
Someone claims $\mathbb{R}$ can be constructed as the following intriguing quotient, which is related to Gromov's bounded cohomology. I want to find out if it is true.
$$\frac{\bigl\{f:\mathbb{...
- 5,230
31
votes
3
answers
4k
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Algebras over the little disks operad
Hello,
The so-called "recognition principle" of Boardman-Vogt and May leaves me unsatisfied.
My problem is the following:
The "recognition principle" says that every "group-like" algebra over the ...
- 2,521
31
votes
2
answers
1k
views
Is Lie group cohomology determined by restriction to finite subgroups?
Consider the restriction of the group cohomology $H^*(BG,\mathbb{Z})$, where $G$ is a compact Lie group and $BG$ is its classifying space, to finite subgroups $F \le G$. If we consider the product of ...
- 413
31
votes
1
answer
3k
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What was the error in the proof of Roos' theorem?
Background: In 1961, Roos (who, sadly, apparently passed away just last month) purported to prove [1] that in an abelian category with exact countable products (AB4${}^\ast_\omega$), limits of inverse ...
- 63.9k
31
votes
1
answer
4k
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For which varieties is the natural map from the Chow ring to integral cohomology an isomorphism?
My apologies if this question is too naive.
Let $X$ be a smooth projective complex variety. There is a natural map $A^{\bullet}(X) \to H^{2\bullet}(X)$ of graded rings from the Chow ring of $X$ to ...
- 118k
31
votes
1
answer
2k
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K(r)-localization and monochromatic layers in the chromatic spectral sequence
While preparing some lecture notes, I had a basic point of confusion come up that I haven't been able to settle.
The $BP$-Adams spectral sequence (or $p$-local Adams-Novikov spectral sequence) for ...
- 6,308
31
votes
1
answer
2k
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A modern interpretation of Quillen's computation of the K theory of finite fields
In his beautiful paper On the cohomology and K theory of the general linear group over a finite field, Quillen constructs (if I understand correctly) an isomorphism on connected components of K-theory ...
- 7,952
31
votes
1
answer
1k
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What results about the topology of manifolds depend on the dimension mod 3?
There are a lot of interesting results about the topology of manifolds that depend on the dimension of the manifold mod 2, mod 4, or mod 8. The simplest ones involve the cup product
$$ \smile \colon ...
- 22.3k
31
votes
0
answers
866
views
The central insight in the proof of the existence of a class of Kervaire invariant one in dimension 126
I understand from a helpful earlier MO question that the techniques leading to the celebrated resolution of the Kervaire invariant one problem in the other candidate dimensions yield no insight on ...
- 2,995
31
votes
0
answers
2k
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When is a compact topological 4-manifold a CW complex?
Freedman's $E_8$-manifold is nontriangulable, as proved on page (xvi) of the Akbulut-McCarthy 1990 Princeton Mathematical Notes "Casson's invariant for oriented homology 3-spheres".
Kirby showed that ...
- 3,901
30
votes
6
answers
3k
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Poincare duality and the $A_\infty$ structure on cohomology
If $X$ is a topological space then the rational cohomology of $X$ carries a canonical $A_\infty$ structure (in fact $C_\infty$) with differential $m_1: H^\ast(X) \to H^{\ast+1}(X)$ vanishing and ...
- 6,229
30
votes
5
answers
4k
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The role of ANR in modern topology
Absolute neighborhood retracts (ANRs) are topological spaces $X$ which, whenever $i\colon X\to Y$ is an embedding into a normal topological space $Y$, there exists a neighborhood $U$ of $i(X)$ in $Y$ ...
30
votes
3
answers
3k
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Why is homology not (co)representable?
This is in the same vein as my previous question on the representability of the cohomology ring. Why are the homology groups not corepresentable in the homotopy category of spaces?
- 5,548
30
votes
2
answers
2k
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Does there exist any non-contractible manifold with fixed point property?
Does there exist any non-trivial space (i.e not deformation retract onto a point) in $\mathbb R^n$ such that any continuous map from the space onto itself has a fixed point. I highly suspect that the ...
- 3,751
30
votes
5
answers
5k
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What's special about the Simplex category?
I have been wondering lately what makes simplicial sets 'tick'.
Edited
The category $\Delta$can be viewed as the category of standard $n$-simplices and order preserving simplicial maps. The goal of ...
user11863
30
votes
4
answers
3k
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Is the space of diffeomorphisms homotopy equivalent to a CW-complex?
Clarification: My question concerns the homotopy type of the space of $C^k$ diffeomorphisms with the compact-open $C^k$ topology, where $0< k \leq\infty$. I have stated my question below with $k=1$ ...
- 6,210
30
votes
2
answers
2k
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Unstable homotopy groups of spheres beyond Toda's range
In 1962 Toda published his book "Composition methods in homotopy groups of spheres", which contains computations of $\pi_{n+k}(S^n)$ for $k\le 19$ and $n\le 20$. The values of these groups are ...
- 35.9k
30
votes
1
answer
1k
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Are homeomorphic representations isomorphic?
Let $G$ be a finite group. Let $V_1, V_2$ be two finite-dimensional real representations. Suppose $f: V_1 \to V_2$ is a $G$-equivariant homeomorphism. Can one conclude that $V_1$ and $V_2$ are ...
- 803
30
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3
answers
3k
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Mumford conjecture: Heuristic reasons? Generalizations? ... Algebraic geometry approaches?
The Mumford conjecture states that for each integer $n$, we have: the map $\mathbb{Q}[x_1,x_2,\dots] \to H^\ast(M_g ; \mathbb{Q})$ sending $x_i$ to the kappa class $\kappa_i$, is an isomorphism in ...
- 21k
30
votes
1
answer
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Morava K-theories for dummies?
Professor Urs Würgler passed away one year ago, and his wife engraved his tombstone with "the formula he was the most proud of" :
$B(n)_*(X)\cong P(n)_*(K(n))\square_{\Sigma_n}K(n)_*(X)$
However ...
- 403
30
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1
answer
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Two questions on rational homotopy theory
I'm trying to read Quillen's paper "Rational homotopy theory" and am a little confused about the construction. As I understand, he associates a dg-Lie algebra over $\mathbb{Q}$ to every 1-reduced ...
- 25.6k
30
votes
3
answers
3k
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Examples for non-naturality of universal coefficients theorem
Does anyone have good examples of a space $X$ and a map $f: X \to X$ so that $f_*: H_*(X) \to H_*(X)$ is the identity but (e.g.) $f_*: H_*(X; \mathbb{F}_2) \to H_*(X; \mathbb{F}_2)$ is not the ...
- 10.1k
30
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5
answers
2k
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Is the universal covering of an open subset of $\mathbb{R}^n$ diffeomorphic to an open subset of $\mathbb{R}^n$ ?
Is the universal covering of a connected open subset $U$ of ℝn diffeomorphic to an open subset of ℝn (standard differentiable structure)?
If not true in general, is there any condition ...
30
votes
1
answer
3k
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What is, really, the stable homotopy category?
When you try to understand the fuss behind the new good categories of spectra that arose on the 90's, you read things such as the following paragraph written by Peter May (from "The Hare and the ...
- 3,004
30
votes
1
answer
2k
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Identifying the stacks in Devinatz-Hopkins-Smith
I read the Devinatz-Hopkins-Smith proof of the nilpotence conjectures last year, and while I followed along sentence to sentence I don't think I understood much of the motivating reasons for why what ...
- 6,308
30
votes
1
answer
787
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Is a filtered colimit of rational spaces again rational?
Let me first explain the statement of the question and then give some indication why the answer might be 'yes'.
By a space I mean, say, a simplicial set and by rational I mean rational in the sense of ...
- 383
30
votes
1
answer
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Which of the proofs of the fundamental theorem of algebra can actually produce bounds on where the roots are?
One of the old classic MO questions is a big-list of proofs of the fundamental theorem of algebra. Here is a second big-list question about this big list:
Which of the FTA proofs can, even in ...
- 118k
30
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0
answers
1k
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Todd class as an Euler class
Let $X$ be a relatively nice scheme or topological space.
In various physics papers I've come accross, the Todd class $\text{Td}(T_X)$ is viewed as the Euler class of the normal bundle to $X\to LX$. ...
- 5,701
30
votes
0
answers
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Why do Clifford algebras determine $KO$ (and $K$-)-theory?
In the paper "Clifford modules" by Atiyah-Bott-Shapiro, they construct a family of Clifford algebras $C_k$ over the real numbers, so that $C_k$ is the algebra associated to a negative definite form on ...
- 25.6k
29
votes
10
answers
3k
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Are infinite dimensional constructions needed to prove finite dimensional results?
Infinite dimensional constructions, such as spaces of diffeomorphisms, spectra, spaces of paths, and spaces of connections, appear all over topology. I rather like them, because they sometimes help me ...
- 22.1k
29
votes
5
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(Co)homology of the Eilenberg-MacLane spaces K(G,n)
Let $(G, n)$ be a pair such that $n$ is a natural number, $G$ is a finite group which is abelian if $n \geq 1$. It is well-known that $\pi_n(K(G,n)) = G$ and $\pi_i (K(G,n)) = 0$ if $i \neq n$.
Also ...
- 3,699
29
votes
3
answers
3k
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The homotopy category is not complete nor cocomplete
I understand that the homotopy category of (pointed) topological spaces and continuous maps is not complete. Nor is it cocomplete. In particular it neither has all pullbacks nor all pushouts.
What ...
- 5,596
29
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3
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5k
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finite generated group realized as fundamental group of manifolds
This is discussed in the standard textbooks on algebraic topology.
Pick a presentation of the group $G = \langle g_1,g_2,...,g_n|r_1,r_2,...r_m \rangle$
where $g_i$ are generators and $r_j$ are ...
- 291
29
votes
7
answers
4k
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Why does the group act on the right on the principal bundle?
In many textbooks, in fact all textbooks I've seen, the fiberwise group action on the principal bundle is on the right. It seems to me that left actions and right actions are essentially the same. ...
- 1,398
29
votes
4
answers
5k
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Quillen's motivation of higher algebraic K-theory
Almost the same question was already asked on MO Motivation for algebraic K-theory?
However, to my taste, the answers there consider the subject from a more modern point of view.
When I open a book ...
- 21.8k
29
votes
4
answers
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Model structure on Simplicial Sets without using topological spaces
The category of simplicial sets has a standard model structure, where the weak equivalences are those maps whose geometric realization is a weak homotopy equivalence, the cofibrations are ...
- 1,292
29
votes
4
answers
3k
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origin of spectral sequences in algebraic topology
I have the following somewhat vague question. I am not sure if it is appropriate for this forum, please feel free to close (or migrate to stackexchange).
I have been "brought up" as an algebraic ...
- 1,961
29
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6
answers
4k
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Concrete example of $\infty$-categories
I've seen many different notions of $\infty$-categories: actually I've seen the operadic-globular ones of Batanin and Leinster, and the opetopic, and eventually I'll see the simplicial ones too. ...
- 3,349
29
votes
5
answers
2k
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Homotopy groups of spheres in a $(\infty, 1)$-topos
Let $H$ be an $(\infty,1)$-topos (seen as a generalization of the homotopy category of spaces).
You can define the suspension of an object $X$ as the (homotopy) pushout of $*\leftarrow X \to *$, ...
- 3,033
29
votes
2
answers
2k
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A simple proof that parallelizable oriented closed manifolds are oriented boundaries?
So let $M$ be a smooth closed orientable real manifold such that $M$ is parallelizable, i.e., the tangent space $TM$ of $M$ is trivial. From the triviality of $TM$ we get that the Stiefel-Whitney and ...
- 7,609
29
votes
4
answers
3k
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Conceptual proof of classification of surfaces?
Every compact surface is diffeomorphic to $S^2$, $\underbrace{T^2\#\ldots \#T^2}_n$, or $\underbrace{RP^2\#\ldots \#RP^2}_n$ for some $n\ge 1$.
Is there a conceptual proof of this classification ...
- 43.2k
29
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1
answer
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Is there an explicit description of a cobordism between $\mathbb{CP}^n$ and $\mathbb{RP}^n\times\mathbb{RP}^n$?
With a little bit of work, one can show that $\mathbb{CP}^n$ and $\mathbb{RP}^n\times\mathbb{RP}^n$ have the same Stiefel-Whitney numbers, so by a theorem of Thom, they are (unorientedly) cobordant.
...
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