All Questions
9,056 questions
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How much of homotopy theory can be done using only finite topological spaces?
Let $X$ be a finite simplicial complex and let $B$ denote the set of barycenters of the simplices of $X$. McCord constructed a $T_0$ topology on $B$ with the property that the inclusion $B \to X$ is ...
- 29.2k
24
votes
3
answers
5k
views
A list of machineries for computing cohomology
This is not a question, but I just hope to hear more from everyone here on it.
A list of ready-to-use machineries to compute the de Rham / Cech cohomology of a manifold / variety. As far as I know, I ...
24
votes
4
answers
2k
views
How many simplicial complexes on n vertices up to homotopy equivalence?
Fix a number $n$, and define $\gamma(n)$ to be the number of simplicial complexes on $n$ unlabeled vertices up to homotopy equivalence. It is unlikely that an explicit formula exists, but what is ...
- 15.5k
24
votes
5
answers
2k
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Lie groups vs Lie monoids
Does there exist a well developed theory of a class of objects which might rightfully be called Lie monoids? By this I mean with axioms similar to those of Lie groups, but with the axiomatic existence ...
- 2,099
24
votes
1
answer
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Finite-order self-homeomorphisms of $\mathbf{R}^n$
Consider the $n$-dimensional euclidean space $\mathbf{R}^n$. A self-homeomorphism $\phi:\mathbf{R}^n\to \mathbf{R}^n$ is said to be of finite order if $\phi^m = \mathrm{id}_{\mathbf{R}^n}$ for some ...
- 1,121
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1
answer
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Characteristic classes of sphere bundles over spheres in terms of clutching functions
I'm trying to understand Milnor's proof of the existence of exotic 7-spheres.
Milnor finds his examples among $S^{3}$ bundles over $S^{4}$ (with structure group $SO(4)$ ). Such a bundle can be ...
- 6,546
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3
answers
4k
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Subgroups of free abelian groups are free: a topological proof?
There is a well-known topological proof of the fact that subgroups of free groups are free. Many people, myself included, think it is easier and more natural than the purely algebraic proofs which ...
- 65.4k
24
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1
answer
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How to "see" that double suspension of homology 3-sphere is homeomorphic to a sphere?
Is there a good way to think about/understand the result that the double suspension of a homology 3-sphere is homeomorphic to a sphere, to get intuition for why this is true? For instance, what sort ...
- 3,532
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votes
3
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2k
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Are there topological obstructions to the existence of almost quaternionic structures on compact manifolds?
$\DeclareMathOperator\End{End}\newcommand\Id{\mathrm{Id}}\DeclareMathOperator\Sp{Sp}\DeclareMathOperator\SO{SO}$I start with some background, but people familiar with the subject may jump directly to ...
- 2,140
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1
answer
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What topological principle is at work here?
[I'm cross-posting this from MSE. I initially asked there 10 days ago, and the question was well-received, but left unanswered.]
My question is inspired by a problem I discovered in Putnam and Beyond,...
- 956
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2
answers
4k
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Isotopy extension theorems
I'm looking for the origins of the isotopy extension theorem in categories other than the smooth category.
Precisely, in the smooth category, the isotopy extension theorem says that if $f : [0,1] \...
- 44.4k
24
votes
2
answers
2k
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Good functorial model for BG
There are several functorial constructions of the space BG for a topological group (meaning BG plus the universal G-bundle). First, there is the Milnor construction, treated in several textbooks. The ...
- 20.9k
24
votes
1
answer
1k
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When is a TQFT the dimensional reduction of a higher dimensional TQFT?
In Lurie's framework for TQFT's, a TQFT is a symmetric mondoial functor from $Cob_n(n)$ to some symmetric monoidal $n$-category $\mathcal{C}$. One can construct an $(n-1)$-dimensional TQFT from an $n$-...
- 8,167
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1
answer
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What is the Todd class *really*?
My question is about how to think about the Todd class.
Usually this is presented via Grothendieck Riemann Roch (GRR): if $X$ is a smooth projective scheme over a field $\mathbf{C}$, the chern ...
- 5,701
24
votes
1
answer
1k
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Mapping class groups in high dimension
$\DeclareMathOperator\MCG{MCG}\DeclareMathOperator\Diff{Diff}\DeclareMathOperator\Homeo{Homeo}$Let $M$ be a $1$-connected, closed, smooth manifold with $\dim(M)>4$ and let us set $\MCG(M)=\pi_0(\...
- 9,870
24
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1
answer
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Combinatorial spin structures
I would like to know how to define spin structures combinatorially, for an oriented smooth manifold equipped with a triangulation. In the case of a 2d manifold, spin structures correspond to ...
- 1,579
24
votes
1
answer
968
views
Groups whose finite index subgroups of fixed index are isomorphic
I am interested in finitely generated groups $G$ that are residually finite and have the following property: For each $d \geq 1$, $G$ has subgroups of finite index $d$, and all such subgroups are ...
- 11.9k
24
votes
1
answer
466
views
To what extent can we characterise the image of the topological Chern character?
For a finite CW complex $X$, the Chern character gives an isomorphism
of finite-dimensional vector spaces:
$$
ch : K^*(X)\otimes \mathbb{Q} \to H^*(X, \mathbb{Q}).
$$
The vector space $V = H^*(X, \...
- 1,444
24
votes
0
answers
833
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The $(\infty, 1)$-category of all topological spaces, including the bad ones
[Edit: Corrected some false claims and modified questions accordingly.]
Let $\mathcal{S}$ be the cocomplete $(\infty, 1)$-category generated by a point.
This is conventionally known as the $(\infty, 1)...
- 15.9k
24
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0
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p-Adic String Theory and the String-orientation of Topological Modular Forms (tmf)
I am going to ask a question, at the end below, on whether anyone has tried to make more explicit what should be, it seems to me, a close relation between p-adic string theory and the refinement of ...
- 19.8k
23
votes
6
answers
6k
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Why chain homotopy when there is no topology in the background?
Given two morphisms between chain complexes $f_\bullet,g_\bullet\colon\,C_\bullet\longrightarrow D_\bullet$, a chain homotopy between them is a sequence of maps $\psi_n\colon\,C_n\longrightarrow D_{n+...
- 22.1k
23
votes
5
answers
4k
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Fundamental groups of topological groups.
Let $G$ be a topological group, and $\pi_1(G,e)$ its fundamental group at the identity. If $G$ is the trivial group then $G \cong \pi_1(G,e)$ as abstract groups. My question is:
If $G$ is a non-...
- 273
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6
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cohomology of BG, G compact Lie group
It has been stated in several papers that $H^{odd}(BG,\mathbb{R})=0$ for compact Lie group
$G$. However, I've still not found a proof of this. I believe that the proof is as follows:
--> $G$ compact ...
- 1,709
23
votes
5
answers
4k
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Understanding/Mastering Analysis in Topology, necessary?
I have spoken to one professor so far about this, which of course was helpful, and so I am looking for additional opinions: To work with topological tools that were built via analysis, should I be a "...
- 17.5k
23
votes
5
answers
5k
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Stiefel-Whitney Classes over Integers?
An interesting thing happened the other day. I was computing the Stiefel-Whitney numbers for $\mathbb{C}P^2$ connect sum $\mathbb{C}P^2$ to show that it was a boundary of another manifold. Of course, ...
- 2,684
23
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5
answers
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Does anyone know a basepoint-free construction of universal covers?
Let $X$ be a real manifold (for simplicity). The standard construction of the universal cover $\varphi: \widetilde{X} \longrightarrow X$ involves fixing a basepoint $p \in X$ and considering homotopy ...
- 4,164
23
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8
answers
3k
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How many proofs that $\pi_n(S^n)=\mathbb{Z}$ are there?
Offhand I can think of two ways in classical homotopy theory:
Show that $\pi_k(S^n)=0$ for $k\lt n$ by deforming a map $S^k\to S^n$ to be non-surjective, then contracting it away from a point not in ...
23
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3
answers
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What are some toy models for the stable homotopy groups of spheres?
The graded ring $\pi_\ast^s$ of stable homotopy groups of spheres is a horrible ring. It is non-Noetherian, and nilpotent torsion outside of degree zero.
Question: What are some "toy models" ...
- 63.9k
23
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3
answers
6k
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Does homology detect chain homotopy equivalence?
Is the following true: If two chain complexes of free abelian groups have isomorphic homology modules then they are chain homotopy equivalent.
- 425
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1
answer
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Is Lemma A.1.5.7 in Higher Topos Theory correct?
Hello to everyone,
I am studying the properties of combinatorial model categories, following the exposition given by Jacob Lurie in Higher Topos Theory ([HTT] from now on), in section A.2.6.
At some ...
- 233
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3
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Brauer Groups and K-Theory
Is there some a priori reason why we should expect the Brauer group of real [complex] super vector spaces to be closely related to periodicity in real [complex] K-theory? By "a priori" I mean a proof ...
- 12.8k
23
votes
2
answers
2k
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Uniqueness of compactification of an end of a manifold
Let $M$ be an $n$-dimensional manifold (smooth or topological). I call $\bar{M}$ a compactification of $M$ if it is an $n$-dimensional compact manifold with boundary $\partial \bar{M}$, an $(n-1)$-...
- 21.5k
23
votes
2
answers
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Latest results in chromatic homotopy theory
I started a PhD in chromatic homotopy three years ago, but I had to quit it due to personal reasons after one year. Last week I was looking at all my notes from that period and I was wondering where ...
- 899
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5
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The "right" topological spaces
The following quote is found in the (~1969) book of Saunders MacLane,
"Categories for the working mathematician"
"All told, this suggests that in Top we have been studying
the wrong mathematical ...
- 18.7k
23
votes
9
answers
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What methods exist to prove that a finitely presented group is finite?
Suppose I have a finitely presented group (or a family of finitely presented groups with some integer parameters), and I'd like to know if the group is finite. What methods exist to find this out? I ...
- 1,861
23
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4
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3k
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What are Picard categories, where can I learn more about them, and why should I care to?
I have the category-theoretic background of the occasional stroll through MacLane's text, so excuse my ignorance in this regard. I was trying to learn all that I could on the subject of tensor ...
- 1,539
23
votes
2
answers
2k
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Why do people say DG-algebras behave badly in positive characteristic?
It seems to be a common wisdom in derived algebraic geometry that commutative DG-algebras are not, in general, a good model for derived rings, with the stated reason that they behave badly in positive ...
- 231
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3
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3k
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Homology theory constructed in a homotopy-invariant way
Singular homology sends homotopic morphisms on equal morphisms and weakly equivalent spaces on isomorphic objects. So singular homology is in fact defined on the homotopy category of topological ...
- 3,033
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3
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A homology theory which satisfies Milnor's additivity axiom but not the direct limit axiom?
Let us agree on the following: a "homology theory" means a functor $h_*$ from the category of pointed CW complexes to the category of graded abelian groups, together with natural isomorphisms $h_{*+1}(...
- 3,004
23
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6
answers
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Pathological Examples of Dimension
I am trying to wrap my head around all the different notions of dimension (and their equivalences). To get a sense of this, it would be nice to know the subtle difficulties that arise. I thus think it ...
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3
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Is it possible to construct an action of an $E_\infty$ operad on $BU$ that respects filtration by $BU(n)$?
It is well known that $BU$ is an infinite loop space, and as such it has an action of an $E_\infty$ operad. An explicit construction of such an action is given, for example, in an answer to this MO ...
- 10.9k
23
votes
2
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Massey Products vs. $A_\infty$-Structures
Does anyone know a good reference for a proof of the fact that given a dga $A$, an $A_\infty$-structure on $HA$ is ''the same'' as coherent choices for all of the higher Massey products of $HA$? More ...
- 2,283
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1
answer
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A property of even continuous functions on the sphere
This question is inspired by On moments of inertia of planar and 3D convex bodies.
Let $f:{\mathbb R}^3\setminus\{0\}\to{\mathbb R}$ be an even homogeneous ($f(kx)=f(x)$ for all real $k\neq 0$) ...
- 91.8k
23
votes
2
answers
467
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Which homotopy 2-types are H-spaces?
Let $X$ be a connected CW-complex with $\pi_k(X)$ trivial for $k >2$. Is it known under which circumstances $X$ is an $H$-group?
I have been able only to deduce the necessary condition that $\pi_1(...
- 344
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4
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De Rham decomposition theorem, generalisations and good references
De Rham decomposition theorem states that every simply-connected Riemannian manifold $M$
that admits complementary sub-bundles $T'(M)$ and $T''(M)$ of its tangent bundle parallel with respect to the ...
- 28.9k
23
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2
answers
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Representing elements of $\pi_2(M)$ by embedded spheres in 3-manifolds
I am sorry that this question is probably too basic - I could not seem to find the answer though. I know the following - let $S$ be a closed orientable surface, an element of $H_1(S;\mathbb{Z})$ is ...
- 5,349
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2
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fake $S^{2k}\times S^{2k}$
Let $X$ be a fixed closed manifold,$S(X)$ the structure set and $Aut(X)$ the group of self homotopy equivalence of $X$.
surgery theory tells us that $\mathcal{M}(X):=S(X)/Aut(X)$ is in bijection ...
- 593
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1
answer
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Which spaces have the (weak) homotopy type of compact Hausdorff spaces?
Inspired by the discussion in the comments of this question, I'd like to ask the following question: is it possible to characterize the class of spaces that are homotopy equivalent (or weak equivalent)...
- 31.2k
23
votes
2
answers
6k
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Definition of fiber bundle in algebraic geometry
If we have a map p: X --> Y of topological spaces, we can make a definition expressing that the topological type of the fibers of p varies continuously (edit: better to say "locally constantly", ...
- 9,283
23
votes
1
answer
949
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Fundamental theorem of K-theory for loop groups over $\mathbb{F}_1$?
As the title says, I would like to know what the fundamental theorem of algebraic K-theory would say over the field with one element. Recall that the fundamental theorem of K-theory provides a ...
- 17.4k