All Questions
9,056 questions
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Did Peter May's "The homotopical foundations of algebraic topology" ever appear?
In the monograph Equivariant Stable Homotopy Theory, Lewis, May, and Steinberger cite a monograph "The homotopical foundations of algebraic topology" by Peter May, as "in preparation." It's their [107]...
- 30.3k
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3
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What tools cannot work for orbifolds?
Consider all of your basic constructions/tools/theorems for manifolds: fundamental group, Euler characteristic, triangulations, orientation, smoothness, bundle structure, cobordisms, etc.. Viewing ...
- 17.5k
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2
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Adams Operations on $K$-theory and $R(G)$
One of the slickest things to happen to topology was the proof of the Hopf invariant one using Adams operations in $K$-theory. The general idea is that the ring $K(X)$ admits operations $\psi^k$ that ...
- 2,327
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3
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Non trivial vector bundle over non-paracompact contractible space
The proof that the set of classes of vector bundles is homotopy invariant relies on the paracompactness and the Hausdorff property of the base space. Are there any known examples of:
Non trivial ...
- 281
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1
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Endomorphism ring spectrum of the Eilenberg-MacLane spectrum
Consider the endomorphism ring spectrum $R = \mathrm{End}_S(H\mathbb{F}_p)$ of the mod $p$ Eilenberg-MacLane spectrum $H\mathbb{F}_p$. The homotopy groups of $R$ are the Steenrod algebra $A^*$ with ...
- 1,243
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The number of polynomials on a finite group
A function $f:X\to X$ on a group $X$ is called a polynomial if there exist $n\in\mathbb N=\{1,2,3,\dots\}$ and elements $a_0,a_1,\dots,a_n\in X$ such that $f(x)=a_0xa_1x\cdots xa_n$ for all $x\in X$. ...
- 41.9k
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On a curious map from the complex projective plane into $S^5$
I have heavily edited the post (including the title), based on a comment by @GregoryArone that my map $f$ is not injective. In an earlier version of this post, I had thought to have constructed a ...
- 5,215
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4
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A Peculiar Model Structure on Simplicial Sets?
I'm wondering if there is a Quillen model structure on the category of simplicial sets which generalizes the usual model structure, but where every simplicial set is fibrant? I want to use this to do ...
- 27.5k
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1
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Is every degree 1 self-map a homotopy equivalence?
In a rather obscure article, I found (without proof) the following statement:
If $M$ is a closed orientable manifold, every degree $1$ map $f: M \rightarrow M$ is a homotopy equivalence.
Is this ...
- 11.9k
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Homotopy type of Diff(ℝP³)
$\DeclareMathOperator\Diff{Diff}$I am looking for the paper computing the homotopy type of the group $\Diff(\mathbb{R}P^3)$ of diffeomorphisms of the $3$-dimensional real projective space $\mathbb{R}P^...
25
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4
answers
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Algorithmically unsolvable problems in topology
This question is inspired by a paper by B. Poonen that appeared on the arxiv some time ago: http://arxiv.org/abs/1204.0299. The paper gives a sample of algorithmically unsolvable problems from various ...
25
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2
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Are "most" spaces aspherical?
There's a heuristic idea that "most" closed manifolds $M$ are aspherical (i.e. $\pi_{\geq 2}(M) = 0$). Does this heuristic extend usefully to all spaces -- or at least to all finite CW complexes?
To ...
- 63.9k
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What can we say about the Cartesian product of a manifold with its exotic copy?
Let $M$ be a smooth oriented manifold, and let $M^E$ be an exotic copy, i.e homeomorphic but not diffeomorphic to $M$.
Is it true that $M\times M$ is diffeomorphic to $M\times M^E$?
I am ...
- 3,751
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3
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Persistent homology of Gaussian fields in Euclidean space
If you generate points in $\mathbb R^n$ via a process that respects a Gaussian normal distribution, then compute the persistent homology / barcodes, to my eye something fairly regular seems to be ...
- 44.4k
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2
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The Dold-Thom theorem for infinity categories?
Let $\mathcal{M}$ denote the category of finite sets and monomorphisms, and let $\mathcal T$ denote the category of based spaces. For a based space $X \in \mathcal T$, one has a canonical funtor $S_X ...
- 1,484
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2
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Conceptual definition of the extension of a connection to 1-forms
I have a question that arose while reading Milnor's "Characteristic Classes". I will use the notation from that book.
Let $M$ be a smooth manifold and let $\zeta$ be a complex vector bundle ...
- 269
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1
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Is there a $4$-manifold which Immerses in $\mathbb{R}^6$ but doesn't Embed in $\mathbb{R}^7$?
I'm interested in both version of the question in the title, i.e. in the topological category and in the smooth category. By a topological immersion I mean a local embedding. I was asking in ...
- 439
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2
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Why not $\mathit{KSO}$, $\mathit{KSpin}$, etc.?
If $X$ is a compact Hausdorff space, we can consider the Grothendieck ring of real vector bundles on $X$,
$\mathit{KO}^0(X)$, and this extends to a generalized cohomology theory represented by a ring ...
- 6,881
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1
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Action of the degree 2 map on $\pi_8(S^4)$
I am currently reading Sullivan's Geometric Topology: Localization, Periodicity, and Galois Symmetry, on page 34 Sullivan claims that the degree 2 map $2:S^4 \to S^4$ induces the map $\left(\begin{...
- 353
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1
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Example of fiber bundle that is not a fibration
It is well-known that a fiber bundle under some mild hypothesis is a fibration, but I don't know any examples of fiber bundles which aren't (Hurewicz) fibrations (they should be weird examples, I ...
- 253
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generalisations of the Seifert-van Kampen Theorem?
I have been reading Jacob Lurie's book "Higher Algebra", version May 8, 2011. One is grateful to him for covering such a lot of ground and for making it all so readily available.
My attention was ...
- 12.3k
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CW complexes and paracompactness
It seems like when we assume "niceness" in homotopy theory we assume that $X$ has the homotopy type of a CW complex, and in fiber bundle theory we assume that $X$ is paracompact. How do these two ...
- 1,207
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Is there a ring stacky approach to $\ell$-adic or rigid cohomology?
Ever since Simpson's paper [Sim], it was observed that many different cohomology theories arise in the following way: we begin with our space $X$, we associate to it a stack $X_\text{stk}$ (which ...
- 711
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2
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844
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Which homotopy classes $S^3 \to S^2$ lift to embeddings $S^3 \to S^2 \times D^3$?
The question is, for a smooth embedding
$$f : S^3 \to S^2 \times D^3$$
one can compose the map $f$ with projection $\pi : S^2 \times D^3 \to S^2$, giving the map $\pi \circ f : S^3 \to S^2$.
Which ...
- 44.4k
25
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2
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Interplay between Loop Quantum Gravity and Mathematics
It is known that there are many interesting connections between String Theory and modern Mathematics, with a rich feedback going on in both directions: there have been advances in mathematics thanks ...
- 2,816
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1
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When are fiber bundles reversible?
My question, in its most general form is this:
Given a fiber bundle $F\rightarrow E\rightarrow B$, when is there a fiber bundle $B\rightarrow E\rightarrow F$?
Here, F,E, and B can lie in whichever ...
- 6,546
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Does every oriented $3$-dimensional submanifold of $\mathbb{R}^6$ bound an oriented $4$-dimensional submanifold?
In my recent research, I encountered the following problem about embeddings. Let $M^3$ be a closed compact oriented smooth $3$-dimensional submanifold of $\mathbb{R}^6$. Does there exist a compact ...
- 587
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0
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Does the Tate construction (defined with direct sums) have a derived interpretation?
Any abelian group M with an action of a finite group $G$ has a Tate cohomology object $\hat H(G;M)$ in the derived category of chain complexes. There are several ways to define this. One is as the ...
- 52.6k
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Why localize spaces with respect to homology?
A basic construction in algebraic topology is the localization of spaces or spectra with respect to a homology theory: one formally inverts the $E$-homology isomorphisms, reflecting each space into ...
- 66.8k
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Can both G and BG be finite CW complexes?
Here's a question out of idle curiosity. Let $G$ be a topological group. Is it possible for both $G$ and (a model of) $BG$ to be finite CW complexes? (Apart from the obvious example of $G$ being [up ...
- 40.2k
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5
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Why should an algebraic geometer care about singular / simplicial (co)homology?
I am a PhD student in algebraic / arithmetic geometry and I never took a formal course in algebraic topology, even though I have some basic knowledge.
In algebraic geometry we deal exclusively with ...
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Plus construction considerations.
In order to realise the K-groups of a ring as the homotopy groups of some space associated to that ring, Quillen proposed the following (roughly-sketched) construction:
Recall that $K_1(R) = GL(R)/E(...
- 1,105
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Can surfaces be interestingly knotted in five-dimensional space?
It's possible this question is trivial, in which case it will be answered quickly. In any case, I realized that it's a basic question the answer to which I should know but do not.
Everybody loves ...
- 54.6k
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3
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Is the $H$-space structure on $S^7$ associative up to homotopy?
Endow $S^7$ with a structure of an $H$-space induced from multiplication in octonions $\mathbb{O}=\mathbb{R}^8$. It is not associative as octonion multiplication is not associative.
Is it associative ...
- 7,377
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What do cohomology operations have to do with the non-existence of commutative cochains over $\mathbb{Z}$?
Let $X$ be a topological space. In elementary algebraic topology, the cup product $\phi \cup \psi$ of cochains $\phi \in H^p(X), \psi \in H^q(X)$ is defined on a chain $\sigma \in C_{p+q}(X)$ by $(\...
- 25.6k
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6
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Multiplicative Structures on Moore Spectra
The motivation for this question is that I want "toy examples" of how to prove/disprove the existence of multiplicative structures on examples of spectra. The class of examples I am thinking of is the ...
- 2,327
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3
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Is there a map of spectra implementing the Thom isomorphism?
A well known theorem in algebraic topology relates the (co)homology of the Thom space $X^\mu$ of a orientable vector bundle $\mu$ of dimension $n$ over a space $X$ to the (co)homology of $X$ itself: $...
- 8,167
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Non-abelian Grothendieck group
By general nonsense the forgetful functor from groups to monoids has a left adjoint. It maps a monoid $(X,\cdot,1)$ to the free group on $\{\underline{x} : x \in X\}$ modulo the relations $\underline{...
- 63.1k
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3
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Euler characteristic of a manifold and self-intersection
This is probably quite easy, but how do you show that the Euler characteristic of a manifold M (defined for example as the alternating sum of the dimensions of integral cohomology groups) is equal to ...
- 5,530
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6
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Simplicial model of Hopf map?
The Hopf fibration is a famous map $S^3\to S^2$ with fiber $S^1$, which is the generator in $\pi_3(S^2)$. We can model this map in terms simplicial sets by taking the singular simplicial sets of these ...
- 27.5k
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3
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Generalization of Borsuk-Ulam
Let $n$ be a positive intger. Is the following true? For continuous maps $f: \mathbb S^n \rightarrow \mathbb S^n$ and $g: \mathbb S^n \rightarrow \mathbb R^n$, there exists a point $x \in \mathbb S^n$ ...
- 11.9k
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Do "surjective" degree zero maps exist?
Is there a map $f\colon X \to Y$ of closed, connected, smooth and orientable $n$-dimensional manifolds such that the degree of $f$ is 0 but $f$ is not homotopic to a non-surjective map?
Added: The ...
- 2,590
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3
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How we do actually compute the topological index in Atiyah-Singer?
This is migrated by math.stackexchange as I did not receive an answer. I do not know if it is too naive for this site.
I am taking a lectured class in Atiyah-Singer this semester. While the class is ...
- 6,259
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1
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All fiber bundles over $S^2$ extendable to $\mathbb{C}P^\infty$?
I ran into the following sanity check. Is the following statement true?
Every smooth fiber bundle (with compact fiber) over $S^2$ can be extended to a smooth fiber bundle over $\mathbb{C}P^\infty$ (...
- 707
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2
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Are homeomorphic open subsets of $\mathbb{R}^n$ also diffeomorphic?
Let $U_1, U_2$ be open subsets of $\mathbb{R}^n$. Both are naturally differentiable submanifold, getting the differentiable structure from $\mathbb{R}^n$. Further, both are natural topological ...
- 243
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complement of a totally disconnected closed set in the plane
While preparing a course in complex analysis, I stumbled over a remark in Dudziak's book on removable sets, namely that any totally disconnected $K \subset\subset {\mathbb C}$ must have a connected ...
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2
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Can one deform an immersion of a 3-manifold in $\mathbb R^4$ to an embedding in $\mathbb R^6$?
Let $M^3$ be an oriented 3-manifold, and let $f:M^3\looparrowright \mathbb R^4$ be a codimension one immersion. Is it possible to find a small deformation of the composite map
$$
M^3 \to \mathbb R^4 \...
- 43.2k
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2
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Research directions in persistent homology
I am interested in what are the possible directions for new research in persistent homology (more of the mathematical theoretical aspects rather than the computer algorithm aspects).
So far from ...
24
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2
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Roadmap to Hill-Hopkins-Ravenel
How does one go from an understanding of basic algebraic topology (on the level of Allen Hatcher's Algebraic Topology and J.P. May's A Concise Course in Algebraic Topology) to understanding the paper ...
- 3,309
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Periodic orbit property
A topological space $X$ satisfies the "Periodic orbit property", briefly POP, if for every continuous map $f:X \to X$, there exist a natural number $n$ and a point $x_{0}\in X$ such that $...
- 356