All Questions
8,725 questions
6
votes
0
answers
357
views
On the nilpotence of the attaching maps for $\mathbb C \mathbb P^\infty$
Consider the usual cell structure on $\mathbb C \mathbb P^\infty$. The skeleta are the $\mathbb C \mathbb P^n$’s, and there is one cell in each even degree. So we have cofiber sequences $S^{2n+1} \to \...
- 64k
35
votes
3
answers
7k
views
Higher Topos Theory- what's the moral?
I've often seen Lurie's Higher Topos Theory praised as the next "great" mathematical book. As someone who isn't particularly up-to-date on the state of modern homotopy theory, the book seems ...
13
votes
1
answer
3k
views
Voevodsky's six functor formalism VS Lucas Mann's
Decades ago, Voevodsky constructed the six-functor formalism in motivic homotopy theory [Ayoub's thesis]. This construction seems very technical, long and "hard".
Very recently [Mann's ...
- 705
8
votes
1
answer
439
views
Model categories as a tool to resolve size issues for localizing categories
I have a rather basic question about one motivation for introducing model categories in David White's notes, as a possible way to overcome troubles with size issues appearing when localizing a ...
- 6,048
5
votes
1
answer
287
views
Is the wildness of 4-manifolds related to the diversity of their fundamental groups?
$n = 4$ is the smallest dimension such that the fundamental group of a closed $n$-manifold can be any finitely-presentable group (leading e.g. to various undecidability results stemming from the ...
- 64k
2
votes
0
answers
75
views
Action of $V$ on the homology of a subposet of the poset of affine subspaces of $V$
Let $(V,Q)$ be a pair, with $V=\mathbb{F}_2^{2n}$ ($n\geq 2$) and $Q$ a nondegenerate quadratic form on $V.$ We consider the poset $\mathcal{P}_n$ of affine totally isotropic (with respect to $Q$) ...
- 245
9
votes
1
answer
325
views
$\operatorname{Spaces}/BG$ $\sim$ $\operatorname{Spaces}^G$ $\sim$ $??(\Omega G)$
This is a crosspost (with minor alterations).
For a topological group $G$, assigning to a $G$-space $X$ the (canonical) map $EG\times_GX\to BG$ establishes an equivalence between the homotopy category ...
- 18.4k
4
votes
1
answer
215
views
Co-index of a Space
I am going through this paper by Tanaka, but I got stuck at Proposition 2.4, given below.
He does not provide any proof, instead referring to Theorem 6.6 of this paper, given below.
Unfortunately, I ...
5
votes
0
answers
231
views
Identifying a map in a fiber sequence
Let $Q = \Omega^{\infty} \Sigma^{\infty}$ be the stabilization functor. Suppose we have a sequence of maps $Q \mathbb{RP}^{n-1} \to Q \mathbb{RP}^{n} \to QS^n$
and suppose we know that it is a fiber ...
- 348
13
votes
1
answer
518
views
Impossibility of realizing codimension 1 homology classes by embedded non-orientable hypersurfaces
Suppose we have an $n+1$-dimensional compact closed oriented manifold $M$ and an $n$-dimensional integral homology class $[\Sigma]\in H_n(M,\mathbb{Z})$ on $M.$ Then is it true that $[\Sigma]$ mod $2$ ...
- 587
0
votes
1
answer
327
views
what is this simple topological space?
Take $M_p$ the mapping cylinder (MC) of the $p=3$-fold cover of $S^1$, $M_q$ the MC of the $q=2$-fold cover of $S^1$, where for both, the identification of the MC is done on the side $\{0\}$ of $[0,1]$...
7
votes
2
answers
836
views
Holonomy as integration of curvature for principal $G$-bundles?
Holonomy and curvature may seem to be slightly advanced topics in
geometry. However, their origins are easily imaginable. Namely,
picture the surface of earth $S$, and pick an arbitrary
contractible ...
- 5,230
1
vote
0
answers
48
views
Connected pre-images spanning $n$-cubes under dimension reducing maps
Let $I^n = [0,1]^n$ be the $n$-dimensional hypercube. For a continuous function $f: I^n \to \mathbb{R}^m$ with $m < n$, we're interested in the existence of points $p \in \mathbb{R}^m$ whose ...
user530766
5
votes
1
answer
436
views
Does coproduct preserve cohomology in differential graded algebra category
Consider two cochain DGA (differential graded algebras) named $A$ and $B$. By "coproduct" of two DGA I mean the category theory coproduct, not the coalgebra coproduct. It is defined in "...
- 159
48
votes
8
answers
5k
views
Ideas for introducing Galois theory to advanced high school students
Briefly, I was wondering if someone can suggest an angle for introducing the gist of Galois groups of polynomials to (advanced) high school students who are already familiar with polynomials (...
0
votes
0
answers
68
views
Large volume growth of covering space
Let $(M,g)$ be a Riemannian manifold with non-negative Ricci curvature. The Bishop-Gromov volume comparison says that: if
$$\alpha_M=\lim_{r\rightarrow\infty}\frac{VolB^M(p,r)}{\omega_nr^n},$$
then $0\...
3
votes
1
answer
402
views
The derived category of $p$-complete abelian groups is comonadic over the derived category of $\mathbb F_p$-vector spaces?
$\DeclareMathOperator\Mod{Mod}\DeclareMathOperator\Ext{Ext}$Let $p$ be a prime. The adjunction
$$\mathbb F_p \otimes_\mathbb{Z} (-) : \Mod(\mathbb Z) \rightleftarrows \Mod(\mathbb F_p) : U $$
descends ...
- 64k
0
votes
0
answers
64
views
Can an upper hemicontinuous correspondence be discountinuous everywhere?
Let $\phi: X \rightrightarrows Y$ be an upper hemicontinuous correspondence. If $K \subset X$ is a compact and convex set, $K$ contains an open set $U$, and $\phi(x)$ is nonempty, compact, and convex ...
- 101
0
votes
1
answer
328
views
Relationship between quotient CW-complexes after attaching cells
I have been trying to prove the following simple-looking result which I require for some work in low-dimensional topology. I expect it is likely true and in a textbook somewhere so any reference or ...
- 187
4
votes
2
answers
293
views
$\mathrm{String}/\mathbb{CP}^{\infty}=\mathrm{Spin}$ or a correction to this quotient group relation
We know that there is a fiber sequence:
$$
\dotsb \to B^3 \mathbb Z \to B \mathrm{String} \to B \mathrm{Spin} \to B^4 \mathbb Z \to \dotsb.
$$
Is this fiber sequence induced from a short exact ...
- 447
2
votes
0
answers
44
views
Link invariants on a thickened surface
Let $\Sigma$ be an oriented surface. I want to know about link invariants in $\Sigma\times [0,1]$. I already know the Ozawa polynomial introduced in this paper, but I couldn’t find any other than that....
- 21
2
votes
1
answer
482
views
homotopic to a constant map
Let $X$ and $Y$ be topological spaces and more precisely connected finite CW complexes.
Let $f\colon X \to Y$ be a continuous map such that there exist a second continuous map $F\colon X^3 \to Y$ and
$...
- 185
3
votes
1
answer
180
views
Connective covers of based spaces
I know that n-connective spectra form a coreflective subcategory of spectra. Namely, there is a n-connective cover functor $\tau_{>k}: Spectra\rightarrow Spectra^{n-connective}$ which has a fully ...
- 719
10
votes
1
answer
493
views
Structure of second homotopy group of a compact CW complex
I am interested in the second (and higher as well) homotopy groups of compact CW complexes. I know these groups don't need to be finitely generated (e.g. for $S^1 \vee S^2$ they are not), but I'd like ...
- 275
85
votes
23
answers
11k
views
Solving algebraic problems with topology
Often, topologists reduce a problem which is - in some sense - of geometric nature, into an algebraic question that is then (partiallly) solved to give back some understanding of the original problem.
...
1
vote
1
answer
144
views
Confusion regarding the vanishing of certain relative cohomology groups
Let $X$ be a projective variety of dimension $n$ and $D \subset X$ is a proper subvariety. Embed $X$ into a projective space $\mathbb{P}^{3n}$. The following argument implies that $H^i(X,X\backslash D)...
- 2,323
3
votes
0
answers
151
views
Reference for homotopy and homology theory of topological groups
I am looking for references which deal with the homotopy theory and homology theory of general topological groups, not necessarily compact, or anything. I am eyeing towards certain infinite-...
- 219
9
votes
2
answers
773
views
Pullbacks of classifying spaces
In what follows all the groups will be discrete, not necessarly finite.
Let $f:G\to H$ be a morphism of groups and $H'\to H$ be the inclusion of a subgroup. It seems to me (but correct me if I am ...
- 641
13
votes
2
answers
2k
views
Can a simply connected manifold satisfy $𝑀\simeq 𝑀\times 𝑀$?
Let $M$ be a simply connected, (finite dimensional) smooth manifold. Is it possible that $M$ is homotopy equivalent to $M\times M,$ without $M$ being contractible? This would imply $\pi_n(M)\times\...
- 1,198
2
votes
1
answer
628
views
Does some published texbook take a particular approach (described here) to the transition from discrete to continuous probability distributions?
(I posted this question at matheducators.stackexchange.com and it seems to be considered an inappropriate question for that site. I don't understand why.)
Imagine an introductory probability course ...
9
votes
1
answer
673
views
Homotopy groups of finite CW complex finitely generated as Lie algebra
This is probably a well-known question, but I haven't found the answer on MO or MSE.
It is well-known that the homotopy groups of a finite CW complex $X$ need not be finitely presented, even as $\...
- 28.7k
4
votes
1
answer
274
views
Comparing Kummer maps to étale homotopy at finite level
$\DeclareMathOperator\Mor{Mor}\DeclareMathOperator\Hom{Hom}\newcommand{\et}{\mathrm{et}}\newcommand{\top}{\mathrm{top}}$In Voevodsky's paper "Étale topologies of schemes over fields of finite ...
- 544
2
votes
0
answers
136
views
Question about the proof of the Duistermaat-VanDerKallen-Theorem, concerning the meaning of intersecting a chain with a set
On page 228 of the paper "Constant terms in powers of a Laurent polynomial" (by J.J. Duistermaat and Wilberd van der Kallen) a 'chain' $\Delta_\tau$ is defined as $f^{-1}([0,\tau])\cap \...
- 360
13
votes
0
answers
318
views
Is there an analogue of Steenrod's problem for $p>2$?
An element $\alpha \in H_k(X; \mathbb{Z})$ is said to be realisable if there is a $k$-dimensional connected, closed, orientable $k$-dimensional submanifold $Y$ such that $\alpha = i_*[Y]$. The ...
- 327
63
votes
2
answers
5k
views
Thomason's "open letter" to the mathematical community
In 1989, Bob Thomason left his CNRS position in Orsay and moved to Paris VII. It was during this period that he composed his "Open Letter" to the mathematical community. The letter explained ...
- 18.9k
4
votes
0
answers
110
views
Bar construction of a commutative monoid
Let $M$ be a commutative monoid. Define the bar construction $BM$ as the thin geometric realization of $[p] \mapsto M^p$. I am looking for a reference for the fact that $BM$ is again a commutative ...
- 965
3
votes
2
answers
364
views
Is the mapping cylinder a replacement for morphism by cofibration in model categories?
Let $M$ be a model category, consider a very good cylinder object $X \coprod X \to X \times I \overset{\operatorname{pr}}{\to} X$ (here $X \times I$ is just a notation, no object $I$ is implied), that ...
- 6,962
9
votes
2
answers
621
views
Generalization of the sphere theorem in dimension at least 4
In 1956, Papakyriakopoulos proved Dehn's lemma, loop theorem and the sphere theorem. The proofs are based on a clever technique called "tower construction". Later, Whitehead, Shaprio, ...
- 2,083
3
votes
0
answers
167
views
Suitability of formal type theory for mathematical thinking (vs. traditional set theory)
Type theory has advantages over set theory for the (computer) formalisation of mathematics, but has anybody who does mathematics with pen and paper found proof assistants or automated theorem provers, ...
2
votes
0
answers
101
views
A roof genus of high dimensional lens space
Let $p$ be a natural number, and for $i\in \{0,
..., p-1\}$,
denote the irreducible rank one complex representation of $\mathbb{Z}/p$. by $\rho_{i}$.
Let $a=(a_{1},\ldots a_{d}) $ ...
- 2,630
33
votes
2
answers
2k
views
What happened to the last work Gaunce Lewis was doing when he died?
In 2006, Gaunce Lewis died at the age of 56. He'd done important work setting up equivariant stable homotopy theory, and I think it's fair to say his work was far ahead of its time. In recent years, ...
- 30.3k
2
votes
0
answers
185
views
Compute the Euler class of tautological $C$-bundle over $CP^1$
$\DeclareMathOperator\SO{SO}$This might be an old question. But since I have not found an explicit answer to this question, I put the question here.
The background is that we need to use a similar ...
- 21
33
votes
3
answers
2k
views
The probabilistic method outside of discrete mathematics
The probabilitic method is a genius idea in combinatorics, graph theory etc, where instead of constructing something by hand, you construct the thing randomly and show that there is a positive ...
0
votes
0
answers
32
views
Morse Theory for Time-Periodic Constrained Path Spaces
Let $(M,g)$ be a smooth, compact Riemannian manifold of dimension $n \geq 2$. Define a time-periodic constraint field $\Phi: M \times \mathbb{R} \to \{0,1\}$ with period $T > 0$, where $\Phi(x,t) = ...
7
votes
1
answer
292
views
Computing $\pi_1$ of the complement of a non-singular plane curve
The following is a well-known fact:
Theorem. The fundamental group of the complement of a non-singular curve of degree $d$ in the complex projective plane is cyclic of order $d$.
This was further ...
- 10.9k
2
votes
0
answers
148
views
Equivalence of cohomology with compact support
Let $𝑋$ be a connected CW complex, $𝜌:\pi_1(𝑋)→\mathrm{Aut}(G)$ a representation, and $G_\rho$ the associated sheaf. It follows from here, that the following two cohomologies are isomorphic.
(1)The ...
- 241
6
votes
0
answers
129
views
Are there isospectrally equivalent exotic spheres?
Let $X$ and $Y$ be two different exotic spheres. Are there metrics $g$ and $h$ on $X$ and $Y$, respectively, such that the laplacians of $(X,g)$ and $(Y,h)$ have the same spectrum?
I would be happy ...
5
votes
1
answer
270
views
Are Euclidean spaces $\Delta$-generated?
From the definition of $\Delta$-generated it seems like $\mathbb R$ should be $\Delta$-generated, as $\mathbb R$ is final with respect to all continuous maps $\mathbb R^n \to \mathbb R$.
However, the ...
- 53
3
votes
1
answer
263
views
Original proof of Lefschetz's theorem on $(1,1)$ classes
Is there a "modern" account of Lefschetz proof of his theorem about $(1,1)$ classes for projective surfaces ?
I believe that would be very interesting to understand the original arguments ...
2
votes
0
answers
116
views
Generalized Stokes's theorem and the relationship of volume to surface area of objects of arbitrary genus
In this post I saw that it could be explained with the Generalized Stokes's theorem why the derivative of the area of a circle is equal to the boundary of the circle (the circumference):
$$C=\frac{d}{...
- 131