All Questions
9,056 questions
3
votes
1
answer
623
views
Homotopy colimit commutes with homotopy groups
I'm interested in something built upon the construction laid out in nlab article on Snaith's theorem
Let $(E, \mu, \iota)$ be a ring spectrum.
For $\beta \in \pi_n(E)$ an element of the $n$th stable ...
- 301
1
vote
0
answers
77
views
Contractible orbit space of action of compact Lie group on Euclidean space
R. Oliver proved that the following in https://www.jstor.org/stable/1970955
Theorem: Any action of a compact Lie group on a Euclidean space has contractible orbit space.
My question is that this ...
- 1,367
5
votes
0
answers
150
views
Analytical Dold-Thom
Let's $X$ be a projective smooth variety over a field that has an embedding into $\mathbb{C}$. Let's denote the infinite symmetric power of $X$ by $\text{Sym}^{\infty}(X)$. Denote the algebraic ...
- 5,901
1
vote
0
answers
380
views
G-local systems via the classifying stack BG
First, let $BG$ be the classifying stack of a Lie group $G$ in either Top or Diff (compactly generated topological spaces or differentiable manifolds). A map $f: X \to BG$ determines a principal $G$-...
user501319
6
votes
1
answer
248
views
How small need a perturbation be to not change the diffeomorphism type of a variety?
Let $f,g \in \mathbb{R}[x_0,\dots,x_k]$ be homogeneous polynomials and $X:=Z(f) \subset \mathbb{RP}^k$ be the projective variety defined by $f$.
Assume that $X$ is smooth and has codimension $1$.
Then ...
- 577
3
votes
0
answers
50
views
Natural morphisms between stable unitary, orthogonal, and (compact) symplectic groups
I am a physicist knowing a bit of algebraic topology, and trying to answer the following question.
This is perhaps not appropriate as a question on MO, in which case I apologize.
I posted this ...
- 131
7
votes
1
answer
183
views
Is lambda calculus polymorphism a type of generalized monad?
Let $\mathbf{C}$ be a Cartesian closed category. Then simply typed lambda calculus in $\mathbf{C}$ in one type variable can be interpreted as a category $\mathbf{STLC}_{\mathbf C}$ where the objects ...
1
vote
0
answers
155
views
Lifting action of torus to torus bundle
Preamble: Let $X$ be a simply connected smooth manifold and $P \to X$ be a principal $T^\ell$ bundle on it.
Let $\phi$ be a smooth action of $T^k$ on $X$.
The paper "Lifting compact group actions ...
- 205
7
votes
1
answer
571
views
Different definitions of homotopy colimits
I was reading about the definition of homotopy limits and colimits, and I have seen two different approaches in "Homotopy theories and model categories" by Dwyer and Spalinski, and in "...
- 119
7
votes
1
answer
458
views
Geometric realization of a poset
Consider the finite Boolean lattice $B_n$ of subsets of $[n]:=\lbrace 1,\dots,n\rbrace$ ordered by inclusion, let $1\leq j,k\leq n$ and consider the poset:
$$A_{j,k}=\lbrace\emptyset\neq U\in B_n\mid (...
- 911
80
votes
10
answers
11k
views
What are the uses of the homotopy groups of spheres?
Pete Clark threw down the challenge in his comment to my answer on Why the heck are the homotopy groups of the sphere so damn complicated?:
Have the homotopy groups of spheres ever been applied to ...
- 26.8k
11
votes
1
answer
302
views
Koszul duals of n-fold loop spaces
Suppose $X$ is a finite $n$-connected CW complex, then the function spectrum $F(\Sigma^\infty_+ X,S^0)$ is an $E_\infty$-ring spectrum, induced by the diagonal of $X$. I believe it is known that if we ...
- 5,849
2
votes
0
answers
147
views
Extension of isotopies
In what follows $M$ will be a manifold (without boundary, for simplicity) and $C\subseteq M$ will denote a compact subset. In the paper Deformations of spaces of imbeddings Edwards and Kirby prove the ...
- 641
0
votes
0
answers
161
views
Gluing faces of n-cube
Assuming $C_n$ be the $n$-cube, the intersection of $C_n$ with a supporting hyperplane $H(P, v)$ is called a face or more precisely a $d$-face if the dimension is $d$.
Let $f_0$ and $f_1$ be faces ...
- 53
6
votes
1
answer
393
views
Algebra generated by transformation matrices
Let $T_n$ be the full transformation monoid of an $n$-set $N_n$ with elements 1,...,n consisting of all functions $f: N_n \rightarrow N_n$.
We can associate to each function $f$ a matrix $M_f$ in the ...
- 26.5k
9
votes
0
answers
147
views
degree 1 maps for bordism homology
Let $f\colon X \to Y$ be a degree 1 map between closed oriented manifolds. Then the induced homomorphism between the homology groups is surjective up to torsion.
Can one say something similar about (...
2
votes
0
answers
104
views
Unordered configuration space with non-distinct points
Consider a topological space $X$, a natural number $n>0$ and
the quotient topological space $Q_n(X)$ of $X^n$ by the equivalence relation : $x\sim y$ if and only if
there is a permutation $\sigma$ ...
- 1,035
12
votes
1
answer
500
views
$\pi_k(\mathbb{S}^n\vee\mathbb{S}^n)$
According to Hilton-Milnor theorem for $n\geq 2$
$$
\pi_k(\mathbb{S}^n\vee\mathbb{S}^n)=
\pi_k(\mathbb{S}^n)\oplus
\pi_k(\mathbb{S}^n)\oplus
\bigoplus_{i=1}^\infty
\pi_k(\mathbb{S}^{m_i}),
$$
where $...
- 28k
21
votes
1
answer
754
views
Is $\mathbb{CP}^3$ minus two points the universal cover of a compact manifold?
After reading some recent questions on mathoverflow about universal coverings, I am curious about the following:
Is it possible to construct a closed $6$-manifold $M$, with universal cover ...
- 6,995
2
votes
0
answers
92
views
Explicit CW-complex replacement of the space of reparametrization maps
Let $P$ be the space of nondecreasing surjective maps from $[0,1]$ to itself equipped with the compact-open topology: $P$ is contractible. There exists a trivial fibration $P^{cof} \to P$ from a CW-...
- 3,901
21
votes
7
answers
1k
views
Reference for topological graph theory (research / problem-oriented)
I would be interested in recommendations for topological graph theory texts. I think Gross and Yellen has a great chapter on topological graph theory, and I find Mohar and Thomassen's Graphs on ...
13
votes
1
answer
513
views
Local systems arising from higher rational homotopy groups
I should mention I have very little background in algebraic topology and don't really know much about homotopy groups besides the definition.
I am aware that for a topological space $X$ and a point $x ...
- 233
13
votes
2
answers
2k
views
Why doesn't local cohomology seem to be used as much in algebraic geometry?
In algebraic topology, relative (co)homology is very useful. For example, we have a long exact sequence which is often helpful for lots of calculations.
In algebraic geometry, we have local cohomology,...
- 721
1
vote
0
answers
160
views
Higher dimensional Seifert surfaces and link numbers of higher knots
In 3-manifold topology, the notion of Seifert surface is well known. It is then used to define link numbers of knots.
Question: Consider embedding $N^n \rightarrow M^{2n+1}$ of n-dimensional manifold $...
- 593
89
votes
5
answers
16k
views
Why higher category theory?
This is a soft question.
I am an undergrad and is currently seriously considering the field of math I am going into in grad school. (perhaps a little bit late, but it's better late then never.) I ...
10
votes
1
answer
749
views
How Mayer-Vietoris follows from a six-functor formalism
In many reasonable six-functor formalisms, open and closed immersions satisfy the so-called recollement conditions. (This holds in all the "constructible" formalisms. For example, in the ...
- 721
5
votes
0
answers
133
views
Division of fibration by $\Sigma_{n}$ gives Serre fibration
This is related to a question posted on StackExchange: https://math.stackexchange.com/questions/4776877/left-divisor-of-a-fibration-by-compact-lie-group-is-a-fibration. The question there had received ...
- 311
6
votes
1
answer
237
views
Canonical reference for dictionary between $G$-spaces and fiber bundles over $BG$?
I'm looking for a comprehensive reference (for citation purposes) laying out the basic facts of the equivalence between $G$-spaces and bundles over $BG$ for a discrete group $G$. I'd like it to also ...
- 2,044
17
votes
1
answer
507
views
Topology of the space of embedded genus $g$ surfaces in $S^3$
Consider the space of smoothly embedded genus $g$ surfaces in the 3-sphere under the $C^\infty$ topology:
$$\mathcal E_g:=\operatorname{Emb}(\Sigma_g,S^3)/{\operatorname{Diff}(\Sigma_g)}$$
where $\...
- 525
2
votes
0
answers
146
views
Explicit S-duality map
$\DeclareMathOperator{\Th}{Th}$
The Thom space of a closed manifold $M$ ($\Th(M)$) is the $S$-dual to $M_+ (=M \cup \{pt\})$. Let $M$ be embedded in $\mathbf R^{n+k}$. I found the duality map from $S^...
5
votes
1
answer
393
views
Computation of the linking invariant on Lens spaces
Let $L_n(p)$ be the $2n+1$ dimensional Lens space
$$
S^{2n+1}/\mathbb{Z}_p
$$
where the action is given as $z_i\rightarrow e^{\frac{2\pi}{p}}z_i$, $i=1,...,n+1$, with $z_i$ the coordinates of $\mathbb{...
- 813
11
votes
1
answer
513
views
Computing KO^-1 of RP^3 without AHSS
I wanted to compute $\mathit{KO}^{-1}(\mathbb{R}P^3)$ and regrettably I could only think of using the Atiyah Hirzebruch spectral sequence, which seemed like a big overkill but looking at similar ...
- 45
1
vote
0
answers
90
views
Cohomological dimension of the kernel of a homomorphism induced by a singular fibration
I have a very concrete question. Let $N={\Bbb C}-\{\pm 1\}$, and ${\Bbb Z}_2$ is acting on $N$ by rotation around $0$. Consider $M=\{(x,y)\in N^2\ |\ {\Bbb Z}_2x\neq {\Bbb Z}_2y\}$. Let $p:M\to N$ be ...
- 585
13
votes
1
answer
855
views
Applications of equivariant homotopy theory to representation theory
Equivariant homotopy theory focuses on spaces together with some group action on them. Jeroen van der Meer and Richard Wong have a paper where they use equivariant methods to compute the Picard group ...
- 404
4
votes
1
answer
208
views
What should be required from a model category so that the category of algebraic objects in it has the natural model structure?
I have two reference questions
What should be required of a category with finite products so that a (multi-sorted, finitary) Lawvere theory induces a monadic adjunction on it? This should be ...
- 6,962
3
votes
0
answers
125
views
Does symmetric product functor preserve fibrations?
I know that the symmetric product is a functor, cf: https://en.wikipedia.org/wiki/Symmetric_product_(topology)#Functioriality.
My question is, does it preserve fibrations in the category of ...
- 506
5
votes
1
answer
246
views
Gysin isomorphism in de Rham cohomology using currents
I'd like to find a reference for the following fact.
First, some background: we can define de Rham cohomology of a smooth manifold $X$ of dimension $d$ using the de Rham complex
$$
\Omega^0_X\to \...
- 2,044
3
votes
0
answers
639
views
What are some of the big open problems in $4$-manifold theory?
I've recently been studying some Manifold Theory and got very interested in their topological as well as geometric properties. From my understanding of the current literature, most the big and ...
- 139
5
votes
0
answers
154
views
Hochschild cohomology of path algebra as a cohomology of simplicial complex
M. Gerstenhaber and S. D. Schack have shown that a cohomology of simplicial complex can be expressed as a Hochschild cohomology of path algebra constructed from this complex (link).
Is the opposite ...
- 91
-2
votes
1
answer
440
views
Relation between $\mathbb{Z}\pi_1 (X)$-module $\pi_n (X)$ and $\mathbb{Z}$-module $H_n (X)$ or $\mathbb{Z}\pi_1 (X)$-module $H_n (\tilde{X})$
Let $X$ be a finite CW-complex of $n$.
For $i\geq 2$, $\pi_i (X)$ is a $\mathbb{Z}\pi_1 (X)$-module.
for $i\geq 2$, $H_i (\tilde{X})$ is a finitely generated $\mathbb{Z}\pi_1 (X)$-module, where $\...
- 1,182
2
votes
1
answer
147
views
Is a left Bousfield localization of simplicial presheaves a locally cartesian closed model category?
Let $\mathcal{C}$ be a small category and let $\mathcal{M} = \operatorname{sPre}(C)$ be the model category of simplicial presheaves on $\mathcal{C}$ with the injective model structure.
Let $S$ be a ...
- 608
5
votes
0
answers
249
views
Algebraic de Rham cohomology with torus coefficients
Let $X$ be a smooth projective variety over $\mathbb{C}.$
On page 3 in this preprint of Simpson, it is stated that
Notice first of all that the algebraic de Rham theory is not going to work well in ...
- 123
1
vote
0
answers
107
views
A question about the Conner Conjecture
In some sources, Conner conjecture is expressed as follows:
Theorem [Conner Conjecture] Let $G$ be a compact Lie group, and let $X$ have
the homotopy type of a finite dimensional $G$-CW complex with ...
- 1,367
10
votes
1
answer
638
views
Submersion vs fiber bundle
If one starts with a fiber bundle $f: X \to Y$ so that fibers having trivial integral homology by using spectral sequence one can get the induced map $f_*: H_*(X;\mathbb{Z}) \to H_*(Y;\mathbb{Z})$ is ...
- 1,177
4
votes
1
answer
244
views
Homotopy groups of quotient of SU(n)
Let $X$ be the quotient topological space obtained by identifying the matrices $A$ and $\overline{A}$ in the topological group $\mathrm{SU}(n)$ (here $\overline{A}$ denotes entry-wise complex ...
- 271
0
votes
0
answers
120
views
Topological transversality by dimension
We know that to achieve transverality in the topological category, for example to make a continuous map into a manifold transverse to a topological submanifold, we need the existence of micro normal ...
- 803
119
votes
6
answers
10k
views
What properties make $[0,1]$ a good candidate for defining fundamental groups?
The title essentially says it all. Consider the category $\mathfrak{Top}_2$ of triples $(J,e_0,e_1)$ where $J$ is a topological space, and $e_i \in J$. There is an obvious generalization of the ...
- 5,759
2
votes
1
answer
243
views
Subdivision of geometric simplicial complex
Let $\{v_0,v_1,\cdots,v_n\}$ be $n+1$ points in $\mathbb{R}^N$ which are geometrically independent. We define their convex hull to be a geometric simplex. Using this we can define geometric simplicial ...
- 141
50
votes
6
answers
7k
views
Total spaces of $TS^2$ and $S^2 \times R^2$ not homeomorphic
Hello,
I'm looking for an invariant to distinguish the homeomorphism types of homotopy equivalent spaces. Specifically, how does one show that the total spaces of the tangent bundle to $S^2$ and the ...
- 931
4
votes
1
answer
168
views
Sheaves and gratings
A grating is a notion in algebraic topology from the 1940 introduced by Alexander. Cartan extended it as follows.
A grating (carapace in french) is defined by a topological space $X$, a module (or a ...
- 18.7k