All Questions
9,056 questions
4
votes
2
answers
374
views
Knot theory in handlebodies of arbitrary genus
It is well known that not all graphs embed on the plane (e.g. the graph $K_{3,3}$). However, every finite graph embeds into a surface of some genus. One can think of this procedure as starting with a ...
5
votes
1
answer
191
views
Regular polyhedral spaces
By symmetrically gluing together opposite faces of a dodecahedron together, one of three spaces can be obtained, depending on the angle the faces are rotated by before twisting. In fact, this can be ...
- 2,773
11
votes
1
answer
367
views
Lower bounds for Betti numbers of a manifold given its boundary?
Let $B$ be some compact, path connected $n$-manifold without boundary such that its cobordism class is trivial, so that there exists some other $n+1$ manifold $M$ with $\partial M= B$. While there is ...
9
votes
1
answer
339
views
Stably-framed cobordism $(\infty,n)$-category
In Lurie's treatment of the cobordism hypothesis, the domain is $\mathsf{Bord}^{fr}_n$, the symmetric monoidal $(\infty,n)$-category of $k$-bordisms with $n$-framing for $0\leq k\leq n$.
If I ...
- 663
14
votes
2
answers
873
views
sSet-enriched categories, quasi-categories and the model-independent theory
sSet-enriched categories are one possible model for $(\infty,1)$-categories, by the work of Bergner and others. They are probably the most important model from the point of view of getting actual ...
- 1,514
2
votes
0
answers
433
views
Classification of homotopy types of topological spaces
Do higher groups classify the homotopy types of topological spaces?
We may assume $\pi_n$ of the topological spaces are all finite
and $\pi_n =0$ for large enough $n$.
For example, if only $\pi_1 \neq ...
- 4,826
5
votes
0
answers
179
views
Zigzag vs direct map in rational homotopy theory
I was reading these notions from "Rational Homotopy Theory" by Felix, Halperin, and Thomas.
The notion of weak homotopy type is as follows: two spaces $X$ and $Y$ are said to be weak ...
- 349
6
votes
1
answer
626
views
Geometric interpretation of transfer map on homology
Let $f\colon M\to N$ a smooth surjective map of compact oriented manifolds of the same dimension. Then there is a map $f_!\colon H_i(N)\to H_i(M)$ obtained from the induced map on cohomology combined ...
- 3,031
0
votes
0
answers
69
views
Number of connective orbit types of torus actions
Suppose that topological group $G$ acting on topological space $X$. If the
set $\left\{ \left[ G_{x}\right] :x\in X\right\} $ is finite, where $\left[
G_{x}\right] $ denotes the conjugacy class of the ...
- 1,367
3
votes
0
answers
90
views
When does homology preserve inverse limits of Eilenberg-MacLane spaces?
Let $... \to G_3 \to G_2 \to G_1$ be an inverse system of abelian groups and $G$ the limit of the system. By a theorem of Goerss the integral homology of the Eilenberg-MacLane space $K(G,n)$ for $n &...
- 499
80
votes
15
answers
15k
views
Why torsion is important in (co)homology ?
I've once been told that "torsion in homology and cohomology is regarded by topologists as a very deep and important phenomenon". I presume an analogous statement could be said in the context of ...
15
votes
1
answer
954
views
Extending diffeomorphisms
Suppose we have a diffeomorphism $f:{\mathbb{S}}^n_{+}\to\mathbb{S}^n$ of class $C^1$ of the closed upper hemisphere onto a submanifold of $\mathbb{S}^n$ with boundary.
Question. Is it possible to ...
- 28k
8
votes
1
answer
217
views
Integral homology classes of which no multiples admit embedded representatives with trivial normal bundle
Let $M$ be a closed smooth manifold of dimension $n$ and $z\in H_l(M,\mathbb{Z})$ a $k$-dimensional integral homology class. Theorem II.4 of Thom's classical 1954 paper states that for $l< n/2$ or $...
- 587
1
vote
1
answer
152
views
Compact locus in (ordered) configuration spaces
Let $\mathit{Conf}_n(\mathbb{R}^2)$ be the configuration space of $n$ ordered distinct points in the plane. I'd like to know if the topological subspace $C_n$ consisting of points $(p_1,...,p_n)$ with ...
- 1,125
6
votes
1
answer
479
views
Abstract simplicial complexes - Reference for an elementary definition of mapping degree for simplicial maps?
I am interested to use the mapping degree for simplicial maps between (oriented) abstract simplicial complexes. What I mean by "elementary": My preference would be to use a definition of ...
- 6,937
2
votes
0
answers
190
views
Completeness of the Infinity Category of A-infinity Categories
Is the infinity category of $A_{\infty}$-categories complete? By complete I mean do there exist arbitrary homotopy limits in the infinity category of $A_{\infty}$-categories?
I felt like this result ...
- 341
87
votes
11
answers
14k
views
What is Quantization ?
I would like to know what quantization is, I mean I would like to have some elementary examples, some soft nontechnical definition, some explanation about what do mathematicians quantize?, can we ...
4
votes
0
answers
184
views
How to think about Beilinson's gluing data?
Let $X$ be a complex manifold, $D$ a divisor (that is globally the zero locus of a function) and $U$ its complement. Recall Beilinson's "how to glue perverse sheaves":
Given a perverse ...
- 5,711
5
votes
1
answer
387
views
When the Pontryagin square is an even class?
Let $n$ be an even integer and $X$ a manifold. Given a cohomology class $B \in H^k(X,\mathbb{Z}_n)$, the Pontryagin square is a class $\mathfrak{P}(B)\in H^{2k}(X,\mathbb{Z}_{2n})$. Is it true that if ...
- 813
2
votes
1
answer
287
views
How does hyperelliptic involution act on the standard generators of the fundamental group of surfaces of genus g with n punctures?
Let $S_{g,n}$ be the surface of genus $g$ with $n$ punctures. We know that $\pi_1(S_{g,n})$ admits a presentation:
$$\left\langle~ \alpha_1,\beta_1,\dots, \alpha_{g},\beta_{g},\gamma_{1},\dots,\gamma_{...
- 79
68
votes
12
answers
29k
views
Algebraic topology beyond the basics: any texts bridging the gap?
Peter May said famously that algebraic topology is a subject poorly served by its textbooks. Sadly, I have to agree. Although we have a freightcar full of excellent first-year algebraic topology texts ...
3
votes
0
answers
103
views
An isomorphism problem for semigroups of ideals
An ideal of a semigroup $S$ (written multiplicatively) is a set $I \subseteq S$ such that $IS$ and $SI$ are both contained in $I$ (here, $XY$ means, for all $X, Y \subseteq S$, the setwise product of $...
- 10.5k
122
votes
7
answers
15k
views
Topology and the 2016 Nobel Prize in Physics
I was very happy to learn that the work which led to the award of the 2016 Nobel Prize in Physics (shared between David J. Thouless, F. Duncan M. Haldane and J. Michael Kosterlitz) uses Topology. In ...
93
votes
3
answers
11k
views
What is homology anyway?
Disclaimer: I don't feel qualified to ask this question and yet it's been troubling me for some time now and I lost my patience and decided to ask to get some kind of answer. If there are any stupid ...
- 7,789
10
votes
1
answer
259
views
Space with compactly closed diagonal but which is not weak Hausdorff
Using the definitions from Peter May's A Concise Course in Algebraic Topology, a topological space $X$ is weak Hausdorff if for every compact Hausdorff space $K$ and continuous function $f:K\to X$, $f(...
- 317
59
votes
4
answers
5k
views
When can one continuously prescribe a unit vector orthogonal to a given orthonormal system?
Let $1 \leq k < n$ be natural numbers. Given orthonormal vectors $u_1,\dots,u_k$ in ${\bf R}^n$, one can always find an additional unit vector $v \in {\bf R}^n$ that is orthogonal to the preceding ...
- 114k
3
votes
1
answer
260
views
Can such a set be simply connected?
$\newcommand\R{\mathbb R}$Let $U$ be an open subset of $\R^2$ such that the point $(0,0)$ is on the boundary of $U$. Let $f\colon[0,1]\to\R^2$ be the path that starts at $(0,0)$ and moves with a (say) ...
- 128k
0
votes
1
answer
274
views
Universal covering of symmetric product
Let $C$ be a 1-dimensional complex manifold whose universal covering is provided by the half-plane $\mathcal{H}=\{z \in \mathbb{C} \mid \operatorname{Im}z>0\}$. The symmetric product $C^{(n)} = C^n ...
- 153
3
votes
1
answer
215
views
Restriction of a fibration to an open subset with diffeomorphic fibers
Let $p:E\to B$ be a locally trivial fibration of connected, non-compact smooth manifolds. Let $U\subset E$ be a connected open subset and $p|_U:U\to p(U)$ has diffeomorphic fibers.
Can we conclude ...
- 585
14
votes
1
answer
360
views
The first two $k$-invariants of $\mathrm{pic}(KU)$ and $\mathrm{pic}(KO)$
$\DeclareMathOperator\Pic{Pic}\DeclareMathOperator\pic{pic}$Real and complex topological $K$-theories, $KO$ and $KU$, have Picard spectra $\pic(KO)$ and $\pic(KU)$ built from the $\mathbb{E}_\infty$-...
- 10.4k
7
votes
1
answer
197
views
Are morphisms in a stable $\infty$-category generated by split injections?
I've seen it stated in the $\infty$-categorical literature (without proof or reference) that every object in the $\infty$-category $\operatorname{Fun}(\Delta^1, \mathcal{C})$ of morphisms in a stable $...
2
votes
1
answer
135
views
Compact objects in persistence modules and interval decomposition
$\newcommand\Mod{\mathrm{Mod}}\DeclareMathOperator\Fun{Fun}$If $k$ is a field, a persistent $k$-module is a functor $\mathbb{R}\to \Mod_k$ where $\mathbb{R}$ is a poset under the natural ordering of $\...
- 213
5
votes
0
answers
171
views
Spectral sequence construction of Euler class of group extension
Let $A$ be an abelian group equipped with an action of a group $G$ and let
$$1 \longrightarrow A \longrightarrow \Gamma \longrightarrow G \longrightarrow 1$$
be an extension of group inducing the ...
- 51
3
votes
3
answers
423
views
Pairing between cohomology and the image of the Hurewicz homomorphism
Let $X$ be a compact manifold of dimension $\geq k$. Denote by
\begin{equation}
h: \pi _k(X) \rightarrow H_k(X,\mathbb{Z})
\end{equation}
be Hurewicz homomorphism and by $\Gamma _k(X)\subset H_k(X,\...
- 813
1
vote
1
answer
256
views
Reference request and prerequisites for understanding the Sphere Theorem and the Loop Theorem in 3-manifold theory
As part of my directed studies project, my advisor has suggested that I completely understand the proof of the Sphere Theorem and the Loop Theorem in 3-manifold theory and explain it to him. I have ...
- 131
1
vote
1
answer
149
views
Relative $G$-equivariant homology groups
Let $X$ be a free $G$-CW-complex with $G$-equivariant cell filtration by
$n$-skeleta $X_0 \subset \dots \subset X_n \subset \dots \subset X$ (for
rigorous definition see
Chap. II, p. 98 in linked ...
- 6,038
3
votes
2
answers
257
views
Cancelable commutative monoids with finite maximal subgroups
Suppose $\mathcal{M} = (M, +, 0)$ is a cancelable commutative monoid. Let $G$ be the maximal subgroup of $M$, i.e.
$$G = \{ a \in M \colon (\exists b \in M)\, a + b = 0 \}.$$
For $a, b \in M$ say $a \...
- 2,221
5
votes
0
answers
165
views
Equivalent descriptions of equivariant K-theory
I am looking at references for computing $$K_{T}(G/H)$$ where $G$ is a compact connected Lie group with maximal torus $T$, and $H\subset G$ is a corank one Lie subgroup such that $G/H\cong S^{2k-1}$ ...
- 51
7
votes
2
answers
500
views
Is the localised $S^1$-equivariant cohomology of the free loop space of a space $X$ isomorphic to that of $X$ itself?
A well-known theorem of Atiyah and Bott states that given a finite dimensional oriented manifold $M$ with circle action, the $S^1$-equivariant cohomology of $M$ (with $\mathbb{Q}$ coefficients) is ...
- 489
4
votes
0
answers
93
views
What is the Goldie dimension of the ring of stable stems?
Let $p$ be a prime, and let $\pi_\ast^{(p)}$ be the ring of stable homotopy groups of spheres localized at the prime $p$. This is a nonnegatively-graded-commutative ring with $\mathbb Z_{(p)}$ in ...
- 64k
10
votes
1
answer
362
views
Example of non-transitive homotopy relation
$\DeclareMathOperator{\Hom}{Hom}$
Dear all,
The question is for teaching purposes and rather basic, so I hope that it also allows (relatively) easy answer.
By abstract homotopy theory we know that if ...
- 1,759
2
votes
1
answer
130
views
Space of the trivial long knot in the thickened surface
Let $F$ be a compact oriented surface and $x_0\in F$ a basepoint. Consider the set $\mathcal E=Emb_0(I,F\times I)$ of embeddings $\sigma\colon I\hookrightarrow F\times I$, $\sigma(\partial I)=\{x_0\}\...
- 357
16
votes
3
answers
797
views
"Phantom" non-equivalences of spectra?
I would like an example of the following situation, or a proof that no such example exists.
$\textbf{Situation}$: Two connective (EDIT: I'm fine with dropping this condition) spectra $X$ and $Y$ such ...
- 2,052
10
votes
1
answer
332
views
Which spectra have a universal connective quotient?
Consider the homotopy category $\mathrm{hoSp}$ of spectra. It has a full subcategory $\mathrm{hoSp}_{\geq 0}$ of connective spectra, equivalently of infinite loop spaces, equivalently $E_\infty$-group ...
- 54.6k
8
votes
1
answer
357
views
Stable splitting of $\Omega SU(n)$
The space $\Omega SU(n)$ is homotopy-equivalent to $SL_n(\mathbb{C}[z,z^{-1}])/SL_n(\mathbb{C}[z])$. Using this, Steve Mitchell introduced a filtration of $\Omega SU(n)$ by subspaces $F_k$ which can ...
- 56.9k
6
votes
1
answer
312
views
A Tate resolution for $\Sigma_p$ - Reference request
Below I will describe a mod $p$ Tate resolution for the symmetric group $\Sigma_p$, i.e. a $\mathbb{Z}$-graded periodic acyclic chain complex $C^*$ of finitely generated modules over $\mathbb{F}_p[\...
- 56.9k
128
votes
12
answers
12k
views
Spectral sequences: opening the black box slowly with an example
My friend and I are attempting to learn about spectral sequences at the moment, and we've noticed a common theme in books about spectral sequences: no one seems to like talking about differentials.
...
- 13.5k
8
votes
1
answer
350
views
When can I extend a map of spectra?
Suppose I have a commutative ring $R$. Given an element $(x_1,x_2)\in R^2$ there exists a homomorphism $\mathbb{Z} \to R\otimes R$ taking $1$ to $x_1\otimes x_2$, so there exists a map $f:S^0 \to HR \...
- 1,025
76
votes
9
answers
15k
views
understanding Steenrod squares
There is a function on $\mathbb{Z}/2\mathbb{Z}$-cohomology called Steenrod squaring: $Sq^i:H^k(X,\mathbb{Z}/2\mathbb{Z}) \to H^{k+i}(X,\mathbb{Z}/2\mathbb{Z})$. (Coefficient group suppressed from ...
- 6,069
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