All Questions
9,056 questions
7
votes
0
answers
218
views
Twisting cochain intuition
I'm currently reading through Ed Brown's paper "Twisted tensor products, I", (MR105687, Zbl 0199.58201) and I couldn't find any simple examples of twisting cochains. I understand all ...
- 171
4
votes
0
answers
121
views
$E_k$-operads and actions on objects inside $k$-tuply monoidal $n$-category
I understood more or less the claim that $k$-tuply monoidal $n$-categories can be seen as $n$-categories equipped with an action of the $E_k$-operad.
For $k=2$, we have a homotopy equivalence $E_2(r) \...
- 1,813
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
4
votes
1
answer
253
views
Intersection pairing on non-compact surface
Let $S$ be a smooth oriented connected $2$-manifold. We have an algebraic intersection pairing $\omega\colon H_1(S) \times H_1(S) \rightarrow \mathbb{Z}$. If $S$ is compact, then this is ...
- 43
1
vote
0
answers
58
views
Which sheaves are good for calculating extraordinary restriction?
Let $X$ be a sufficiently nice locally compact Hausdorff space and let $i:Y\subset X$ be the inclusion map of a sufficiently nice closed subspace. For example, one could take $X$ to be a locally ...
- 23.5k
5
votes
0
answers
160
views
$\infty$-category of spectra and cofibrancy
I have two options for the $\infty$-category of spectra. I would like to know they are equivalent as $\infty$-categories.
Premise: by work of Dwyer and Kan, if we have a simplicial model category, the ...
- 410
4
votes
1
answer
164
views
Are monomorphisms in an $\infty$-topos preserved by $0$-truncation?
Let $\mathfrak{X}$ be an $\infty$-topos and let $f\colon X\to Y$ be a morphism of $\mathfrak{X}$. We say that $f$ is a monomorphism if it is $(-1)$-truncated which means that for every $Z\in\mathfrak{...
- 10.4k
1
vote
1
answer
215
views
Reference about cancellation property for semigroups
Have the semigroups with the following cancellation property been studied?
Property: Let $S$ be a semigroup and $x,y\in S$ such that $xz=yz,$ for all $z\in S,$ then $x=y$.
- 313
8
votes
1
answer
232
views
Product structure in Milnor exact sequence
Let $h^*$ be a (multiplicative) generalized cohomology theory. Let $X$ be a CW complex which is a union of an increasing sequence $X_0 \subset X_1 \subset X_2 \subset \cdots$ of subcomplexes. Then ...
- 959
7
votes
1
answer
843
views
Algebraic K-theory and Witt groups
Let $S$ be a ring with involution (with 2 invertible). Suppose that the non connective algebraic K-theory $K(S)$ is 0 (i.e. $K_{n}(S)=0$, for all $n$).
Can we say something about the (higher) Witt ...
- 855
6
votes
1
answer
426
views
Nilpotency of generalized cohomology
$\newcommand\pt{\mathrm{pt}}$Let $(X,\pt)$ be a connected, pointed, finite CW complex and let $h$ be a generalized cohomology theory. Let $\smash{\tilde{h}}^*(X)$ denote the kernel of restriction $h^*(...
- 959
0
votes
0
answers
138
views
Shub Conjecture and polynomial entropy
The Shub conjecture on topological entropy $h(f)$ of self map f on manifold M says that the topological entropy is greater (or equal) than (to) the log of maximum absolute values of the ...
- 356
6
votes
1
answer
245
views
Fundamental group of the homeomorphism group of a compact manifold
Let $X$ be a compact connected manifold and $\mathcal H(X)$ be the group of all homeomorphisms of $X$, equipped with the compact-open topology. Is the fundamental group of $\mathcal H(X)$ countable? ...
14
votes
0
answers
326
views
When can we extend a diffeomorphism from a surface to its neighborhood as identity?
Let $M$ be a closed and simply-connected 4-manifold and let $f: M^4 \to M^4$ be a diffeomorphism such that $f^*: H^*(M;\mathbb{Z})\to H^*(M;\mathbb{Z})$ is the identity map. Moreover, let $\Sigma \...
- 3,751
13
votes
1
answer
518
views
Low dimensional homotopy groups of $\operatorname{Top}(4)$
$\DeclareMathOperator\Top{Top}$I would like to compute $\pi_3\Top(4)$ and $\pi_4\Top(4)$. It is known that $\Top(4)/O(4) \rightarrow \Top/O$ is 5-connected and
$$
\pi_k(\Top/O) =
\begin{cases}
...
3
votes
1
answer
224
views
LS category of 4-manifolds with free fundamental group
In this math overflow question asked almost 5 years ago about the (normalized) Lusternik-Schnirelmann category of $4$-manifolds, the second-last comment (by Jeff Strom) says the following:
A $4$-...
- 311
4
votes
3
answers
322
views
Equivariant cohomology of fixed points using the localisation theorem
I am trying to understand the Smith-Thom inequality for spaces equipped with an action by a cyclic group and also the case, when it's an equality:
In the following, let $G=\mathbb{Z}/p$, $\mathbb{F}$ ...
- 131
2
votes
1
answer
155
views
Unimodular intersection form of a smooth compact oriented 4-manifold with boundary
Let $X$ be a smooth compact oriented 4-manifold with nonempty boundary. Its intersection form $$ Q_X : H^2(X,\partial X;\Bbb Z)/\text{torsion}\times H^2(X,\partial X;\Bbb Z)/\text{torsion}\to \Bbb Z$$
...
- 335
4
votes
1
answer
359
views
Nerve theorem for simplicial sets
There are various kinds of nerve theorems. I am wondering if the following version of nerve theorem for simplicial sets is true:
Let $X:\Delta^{\mathrm{op}}\to \mathrm{Set}$ be a simplicial set. Let $\...
- 173
0
votes
1
answer
91
views
Topological Properties of Subsets of $R^{m}$ induced by Smooth Manifolds in Matrix Spaces
We know that $M_{m \times n } $ is isomorphic to $R^{mn}$. Let's take a smooth manifold $\mathbb{M}$ in $R^{mn}$ and fix a point in $R^{n}$, say, $p$. Now realize the manifold $\mathbb{M}$ as a subset ...
- 101
2
votes
0
answers
116
views
Generalized Stokes's theorem and the relationship of volume to surface area of objects of arbitrary genus
In this post I saw that it could be explained with the Generalized Stokes's theorem why the derivative of the area of a circle is equal to the boundary of the circle (the circumference):
$$C=\frac{d}{...
- 131
0
votes
1
answer
219
views
Finding $\mathbb{C}(u,v)$ such that $\mathbb{C}(u,v,x^p+y^p)=\mathbb{C}(x,y)$, for every prime number $p$
Denote the set of prime numbers by $P$, $P=\{2,3,5,7,\ldots\}$.
Let $F \subseteq \mathbb{C}(x,y)$ be a subfield of $\mathbb{C}(x,y)$, and for $w \in \mathbb{C}[x,y]$ denote by $F(w)$ the subfield of $\...
- 2,837
4
votes
1
answer
523
views
Is automorphism on a compact group necessarily homeomorphism? How about N-dimensional torus？ [closed]
Is automorphism on a compact group necessarily homeomorphism? I don't think so，but I think it is possible on the N-dimensional torus.
- 83
3
votes
1
answer
130
views
Why is the Vietoris–Rips complex $\operatorname{VR}(S, \epsilon)$ a subset of the Čech complex $\operatorname{Čech}(S, \epsilon\sqrt{2})$?
$\DeclareMathOperator\Cech{Čech}\DeclareMathOperator\VR{VR}$I am reading Fasy, Lecci, Rinaldo, Wasserman, Balakrishnan, and Singh - Confidence sets for persistence diagrams (see here for a version of ...
4
votes
1
answer
148
views
Seifert invariants for Brieskorn manifolds $\Sigma(p,q,r)$
I've been studying Brieskorn manifolds $\Sigma(p,q,r)$ for $p,q,r\geq 2$. I know they are defined as the intersection of the complex surface $z_1^p+z_2^q+z_3^r=0$ as a subset of $\mathbb{C}^3$ and $S^...
- 197
2
votes
1
answer
95
views
How to determine the LS category of branched covers?
Define the (normalized) Lusternik-Schnirelmann (LS) category of a space $X$, denoted $\mathsf{cat}(X)$ to be the least integer $n$ such that $X$ can be covered by $n+1$ number of open sets $U_i$ each ...
- 31
8
votes
0
answers
242
views
Tannaka reconstruction for homotopy types
All sorts of things can be reconstructed from their "linear representations". One example is Tannaka (Deligne, Tannaka-Krein, etc.) reconstruction where a group is recovered from its ...
- 12.3k
3
votes
1
answer
240
views
Cohomology of the complement of a subvariety
Let $X$ be a complex manifold, $Y\subset X$ a subvariety, and $U:=X\setminus Y$ of codimension $d$. It is well known that the restriction map
$$
H^i(X,\mathbb Q)\to H^i(U,\mathbb Q)
$$
is an ...
- 178
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
3
votes
0
answers
93
views
References for variations of Seifert–van Kampen's theorem: HNN extensions and "sensible" intersections
A basic consequence of the Seifert–van Kampen theorem is the following.
Theorem: Consider a union of topological spaces $X$, $Y$ whose intersection $X\cap Y = Z$ is open connected and $\pi_1$-...
- 1,033
2
votes
0
answers
27
views
Topological meaning of a "totally recurrent" 1d foliation in 3-manifold
I'm trying to understand Sullivan's "cycles for the dynamical study..":
https://www.math.stonybrook.edu/~dennis/publications/PDF/DS-pub-0033.pdf
which I find very complicated being ...
- 111
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) = ...
3
votes
1
answer
431
views
Detecting a PL sphere and decompositions
Question 1. A PL $n$-sphere is a pure simplicial complex that is a triangulation of the $n$-sphere. I'm looking for a computer algorithm that gives a Yes/No answer to whether a given simplicial ...
- 903
4
votes
0
answers
107
views
Reference request for a theorem of Jaworowski
Jan Jaworowski, in 2000, proved the following theorem (I came to know about it from here)
Jaworowski (2000) : Let $Y$ be a finite simplicial complex of dimension $k$ and let $n\ge 2k$. If $f:S^n\to Y$...
- 141
5
votes
1
answer
245
views
Does a "good" homotopy equivalence between pairs imply homotopy equivalence between quotient spaces?
If $(X,A)$ and $(Y,B)$ are (good) pairs of topological spaces, and $f:X\rightarrow Y$ is a homotopy equivalence such that the restriction $f\restriction_A$ is a homotopy equivalence between $A$ and $B$...
- 213
2
votes
1
answer
174
views
A topological space has the homotopy-type of a CW-complex, then is it locally contractible?
Let $X$ be a topological space which has the homotopy-type of a CW-complex. As well-known, a CW-complex is locally contractible.
Question: Is $X$ locally contractible? If not, can some one give a ...
- 327
2
votes
1
answer
57
views
Are simplicial commutative inverse semigroups fibrant?
Let $X$ be a simplicial object in the category of commutative inverse semigroups (or monoids, if needed). Is the underlying simplicial set of $X$ always a Kan complex? If so, are there some nice ...
- 742
5
votes
0
answers
93
views
Connectivity of a induced map between homotopy pullbacks
Consider cospans of continuous maps $A\stackrel{f}{\rightarrow }C\stackrel{g}{\leftarrow }B$ and $A'\stackrel{f'}{\rightarrow }C'\stackrel{g'}{\leftarrow }B'$ along with maps $\alpha :A\to A'$, $\beta ...
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
2
votes
0
answers
137
views
details of a dévissage argument for constructible sheaves
I am working on the following Künneth-type isomorphism from [SGA5, exposé III, 2,3]:
$\mathrm{Settings}.$ Let $X_1, X_2$ be separated finite type schemes over the spectrum of a field $S=\mathrm{Spec}...
- 375
2
votes
0
answers
193
views
A $\mathbb{Z}_2$-equivariant map from $n$-torus to $2$-sphere that is null-homotopic is $\mathbb{Z}_2$-homotopic to a non-surjective map?
I have been thinking on the problem below for a while and I am not sure if it is correct or not. I am trying to see if there exists a counter-example for the problem below.
Problem: Let $f: (S^1)^n \...
- 21
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\...
8
votes
1
answer
485
views
A question about cohomology of the classifying spaces of compact groups
Let $G$ be a compact group (maybe non-Lie group). Let $B_{G}$ denote the
classifying space of $G$. If $G$ contains a circle group $\mathbb{S}^{1}$,
then I think that $H^{\ast }( B_{G};\mathbb{Q}
)$ is ...
- 1,367
8
votes
0
answers
118
views
Defining convex sums locally on the sphere?
$S^1$ and the torus $T^2$ are spaces in which convex combinations don't make sense globally but do locally. Despite their standard representations in $\mathbf{R}^2$ and $\mathbf{R}^3$ respectively not ...
- 637
3
votes
2
answers
246
views
Explicit description of transfer for $K_1$
Let $R$ be a commutative regular ring and let $s \in R$ be an element such that $R / s$ is also regular. Then we have a long exact localization sequence
$$
\ldots \rightarrow K_i(R/s) \rightarrow K_i(...
7
votes
0
answers
237
views
Relation beween Chern-Simons and WZW levels, and transgression
3d Chern-Simons gauge theories based on a Lie group $G$ are classified by an element $k_{CS}\in H^4(BG,\mathbb{Z})$, its level. Via the CS/WZW correspondence the theory is related with a 2d non-linear ...
- 813
2
votes
1
answer
179
views
Model structures on simplicial presheaves of piecewise-linear manifolds
Let $\mathbf{PL}$ denote the category of piecewise-linear manifolds. The goal is to embed $\mathbf{PL}$ into a category of simplicial presheaves, endow it with a model structure, and then localize it ...
user528837
8
votes
0
answers
229
views
A few questions about Priddy’s construction of $BP$
In A Cellular Construction of BP and Other Irreducible Spectra, Priddy gives an interesting approach to constructing the Brown-Peterson spectrum $BP$. His result is often summarized as
If you start ...
- 63.9k
0
votes
1
answer
73
views
If $\widehat{\Gamma}$ is a simply connected clique complex then $\mathrm{Out}(A_\Gamma)$ is an infinite group
Let $\Gamma$ be a simplicial graph and $\widehat{\Gamma}$ be the corresponding clique complex (the flag complex obtained after adding simplices for each compete graph). We can costruct the right-...
- 911
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