All Questions
8,725 questions
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
522
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
150
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 ...
- 12.4k
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
247
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
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
195
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
230
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 ...
- 64k
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
11
votes
6
answers
2k
views
Hard problems with an easy-to-understand answer
I am very interested by problem in mathematics which are difficult (go at least 10 years without a resolution, say) but which have a solution that is short and elementary.
In this video Launay gave an ...
4
votes
1
answer
276
views
Preserving simple-connectedness under intersection complexes
Given a simplicial complex $X$, and a family of its subcomplexes $\{U_i\}_{i\in I}$, we define the corresponding intersection complex to be the simplicial complex $X_U$ with vertex set $I$ where $A \...
- 1,936
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
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
2
votes
0
answers
179
views
Isomorphic simplicial complexes are simple-homotopy-equivalent: reference?
Any two isomorphic simplicial complexes are simple-homotopy-equivalent.
This is a fairly simple result, but it is not obvious. Yet I have been surprisingly unable to find it in the literature on ...
- 33.8k
1
vote
0
answers
54
views
Estimating the growth rate around singular points of the analytic continuation of functions of Nilsson class defined by an integral
In Lemma 8 of the paper "Constant terms in powers of a Laurent polynomial" (by J.J. Duistermaat and Wilberd van der Kallen) the exponent $\alpha$ in the asymptotic expansion of a function of ...
- 360
18
votes
1
answer
980
views
Are Eilenberg-MacLane spaces limits of manifolds?
In this answer, mme shows that for any compact Lie group $G$, there is a model for its classifying space $BG$ which is the direct limit of closed manifolds. If $G$ is discrete (i.e. $\dim G = 0$), ...
- 19.3k
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
11
votes
1
answer
457
views
Steenrod powers of the Thom class
René Thom in 1952 proved the formula
$$
Sq^i(U_2)=\Phi_2(w_i),
$$
which in modern parlance says that the Steenrod squares of the mod $2$ Thom class of an orthogonal bundle are the images under the mod ...
- 35.9k
4
votes
1
answer
355
views
Bott & Tu differential forms Example 10.1
In Bott & Tu's "Differential forms", Example 10.1 states:
$\textbf{Example 10.1}$ Let $\pi: E \to M$ be a fiber bundle with fiber $F$. Define a presheaf on $M$ by $\mathcal F(U) = H^q(\...
- 51
1
vote
1
answer
199
views
Gluing $n$ $2(n-1)$-simplices
It should be true that $n$ $2(n-1)$-simplices can be glued together, such that $k$ of them intersect in a common $(2n -k -1)$-cell and the resulting object is a "convex $2(n-1)$-disc" whose ...
- 1,813
8
votes
1
answer
557
views
Relation between 16 $\mathbf{CP}^2$ and $\overline{K3}$
In bordism theory and algebraic topology, 4d spin bordism group is generated by $K3$ surface, while 4d $SO$ bordism group generated by $\mathbf{CP}^2$.
$K3$'s 4-manifold signature is $- 16$
and $\...
- 447
4
votes
1
answer
514
views
A question about spectral sequences
In the following proof (from The pontrjagin numbers of an orbit map and generalized G-signature theorem by Hsu-Tung Ku & Mei-Chin Ku https://link.springer.com/chapter/10.1007/BFb0085610), it is ...
- 1,367
4
votes
2
answers
290
views
Loop-space functor on cohomology
For a pointed space $X$ and an Abelian group $G$, the loop-space functor induces a homomorphism $\omega:H^n(X,G)\to H^{n-1}(\Omega X,G)$.
More concretely, $\omega$ is given by the Puppe sequence
$$\...
- 663