All Questions
8,725 questions
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
23
votes
3
answers
2k
views
What are some toy models for the stable homotopy groups of spheres?
The graded ring $\pi_\ast^s$ of stable homotopy groups of spheres is a horrible ring. It is non-Noetherian, and nilpotent torsion outside of degree zero.
Question: What are some "toy models" ...
- 64k
4
votes
1
answer
255
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
1
answer
134
views
Is the product of torus and sphere a cover of the symmetric square of torus?
Let $T$ denote the $2$-dimensional torus and $T^{(2)}$ denote its symmetric square (which is the orbit space of the canonical $\mathbb{Z}_2$ action on the $4$-torus $T \times T$).
One can see $T^{(2)}$...
- 31
5
votes
0
answers
92
views
For spaces $U$ and discrete sets $I,J$, are maps $f\colon U \times I \rightarrow U \times J$ commuting with the projection to $U$ covering spaces?
Let $U$ be a topological space, let $I$ and $J$ be discrete sets, and let $f\colon U \times I \rightarrow U \times J$ be a continuous map that commutes with the projection onto the first factor. In ...
- 395
5
votes
0
answers
159
views
Topologies on the infinite join
Let $G$ be a topological group. Following Milnor one way of defining the total space of the universal bundle $EG$ of $G$ is to form the infinite join
$$
EG = G^{\ast \infty} = G \ast G \ast \dots
$$
...
- 7,637
93
votes
20
answers
10k
views
Short papers for undergraduate course on reading scholarly math
(I know this is perhaps only tangentially related to mathematics research, but I'm hoping it is worthy of consideration as a community wiki question.)
Today, I was reminded of the existence of this ...
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
8
votes
1
answer
508
views
On the definition of stably almost complex manifold
According to Adams' paper "Summary on complex cobordism", a manifold is
stably almost-complex if it can be embedded in a sphere of sufficiently high dimension with a normal bundle which is a ...
- 959
4
votes
1
answer
183
views
When can a generalized connected sum be aspherical
Let $M$ and $N$ be compact $n$-dimensional manifolds with a common (nicely embedded) compact submanifold $S$ (we may assume that the inclusion of $S$ in $M$ and $N$ is $\pi_1$-injective). Let $M\#_S N$...
- 311
1
vote
0
answers
64
views
Energy minimization and boundary homotopy types of compact manifolds
Let $X$ be a connected, compact, smooth manifold of dimension $n$ with a non-empty boundary $\partial X$. Define the boundary homotopy type of $X$ as the homotopy type of the pair $(X, \partial X)$.
...
7
votes
0
answers
192
views
Eulerian posets and order complexes
To every poset $P$ it is possible to associate its order complex $\Delta(P)$. The faces of $\Delta(P)$ correspond to chains of elements in $P$. An Eulerian poset is a graded poset such that all of its ...
- 1,889
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
8
votes
2
answers
600
views
Derivations in the Steenrod algebra
Let $\mathcal A^\ast$ be the (mod 2) Steenrod algebra.
Question 1:
Is there a classification of homogeneous elements $D \in \mathcal A^n$ such that $D^2 = 0$?
Question 2: Is there a classification of ...
- 64k
3
votes
1
answer
118
views
Characterization of self-conjugate spin$^c$ structures
Let $M$ be an oriented Riemannian $n$-manifold. Then we can choose a trivializing open cover $M=\bigcup_\alpha U_\alpha$ for $TM$ and corresponding transition functions $g_{\alpha \beta}:U_\alpha \...
- 335
4
votes
1
answer
469
views
How to calculate $\mathrm{TP}(\mathbb{F}_p[t])$?
$\DeclareMathOperator\TP{TP}$I am trying to learn about topological periodic cyclic homology following the notes:
https://www.uni-muenster.de/IVV5WS/WebHop/user/nikolaus/Papers/Lectures.pdf
https://...
- 959
6
votes
5
answers
953
views
Two arcs in the complement of a disc must intersect?
Let $D=\{z\in \mathbb C:|z|\leq 1\}$ be the unit disc in the complex plane, with interior $U=\{z\in \mathbb C:|z|<1\}$.
Let $A\subset \mathbb C\setminus U$ be an arc intersecting $D$ only at its ...
- 3,317
7
votes
0
answers
219
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
5
votes
2
answers
333
views
Differing monoidal model structures on a fixed model category
Suppose that $\mathcal{C}$ is a model category (with a fixed model structure). Are there any known examples where $\mathcal{C}$ is a (symmetric) monoidal model category in two different ways? I.e., ...
- 195
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
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
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(...
8
votes
1
answer
470
views
Non-triviality of Whitehead products in wedges of CW-complexes
Suppose $X$ and $Y$ are finite, simply connected, based CW-complexes and $m,n\geq 2$. If $a\in \pi_m(X)$ and $b\in \pi_n(Y)$, then one can regard these as elements of the homotopy groups of $X\vee Y$. ...
- 517
5
votes
2
answers
442
views
Is every codimension-one homology class of a closed manifold represented by a $\pi_1$-injective embedded submanifold?
Let $M$ be a connected closed orientable smooth $n$-manifold and $\nu \in H_{n-1}(M, \mathbb{Z})$ a non-trivial codimension-one homology class. It is known that $\nu$ can be represented by an embedded ...
- 302
150
votes
31
answers
70k
views
What are the most misleading alternate definitions in taught mathematics?
I suppose this question can be interpreted in two ways. It is often the case that two or more equivalent (but not necessarily semantically equivalent) definitions of the same idea/object are used in ...
74
votes
51
answers
28k
views
An example of a beautiful proof that would be accessible at the high school level?
The background of my question comes from an observation that what we teach in schools does not always reflect what we practice. Beauty is part of what drives mathematicians, but we rarely talk about ...
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
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
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
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{...
- 10.5k
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
109
votes
28
answers
41k
views
Why should one still teach Riemann integration?
In the introduction to chapter VIII of Dieudonné's Foundations of Modern Analysis (Volume 1 of his 13-volume Treatise on Analysis), he makes the following argument:
Finally, the reader will ...
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
3
votes
0
answers
85
views
de Rham cohomology relative to a closed subset
I am interested whether there exists a versions of de Rham relative cohomology $H^\bullet(M, N)$ in which $N$ does not need to be a manifold. I know there are two main definitions in literature as ...
3
votes
3
answers
537
views
Fundamental group of a generalized connected sum
Let $M$ and $N$ be two $n$ dimensional connected closed manifolds with $n \ge 3$, and let $S$ be a $(n-1)$ dimensional closed submanifold common to $M$ and $N$. Consider the connected sum of $M$ and $...
- 311
25
votes
6
answers
3k
views
What is the standard 2-generating set of the symmetric group good for?
I apologize for this question which is obviously not research-level. I've been teaching to master students the standard generating sets of the symmetric and alternating groups and I wasn't able to ...
- 4,492
1
vote
0
answers
37
views
Asymptotic growth of twisted alexander polynomials and hyperbolic volume for infinite families of knots
Let $\{K_n\}_{n=1}^\infty$ be an infinite family of hyperbolic knots with increasing crossing number, and let $\rho_n: \pi_1(S^3 \setminus K_n) \to SL_N(\mathbb{C})$ be a sequence of irreducible ...
13
votes
1
answer
580
views
Identifying two definitions of orientation on a vector space
Let $V$ be an $n$-dimensional real vector space. Here are two definitions of an orientation on $V$:
A generator of the $1$-dimensional vector space $\wedge^n V$, up to multiplication by positive ...
- 133
13
votes
2
answers
1k
views
Categories on which one can determine all model structures?
Famously, there are exactly nine model structures on the category of sets, which are detailed here. In this case, one can exhaustively determine all six weak factorization systems and then see which ...
- 64k
2
votes
0
answers
100
views
A question about Milnor space
Let $G$ be a compact group and $H$ be a closed subgroup of $G$. I know that
if $G\rightarrow G/H$ is a principal $H$-bundle, then we can choose $E_{H}$ as $%
E_{G}$, where $E_{G}$ is the Milnor space ...
- 1,367
86
votes
44
answers
21k
views
Demystifying complex numbers
At the end of this month I start teaching complex analysis to
2nd year undergraduates, mostly from engineering but some from
science and maths. The main applications for them in future
studies are ...
5
votes
3
answers
497
views
Bar construction in commutative algebras is calculated by pushout
$\DeclareMathOperator\colim{colim}$
Also asked in MathStackexchange here
This is a statement in Lurie's Higher Algebra 5.2.2.4.
Proposition 3.2.4.7 in HA said that the monoidal structure on $\text{...
- 163
7
votes
1
answer
425
views
Does there exist a Bousfield localization of the category of spectra which makes the sphere unbounded below?
Let $Sp$ be the category of spectra. Let $L : Sp \to Sp_L$ be the localization functor onto a reflective subcategory.
Question 1: Is it ever the case that $L(S^0)$ is not bounded below?
Question 2: ...
- 64k
2
votes
2
answers
495
views
Are Chern classes always vertical?
Let $c_k \in H^{2k}(M, \mathbb{Z})$ be the $k$-th Chern class of the tangent bundle of a Hermitian manifold $M$.
Is $c_k$ necessarily vertical, i.e.
$$
c_k = \sum_{i_1,\dots, i_{k}} \alpha_{i_1 \dots ...
- 105
3
votes
1
answer
240
views
Cohomology of the complement of a subvariety
Let $X$ be a complex manifold, $Y\subset X$ a subvariety, and $U:=X\setminus Y$ of codimension $d$. It is well known that the restriction map
$$
H^i(X,\mathbb Q)\to H^i(U,\mathbb Q)
$$
is an ...
- 178
2
votes
1
answer
155
views
Unimodular intersection form of a smooth compact oriented 4-manifold with boundary
Let $X$ be a smooth compact oriented 4-manifold with nonempty boundary. Its intersection form $$ 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
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
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 ...
147
votes
21
answers
23k
views
Are there examples of non-orientable manifolds in nature?
Whilst browsing through Marcel Berger's book "A Panoramic View of Riemannian Geometry" and thinking about the Klein bottle, I came across the sentence:
"The unorientable surfaces are never discussed ...
5
votes
1
answer
318
views
Surjection onto $H_{2}(\mathrm{PGL}(2,\mathbb{C}),\mathbb{Z})$
Let $G \leq \mathrm{PGL}(2,\mathbb{C})$ be the subgroup of upper-triangular matrices. I am interested in the natural morphism on the Schur multiplier (i.e. group homology as discrete groups)
$H_{2}(G,...
- 305