All Questions
8,725 questions
3
votes
0
answers
151
views
Reference for homotopy and homology theory of topological groups
I am looking for references which deal with the homotopy theory and homology theory of general topological groups, not necessarily compact, or anything. I am eyeing towards certain infinite-...
- 219
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
2
votes
0
answers
148
views
Equivalence of cohomology with compact support
Let $𝑋$ be a connected CW complex, $𝜌:\pi_1(𝑋)→\mathrm{Aut}(G)$ a representation, and $G_\rho$ the associated sheaf. It follows from here, that the following two cohomologies are isomorphic.
(1)The ...
- 241
15
votes
1
answer
541
views
Where is the Steenrod Realization problem at?
I'm wondering if there is a more modern reference out there for the Steenrod Realization Problem than the book of Connor and Floyd?
Realizing homology classes in a manifold via embedded submanifolds, ...
- 44.4k
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
0
answers
185
views
Compute the Euler class of tautological $C$-bundle over $CP^1$
$\DeclareMathOperator\SO{SO}$This might be an old question. But since I have not found an explicit answer to this question, I put the question here.
The background is that we need to use a similar ...
- 21
1
vote
0
answers
57
views
extendability of fibre bundles on manifolds with same dimensions
Let $M$ be an $m$-manifold. Let $M'\subseteq M$, where
$M'$ is also an $m$-manifold.
Let $N$ be an $n$-manifold. Let $N'\subseteq N$, where
$N'$ is also an $n$-manifold.
Suppose there is fibre ...
- 123
6
votes
0
answers
129
views
Are there isospectrally equivalent exotic spheres?
Let $X$ and $Y$ be two different exotic spheres. Are there metrics $g$ and $h$ on $X$ and $Y$, respectively, such that the laplacians of $(X,g)$ and $(Y,h)$ have the same spectrum?
I would be happy ...
1
vote
0
answers
103
views
Is the mapping space from $\Delta[k]$ to a simplicial set $X$ weak equivalent to $X$?
Let $X$ be a simplicial set. It is well known that a model for the path object can be given by the mapping space $\mathrm{Hom}(\Delta[1], X)$. In particular this offers a fiber replacement for the ...
- 91
0
votes
0
answers
142
views
Fibration exact sequence in homotopy vs spectral sequence in (co)homology
Perhaps this should be obvious but why is it that one may associate to a fibration exact sequences of topological spaces a long exact sequence of fundamental groups, but in (co)homology, one only has ...
- 2,907
2
votes
0
answers
136
views
Question about the proof of the Duistermaat-VanDerKallen-Theorem, concerning the meaning of intersecting a chain with a set
On page 228 of the paper "Constant terms in powers of a Laurent polynomial" (by J.J. Duistermaat and Wilberd van der Kallen) a 'chain' $\Delta_\tau$ is defined as $f^{-1}([0,\tau])\cap \...
- 360
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
2
votes
0
answers
119
views
Crossed homomorphism as morphism in the ambient category
Suppose we are given a crossed-homomorphism $\phi:G\to A$ (and an action $\alpha$ of $G$ on $A$)
$\phi(ab)=\phi(a)+\alpha(a)(\phi(b))$. Now, unless the action is trivial, this is not a homomorphism ...
- 199
3
votes
1
answer
180
views
Connective covers of based spaces
I know that n-connective spectra form a coreflective subcategory of spectra. Namely, there is a n-connective cover functor $\tau_{>k}: Spectra\rightarrow Spectra^{n-connective}$ which has a fully ...
- 719
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
2
votes
0
answers
112
views
lifting a family of curves to a family of sections of a vector bundle?
This is a question in obstruction theory. It should be basic but I can't find a reference.
Let's stick to the $C^\infty$ category, so all objects mentioned are smooth. Let $\pi: E \to M$ be a vector ...
- 51
6
votes
1
answer
373
views
Fivebrane bordism $\Omega_d^{\mathrm{Fivebrane}}$
$\newcommand{\Fr}{\mathrm{Fr}}\newcommand{\Fivebrane}{\mathrm{Fivebrane}}\newcommand{\String}{\mathrm{String}}\DeclareMathOperator\Spin{Spin}\DeclareMathOperator\SO{SO}\DeclareMathOperator\GL{GL}$What ...
- 10.5k
6
votes
1
answer
328
views
Spectral sequence generalizing Čech cohomology
Let $X$ be a 'nice' topological space. Let $\left(U_i\right)_{i\in I}$ be a finite open covering of $X$. Let $\mathcal{F}$ be a sheaf of abelian groups.
For a subset $A\subset I$ denote $$U_A:=\cap_{...
- 21.8k
1
vote
0
answers
78
views
A question about the localization theorem of Borel-Hsiang and spectral sequence
Suppose that $T$ is a torus acting on a topological space $X
$. Let $T\longrightarrow E_{T}\longrightarrow B_{T}$ be the universal $T$-bundle. Let $X\longrightarrow X_{T}\longrightarrow B_{T}$ be the ...
- 1,367
2
votes
0
answers
156
views
Testing for weak homotopy equivalences with compact Hausdorff spaces
Let $f \colon X \to Y$ be a weak homotopy equivalence between topological spaces. If I am not mistaken, then one can rephrase this by stating that the induced map $[K,X] \to [K,Y]$ between homotopy ...
- 2,998
3
votes
1
answer
246
views
Borel-Moore homology for resolution of singularities
Let $X$ be a singular projective variety. Denote by $Z$ the singular locus of $X$. Consider the resolution of singularities $$\pi: \widetilde{X} \to X$$
Denote by $E$ the exceptional divisor. We know ...
- 2,323
4
votes
1
answer
183
views
Notation for weak derivatives
I remember that, as a student, I felt a bit uncomfortable because I had to use the same notation (say $f'$, $D^\alpha f$, $\frac{\partial f}{\partial x^j}$, $\nabla \cdot f$ etc...) for classical and ...
- 4,459
0
votes
0
answers
60
views
The size of super level sets and the symmetry on a sphere
Let $u$ be a smooth function defined on the sphere $\mathbb{S}^2$, and let $R \in \mathrm{SO}(3)$ be a three-dimensional rotation. Define
$$
S_R = \{x \in \mathbb{S}^2 : u(x) \neq u(Rx)\}.
$$
Suppose ...
- 434
1
vote
1
answer
144
views
Confusion regarding the vanishing of certain relative cohomology groups
Let $X$ be a projective variety of dimension $n$ and $D \subset X$ is a proper subvariety. Embed $X$ into a projective space $\mathbb{P}^{3n}$. The following argument implies that $H^i(X,X\backslash D)...
- 2,323
8
votes
2
answers
730
views
Could there be any homotopy group without "Lebesgue Number Lemma"?
This is about a comment that I have made in my general topology class while I was proving the abovementioned lemma as a consequence of compactness!
As far as I know, essentially, there is only one ...
- 3,173
3
votes
0
answers
167
views
Suitability of formal type theory for mathematical thinking (vs. traditional set theory)
Type theory has advantages over set theory for the (computer) formalisation of mathematics, but has anybody who does mathematics with pen and paper found proof assistants or automated theorem provers, ...
14
votes
1
answer
1k
views
Progress on Gromov's Conjecture of the bound of total Betti numbers
This question is a reference request.
Let $(M,g)$ be a Riemannian manifold of dimension $n$, and $b_i(M) = \dim H_i(M,\mathbb{R})$. Gromov proved it that there are constants $C(n)$ such that, if the ...
- 175
28
votes
3
answers
1k
views
Proofs of Poincaré duality
I know several proofs of Poincaré duality:
The original proof using dual cell complexes. Probably the nicest version of this uses a handle decomposition.
The argument (in Hatcher and many other ...
- 281
7
votes
4
answers
414
views
Why is the first nontrivial $p$-local stable stem cyclic?
Let $\pi_\ast^{(p)}$ be the ring of $p$-local stable homotopy groups of spheres. This is a nonnegatively graded ring, with $\mathbb Z_{(p)}$ in degree $0$. The first nonvanishing positive degree ...
- 64k
1
vote
0
answers
177
views
If $X$ is a strong deformation retract of $\mathbb{R}^n$, then is $X$ simply connected at infinity?
Let $X \subseteq \mathbb{R}^n$, and assume there is a strong deformation retract from $\mathbb{R}^n$ to $X$. Is $X$ necessarily simply connected at infinity?
(Edit) Follow up question: if there is a ...
- 637
2
votes
0
answers
148
views
Variant of Leray-Hirsch for complex-oriented cohomology theory
I am interested in a variant of the Leray-Hirsch theorem. Let $\mathbb{E}^*$ be a complex-oriented cohomology theory and let $\mathbb{E}_*$ the coefficient ring. Suppose that $F \xrightarrow{i} P \to ...
- 959
3
votes
0
answers
132
views
The tautness property and the continuity property of cohomology theory
Let $H^{\ast }$ denote the Čech cohomology or Alexander-Spanier
cohomology.
Definition: (Tautness property of cohomology) Let $X$ be a
paracompact Hausdorff space and $A$ be a closed subspace of $X$. ...
- 1,367
0
votes
1
answer
135
views
Local embedding and disk in domain perturbation
Consider say $M=(\mathbb{S}^1\times\dotsb\times \mathbb{S}^1)-q$ ($n$-times). Assume that $B$ is an $n$ disk in $M$ (for instance, thinking of $\mathbb{S}^1$ as gluing $-1$ and $1$, the cube $B=[-\...
1
vote
0
answers
132
views
The equation of cubic surface
I was studying the nature of singularities of cubic hypersurface on $\Bbb P^{n+1}$. The chosen form was
$$f(x_0,x_1 \dots , x_{n+1})=x_0^3+x_1^3+\dotsb+x_{n+1}^3-c(x_0+x_1+\dotsb+x_{n+1} )^3=0.$$
I ...
- 21
6
votes
2
answers
239
views
Euclidean algorithm for simple closed curves
In the proof of Proposition 6.2 in Farb & Margalit, "A primer on mapping class groups", an analog of the Euclidean algorithm is used to construct a simple, closed representative (...
- 83
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
5
votes
1
answer
287
views
Is the wildness of 4-manifolds related to the diversity of their fundamental groups?
$n = 4$ is the smallest dimension such that the fundamental group of a closed $n$-manifold can be any finitely-presentable group (leading e.g. to various undecidability results stemming from the ...
- 64k
8
votes
0
answers
183
views
Morphisms in cube category $\Box$ = Compositions of morphisms in simplex category $\Delta$?
Let $\Delta$ be the simplex category. For $m \leq c \leq n$, let $[m] \to [c] \to [n]$ be the composition of two injective morphisms in $\Delta$.
We now define a category $\Box$ with same objects as $\...
- 1,813
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
13
votes
5
answers
2k
views
What are some good examples of spectral sequences which degenerate after the first nontrivial differential?
The difference between algebraic geometry and algebraic topology is that in AG, you usually hope that your spectral sequences degenerate immediately at the $E_2$ page. In AT, you often have to live ...
1
vote
0
answers
61
views
Necessary or sufficient conditions for the $k$-fold intersection to be empty in a covering with a "tree structure"
Consider a finite collection of $d$-dimensional balls $\mathfrak{B}=\{B_1,\ldots,B_n\}$ which cover a PL $d$-manifold $M$, i.e. $M=\bigcup_{i=1}^{n}B_i$. Suppose we want to compute the Euler ...
- 159
5
votes
1
answer
430
views
Linking number and intersection number
Consider a disjoint union of two circles $A$ and $B$ smoothly embedded in $\mathbb{R}^3$ with linking number more than $1$. Suppose we know that there exists a disc $D$ in $\mathbb{R}^3$ such that $\...
- 125
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
2
votes
0
answers
55
views
Fundamental group of cyclic branched cover of affine plane
Let $f\in \mathbb{C}[x,y]$ be an irreducible polynomial. Let $n>0$ be an integer such that the hypersurface $S:=\{ (x,y,z)\in \mathbb{C}^3|z^n=f(x,y) \}$ is a connected complex submanifold of $\...
- 615
13
votes
0
answers
318
views
Is there an analogue of Steenrod's problem for $p>2$?
An element $\alpha \in H_k(X; \mathbb{Z})$ is said to be realisable if there is a $k$-dimensional connected, closed, orientable $k$-dimensional submanifold $Y$ such that $\alpha = i_*[Y]$. The ...
- 327
6
votes
1
answer
343
views
Does a ring spectrum with even homotopy and even cells always have a polynomial algebra of homotopy groups?
Let $R$ be a spectrum. Assume that $R$ is bounded-below. Then we can “even-ify” $R$: cone off some generating set of the first nonvanishing odd-dimensional homotopy group of $R$. Do this repeatedly. ...
- 64k
7
votes
2
answers
209
views
Cover the $n$-disc irredundantly with $n+1$ open sets. Suppose that the $(n+1)$-fold intersection is empty. Then is some $n$-fold intersection empty?
Cover the $n$-disc with $n+1$ open sets $D^n = U_0 \cup \dotsb \cup U_n$. Suppose that $U_0 \cap \dotsb \cap U_n = \emptyset$. Suppose moreover that the cover is irredundant in the sense that no ...
- 64k
2
votes
1
answer
162
views
Pullback morphism of a hyperplane inclusion is zero in the derived category
Let $L \subset \mathbb{C}^n$ be a hyperplane and let $i:L \to \mathbb{C}^n$ be the inclusion. Since $i$ is proper, we have induced maps $i^*: H^k_c(\mathbb{C}^n) \to H^k_c(L)$, and these maps are zero ...
0
votes
1
answer
328
views
Relationship between quotient CW-complexes after attaching cells
I have been trying to prove the following simple-looking result which I require for some work in low-dimensional topology. I expect it is likely true and in a textbook somewhere so any reference or ...
- 187
9
votes
1
answer
315
views
Does the Atiyah-Hirzebruch spectral sequence for $E^\ast(X)$ collapse whenever $E$ is complex-oriented and $X$ has even cells?
Let $X$ be a (finite, say — or maybe of finite type) spectrum with even cells (in other words, $H_\ast(X;\mathbb Z)$ is free and concentrated in even degrees). Let $E^\ast$ be a complex-oriented ...
- 64k