Questions tagged [at.algebraic-topology]
Homotopy theory, homological algebra, algebraic treatments of manifolds.
8,228
questions
5
votes
5
answers
888
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
votes
0
answers
210
views
Classifying spaces beyond CW complexes
We know that for a reasonable topological group $G$ (say a compact Lie group) admits a classifying space for $G$-bundles within the category of countable CW complexes. That means, there is a space $BG$...
4
votes
2
answers
279
views
On a generalized homotopy transfer theorem
In the book of Loday and Vallette "Algebraic Operads" a necessary condition for the Homotopy Transfer Theorem is that the starting operad is Koszul. I am interested in a generalization of ...
13
votes
3
answers
825
views
Are negatively pinched manifold locally conformally flat?
One knows that hyperbolic manifolds are locally conformally flat.
How about those negatively pinched manifolds, i.e. the sectional curvature $K$ satisfy:
$$
-\Lambda \le K \le -\lambda$$
for $\Lambda&...
3
votes
2
answers
412
views
A question on the manifold $ \{n\otimes n-m\otimes m:n,m\in S^2,(n,m)=0\} $
Consider a manifold $ N $ defined as follows
$$
N=\{n\otimes n-m\otimes m:n,m\in S^2,\quad(n,m)=0\}\subset M^{3\times 3},
$$
where $ S^2 $ denotes the two dimensional sphere, $ (\cdot,\cdot) $ ...
2
votes
0
answers
142
views
Do the nearby cycle and Beilinson's vanishing cycle functors commute?
Let $X$ be a complex algebraic variety with a pair of regular functions $f_1,f_2$. To these functions we can associate various functors: the nearby cycles functor $\Psi_{f_i}$, the vanishing cycles ...
23
votes
2
answers
3k
views
Calculating Mayer-Vietoris efficiently
This is a question whose motivation and framing seem to involve a lot of topology, but which I suspect comes down to some simple and standard combinatorics that's probably recorded in a book somewhere....
23
votes
3
answers
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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" ...
1
vote
0
answers
141
views
Minimal first Pontryagin class $p_1=1$?
From Hirzbuch theorem,
the signature of 4-manifold $\sigma = p_1/3$ with the first Pontryagin class $p_1$.
I know that the $\sigma=p_1/3 =1$ so $p_1=3$ for complex projective space $\mathbb{P}^2$.
Is ...
39
votes
1
answer
6k
views
Classification of surfaces and the TOP, DIFF and PL categories for manifolds
A surface is simply a 2-manifold. The classification theorem for compact connected surfaces (with boundary) is commonly regarded in the categories TOP, DIFF and PL. Well known proofs (e.g. via ...
4
votes
0
answers
91
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 ...
2
votes
0
answers
107
views
Extension of isotopies
In what follows $M$ will be a manifold (without boundary, for simplicity) and $C\subseteq M$ will denote a compact subset. In the paper Deformations of spaces of imbeddings Edwards and Kirby prove the ...
6
votes
1
answer
224
views
Relationship between the holonomy pseudogroup and holonomy homomorphism (foliation)
This question is surely a duplication of https://math.stackexchange.com/questions/4343635/relationship-between-the-holonomy-pseudogroup-and-holonomy-homomorphism-foliati , however, I got no replies. ...
1
vote
0
answers
112
views
A question about cohomology with local coefficient
Let's consider the next theorem.
Theorem
[The cohomology Leray-Serre Spectral sequence] Let $R$ be a
commutative ring with unit. Given a fibration $F\hookrightarrow E\overset{p}{%
\rightarrow }B$, ...
1
vote
1
answer
138
views
Lie group framing and framed bordism
What is the definition of Lie group framing, in simple terms?
Is the Lie group framing of spheres a particular type of Lie group framing? (How special is the Lie group framing of spheres differed ...
3
votes
1
answer
158
views
Pontryagin product on the homology of cyclic groups
Consider the cyclic group $C_{p^N}$ of order $p^N$, and let $k$ be a field of characteristic $p$. I would like to know what the algebra structure on the homology $H_*(C_{p^N};k)$ induced by the ...
2
votes
0
answers
62
views
Framed bordism and string bordism in 3-dimensions vs topological modular form
In simple colloquial terms, how are the framed bordism and string bordism in 3-dimensions related to the study of the theory of topological modular form TMF? I want to know some simple derivable ...
1
vote
2
answers
151
views
Reference for choosing a path lifting function?
I recall having seen discussion of a Hurewicz or Serre fibration
equipped with a chosen path lifting function. Citation??
34
votes
6
answers
4k
views
Why study finite topological spaces?
In rereading Thurston's essay On Proof and Progress in Mathematics I ran across this passage:
… this means that some concepts that I use freely and naturally in
my personal thinking are foreign to ...
3
votes
2
answers
219
views
Is the free algebra functor over an $\infty$-operad symmetric monoidal?
Suppose $F: \mathcal{O}^\otimes \to \mathcal{P}^\otimes$ is a map of $\infty$-operads, and $\mathcal{C}$ is a symmetric monoidal $\infty$-category that admits small colimits, such that the tensor ...
2
votes
2
answers
160
views
$String/CP^{\infty}=Spin$ or a correction to this quotient group relation
We know that there is a fiber sequence:
$$
... \to B^3 Z \to B String \to B Spin \to B^2 Z \to ...
$$
Is this fiber sequence induced from a short exact sequence?
If so, is that
$$
1 \to B^2 Z = B S^...
1
vote
0
answers
86
views
A question about the Conner Conjecture
In some sources, Conner conjecture is expressed as follows:
Theorem [Conner Conjecture] Let $G$ be a compact Lie group, and let $X$ have
the homotopy type of a finite dimensional $G$-CW complex with ...
9
votes
2
answers
571
views
Generalization of the sphere theorem in dimension at least 4
In 1956, Papakyriakopoulos proved Dehn's lemma, loop theorem and the sphere theorem. The proofs are based on a clever technique called "tower construction". Later, Whitehead, Shaprio, ...
1
vote
1
answer
85
views
Simplicial cochain representing the pullback of a class Poincaré dual of a submanifold
Let $K$ be a simplicial complex of dimension $n$, $M$ be a topological manifold, and $f \colon |K|
\to M$ be a continuous map. Let $X$ be an embedded manifold in $M$ of codimension $n$, such that
$f(|...
0
votes
0
answers
68
views
Topological transversality by dimension
We know that to achieve transverality in the topological category, for example to make a continuous map into a manifold transverse to a topological submanifold, we need the existence of micro normal ...
2
votes
1
answer
142
views
string bordism group and framed bordism group for $d \leq 6$ and $d \geq 7$
Why do the string bordism group and the framed bordism group
coincide the same in dimensions lower than 7 ($d = 0,1,2,3,4,5, 6$)?
Why do the string bordism group and the framed bordism group differ
...
3
votes
1
answer
250
views
Do objects in the derived category behave stackily?
It is well known that derived categories (I'm particularly thinking of constructible derived categories and derived categories of D-modules) don't form a stack. In particular given morphisms in the ...
1
vote
0
answers
125
views
Determine the Eilenberg-MacLane spaces on the right-handed side of this Whitehead tower?
What and how to determine the Eilenberg-MacLane spaces on the right-handed side of this Whitehead tower?
Namely, how do we know
$$
K(Z_2,1)?, \quad K(Z_2,2)?, \quad K(Z,4)?
$$
Naively -- in each step ...
97
votes
10
answers
13k
views
equivalence of Grothendieck-style versus Cech-style sheaf cohomology
Given a topological space $X$, we can define the sheaf cohomology of $X$ in
I. the Grothendieck style (as the right derived functor of the global sections functor $\Gamma(X,-)$)
or
II. the Čech ...
2
votes
0
answers
128
views
Tensor product of objectwise weak homotopy equivalences of $\mathcal{M}$-spaces
I consider the enriched category $[\mathcal{M}^{op},\mathrm{Top}]$ of enriched functors (I call them $\mathcal{M}$-spaces) from the enriched small category $\mathcal{M}^{op}$ to the enriched category $...
1
vote
0
answers
216
views
Examples of when $X$ is homotopy equivalent to $X\times X$
I was thinking about this question the other day: When is a topological space $X$ homotopy equivalent to $X\times X$ (with the product topology)? This is essentially a cross-post of this MSE question.....
3
votes
1
answer
187
views
On infinity-morphisms between algebras over algebraic operads
I posted this question in the "Mathematics" stack exchange, but it hasn't got much attention... I hope it will get more here.
Let $P$ be a Koszul operad.
In the book of Loday-Vallette "...
2
votes
0
answers
89
views
Explicit CW-complex replacement of the space of reparametrization maps
Let $P$ be the space of nondecreasing surjective maps from $[0,1]$ to itself equipped with the compact-open topology: $P$ is contractible. There exists a trivial fibration $P^{cof} \to P$ from a CW-...
8
votes
1
answer
335
views
Telescopic localisation of Eilenberg-MacLane spaces
Fix a prime $p$ and an integer $n>0$. Let $K$ be the corresponding Morava $K$-theory spectrum, and let $T$ be the telescope of a $v_n$-self map of a finite spectrum of type $n$, and let $X$ be the ...
0
votes
0
answers
83
views
What happens if I take a doubly-free simplicial abelian group?
Suppose that I have a simplicial set $X_\bullet$. I can take the free abelian group generated by $X_\bullet$, $\mathbb{Z}X_\bullet$. But then I can forget that this has an abelian group structure, ...
0
votes
0
answers
100
views
Basis of Lambda algebra for a programmer
First of all, I'm not a specialist in alg. top., but I try to apply computational math to it, so if I'm wrong in something you'd be doing a better thing explaining it to me instead of blaming me :) ...
5
votes
1
answer
553
views
Intersection cohomology and Poincaré duality
When trying to learn about perverse sheaves I hand-wavingly thought that intersection cohomology is the ‘minimal’ way of fixing the failure of Poincaré duality. But I am very aware that it is risky to ...
4
votes
0
answers
162
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 ...
2
votes
1
answer
221
views
Comparing the exit path category and the nerve of a stratified space
Let $P$ be a finite poset, and $X$ be a topological space stratified by $P$, in the sense that $X$ is equipped with a continuous map $X \to P$ in the Alexandroff topology (or equivalently with a ...
2
votes
1
answer
226
views
Derived category of local systems of finite type on a $K(\pi,1)$ space: an explicit counterexample
Let $X$ be a nice enough topological space. I am mostly interested in smooth complex algebraic varieties. One may ask whether the bounded derived category of the category $\mathrm{Loc}(X)$ of local ...
25
votes
1
answer
1k
views
What can we say about the Cartesian product of a manifold with its exotic copy?
Let $M$ be a smooth oriented manifold, and let $M^E$ be an exotic copy, i.e homeomorphic but not diffeomorphic to $M$.
Is it true that $M\times M$ is diffeomorphic to $M\times M^E$?
I am ...
2
votes
0
answers
165
views
How a circle $S^1$ acts on the Cayley plane $OP^2$ with exactly three fixed points?
The complex projective $CP^2$, the quaternionic projective space $HP^2$, and the octonionic projective space $OP^2$ each admit a circle action with $3$ fixed
points.
The circle action on $HP^2$ can be ...
8
votes
2
answers
685
views
Pullbacks of classifying spaces
In what follows all the groups will be discrete, not necessarly finite.
Let $f:G\to H$ be a morphism of groups and $H'\to H$ be the inclusion of a subgroup. It seems to me (but correct me if I am ...
4
votes
2
answers
364
views
Classifying space of a non-discrete group and relationship between group homology and topological homology of Lie groups
I have a very soft question which might be very standard in textbooks or literature but I haven't seen it.
To a fixed group $G$ we may attach different topologies to make it different topological ...
13
votes
1
answer
336
views
Is $KU\otimes S^1_+$ isomorphic to $F(S^1_+,KU)$ as $E_\infty$ rings?
There are various ways to construct $KU$ as an $E_\infty$ ring spectrum; I will take that as given. Using this, we can make $KU\otimes G_+$ into an $E_\infty$ ring for any commutative topological ...
2
votes
0
answers
95
views
Unordered configuration space with non-distinct points
Consider a topological space $X$, a natural number $n>0$ and
the quotient topological space $Q_n(X)$ of $X^n$ by the equivalence relation : $x\sim y$ if and only if
there is a permutation $\sigma$ ...
2
votes
1
answer
208
views
Projective objects in chain complexes of an abelian category: Further question
Yes, I see there are other Q&A's on this, for instance here: Projective objects in the category of chain complexes
I am wondering why a level-wise projective chain complex $P$ which is split ...
14
votes
1
answer
547
views
Mednykh's formula for 2d manifold with boundaries? How to count principal G-bundles with prescribed holonomies?
Let S be a closed, orientable 2d manifold and G a finite group. Since a principal G-bundle over S is specified by maps $\phi : \pi_1(S) \rightarrow G$ modulo the adjoint action by G, the way to count ...
6
votes
2
answers
619
views
Removing a submanifold from a closed manifold
Let $M$ be a simply-connected closed manifold. Can we find a closed submanifold $N \subsetneq M$ such that $M\backslash N$ is simply-connected and has finite second homotopy group?
2
votes
0
answers
58
views
Module structure of $\Omega_*(Z_p)$
In Conner-Floyd's book, Differentiable Periodic Maps in (46.1), for p an odd prime and $k=1,2,\dots$, it is posted the identities:
$$p\alpha_{2k+1}+[M^4]\alpha_{2k-3}+[M^8]\alpha_{2k-7}+\dots=0$$
in $\...