Questions tagged [at.algebraic-topology]
Homotopy theory, homological algebra, algebraic treatments of manifolds.
6,875
questions
6
votes
1answer
118 views
On the comparison map $MU^\bullet(X)\otimes_{MU^\bullet(pt)}E^\bullet(pt)\to E^\bullet(X)$ for complex oriented multiplicative cohomology theories
Whatever complex oriented multiplicative cohomology theories are, they come with two basic properties (among many others):
i) a complex oriented multiplicative cohomology theory is a contravariant ...
3
votes
1answer
165 views
Functoriality of Thurston's norm
Let $M$ be a manifold of dimension $3$ and let $N$ be an embedded submanifold of $M$ (also of dimension $3$).
Then, both second homologies $H_2(M)$ and $H_2(N,\delta N)$ are equipped with a norm (...
-6
votes
0answers
60 views
non commutative algebra or operator theory [closed]
Is Hurwitz-Radon theorem is true for the field $ \mathbb{Q} $. Actually I want to know when $M_{n}(\mathbb{Q}) $has n- subspace of dimension n?
links-<PROBLEMS AND THEOREMS
IN LINEAR ALGEBRA,by V. ...
9
votes
0answers
159 views
Is there a homotopy coherent analogue of Dieudonné modules?
Let $K$ be a perfect field with characteristic $>0$ and $\mathcal{H}$ the category of graded connected abelian hopf algebras over $K$.
By a theorem of Schoeller there is a canonical equivalence ...
5
votes
1answer
167 views
Can we construct a filtered chain complex from a spectral sequence?
Suppose $\{(E_r,d_r)\}$ for $r>0$ forms a spectral sequence over a field $\mathbb{F}$, i.e. for any $r$, $(E_r,d_r)$ is a chain complex over $\mathbb{F}$ and $E_{r+1}=H(E_r,d_r)$. For simplicity, ...
5
votes
2answers
325 views
Variety having infinite and perfect $\pi_1$
Question. Does there exist a smooth complex projective variety with infinite and perfect fundamental group?
A group $G$ is perfect if its Abelianisation $G/[G,\, G]$ is the trivial group.
3
votes
0answers
77 views
Homology of a fiber as a cotorsion product
Let $K$ be a field. For any differentially graded coalgebra $A$ over $K$, any differentially graded right $A$-comodule $M$ over $K$ and any differentially graded left $A$-comodule $N$ over $K$ let
$\...
5
votes
0answers
133 views
Borel equivariant cohomology operations
Fix a group $G$. For an abelian coefficient group $A$ let $H^*_G(-;A):=H^*(EG\times_G-;A)$ be the Borel cohomology with $A$ coefficents. This is a functor from $G$-spaces to graded abelian groups ...
3
votes
0answers
79 views
Milnor exact sequence for homology of hopf algebras
Let $K$ be a field and $\mathrm{Hopf}^K_{E_\infty}$ the $\infty$-category of
homotopy-coherent hopf algebras over $K$ that are coherently commutative and cocommutative.
Precisely, $\mathrm{Hopf}^K_{E_\...
13
votes
0answers
218 views
Morava K-theory of loop spaces of spheres
Some time ago I cam across the paper "What we still don't know about loop spaces of spheres" by Ravenel:
https://people.math.rochester.edu/faculty/doug/mypapers/loop.pdf
which concerns ...
4
votes
0answers
93 views
Cofibrancy of a right module over an operad
If I have a right module $M$ over an operad $\mathscr{O}$ in spaces, are there general methods to determine if $M$ is cofibrant with respect to the Reedy model structure? What if I know that my module ...
9
votes
0answers
398 views
+150
Exotic analytic triangulations of $S^5$?
I would like to understand better the nature of bad triangulations of $S^5$, discussed in two Lectures of Jacob Lurie
https://www.math.ias.edu/~lurie/937notes/937Lecture2.pdf
http://www-math.mit.edu/~...
7
votes
1answer
199 views
When do two topoi have the same cohomology of constant sheaves
Recently, I have some questions for some generalizations from algebraic topology.
I learn some homotopy theory in algebraic topology. We know that, if two spaces are homotopy, then they have same ...
19
votes
1answer
732 views
Can the product of a 3-dimensional lens space with a circle be diffeomorphic to another such product when the lens spaces aren't diffeomorphic?
This is a question that I need to answer in order to resolve an issue for my dissertation and I am looking for a reference. Here is the precise statement of the question.
Suppose we have two three-...
7
votes
0answers
93 views
Endofunctors of the surface category
Let $\mathrm{Cob}_2$ be the symmetric monoidal $(\infty,1)$-category whose objects are closed oriented $1$-manifolds and whose morphisms are compact oriented surface bordisms. (Higher morphisms are ...
6
votes
1answer
213 views
Cohomology of BG, G non-connected Lie group, and spectral sequence relating to classifying space of connected component of the identity
Suppose $G$ is a Lie group, with $\pi_0(G)$ not necessarily finite, but might as well assume $G_0$, the connected component of the identity, is compact.
In the case that $\pi_0(G)$ is finite, then we ...
5
votes
1answer
295 views
$\pi_{2n-1}(\operatorname{SO}(2n))$ element represents the tangent bundle $TS^{2n}$, not torsion and indivisible for $n>1$?
Question: Is the element $\alpha$ in $\pi_{2n-1}(\operatorname{SO}(2n))$ representing the tangent bundle $TS^{2n}$ of the sphere $S^{2n}$ indivisible and not torsion?
My understanding so far —
An $\...
6
votes
0answers
455 views
Update on “A Mad day's work” by Cartier
In his paper "A mad day's work: from Grothendieck to Connes and Kontsevich. The evolution of concepts of space and symmetry," Cartier discusses in the last sections 8 and 9 the role of ...
10
votes
1answer
575 views
How to motivate constructible sheaves
I'm writing some notes for some students which just finished a first course in scheme theory. There I would like to talk about constructible sheaves, but I found it hard to give a compelling ...
5
votes
0answers
58 views
Homotopy, contraction mapping and the inverse function theorem on Banach spaces
We all know the "open mapping" part of the inverse function theorem on Euclidean spaces can be proved by either contraction mapping (iterations), or homotopy methods (degree theory / ...
1
vote
0answers
92 views
Metrizable cellular topological spaces
For a CW-complex, locally compact, metrizable, first countable and locally finite are equivalent conditions. A proof is available in https://epub.ub.uni-muenchen.de/4524/1/4524.pdf. I need the same ...
9
votes
2answers
482 views
Stable homotopy groups of complex projective plane
We know that there is a cofiber sequence $S^3\xrightarrow{\eta}S^2\to\mathbb{C}\mathbb{P}^2$. It's easy to know that $\pi_3^s(\mathbb{C}\mathbb{P}^2)=0$ so there is a surjection
$$\partial:\pi_7^s(S^2\...
9
votes
0answers
153 views
Aspherical fibrations and group epimorphisms
Let $\mathsf{Top}$ denote the category of pointed spaces having the pointed homotopy type of a pointed CW-complex. Let $\mathsf{Grp}$ denote the category of groups. It is well documented that for ...
4
votes
1answer
229 views
Surface bundles associated to a short exact sequence of groups
Suppose $S$ is a closed, connected, oriented surface of genus at least two and $G$ is any group. Suppose further that $\Gamma$ is any group that fits into the following short exact sequence:
$$ 1 \to \...
2
votes
1answer
115 views
Is the category of topological operads left proper?
I just learned that there is a model structure on the category $Op_{Top}$ of topological operads, due to Berger-Moerdjik [1], obtained by right transfer of the Quillen model structure on $Top$.
Since $...
3
votes
0answers
216 views
Intuitive (topological) explanation of a proof from the HoTT book [closed]
My friends and I are struggling with understanding intuitively the proof of equivalence between based and free path inductions (HoTT book 1.12.2)
The first major problem is understanding the meaning ...
8
votes
2answers
135 views
Degree one self-map of $\Bbb R^2\big\backslash \big\{(n,0):n\in \Bbb Z\big\}$ not homotopic to any self-homotopy equivalence
Consider the surface $\Sigma=\Bbb R^2\big\backslash \big\{(n,0):n\in \Bbb Z\big\}$. Does there exist a proper
map $f\colon \Sigma\to \Sigma$ of degree $1$ and not homotopic to
any self-homotopy ...
4
votes
1answer
251 views
Is a complex or real algebraic variety homotopically equivalent to a CW complex?
Let $k$ be either the field $\Bbb C$ of complex numbers or the field $\Bbb R$ of real numbers.
Let $X$ be an algebraic variety over $k$, say, quasi-projective and smooth (but not necessarily ...
8
votes
1answer
288 views
Whitehead product and a homotopy group of a wedge sum
Note : this is a crosspost from the Mathematics StackExchange, as suggested by this meta post.
Let $X$ be an $n$-connected ($n\geqslant1$) CW-complex and $Y$ be a $k$-connected ($k\geqslant1$) CW-...
15
votes
1answer
476 views
Is there a $4$-manifold which Immerses in $\mathbb{R}^6$ but doesn't Embed in $\mathbb{R}^7$?
I'm interested in both version of the question in the title, i.e. in the topological category and in the smooth category. By a topological immersion I mean a local embedding. I was asking in ...
5
votes
1answer
137 views
Enriched coends which preserve equivalences
Although this question might be formulated in higher generality, let me try to be concrete:
Let $(\mathbf{Top},\times,*)$ be the monoidal category of compactly generated weak Hausdorff spaces; and let ...
6
votes
2answers
423 views
Deformation of a diagram preserve the homotopy limit
I have been a bit sloppy in the title, but let me be specific. I stepped again into the subtle difference between homotopy limit and limit in the homotopy category, in the following version.
Suppose ...
2
votes
0answers
146 views
Spaces homotopy equivalent over the topologist's sine curve
Consider $$T=\left\{ \left( x, \sin \tfrac{1}{x} \right ) : x \in [-1, 0)\cup(0,1] \right\} \cup \{(0,0)\}\subset \mathbb{R}^2$$
with the subspace topology.
Denote $p=(-1, \sin -1), q=(1, \sin 1)\...
6
votes
1answer
138 views
Homotopy groups of $K(n)$-localization of the Brown-Peterson spectrum
We fix $p$ prime and $n$ a natural number. We let $K(n)$ be the $2(p^{n}-1)$-periodic Morava $K$-theory, i.e. $K(n)_*=\mathbb{F}_p[v_n^{\pm 1}]$ with $|v_n|=2(p^n-1)$. I distinctly recall that we ...
5
votes
0answers
135 views
Construction of equivariant Steenrod algebra
I am reading through the calculations in Hu-Kriz "Real-oriented homotopy theory and an analogue of the Adams-Novikov spectral sequence" and I've got a small problem in understanding the ...
0
votes
1answer
125 views
How many n-dimensional closed submanifolds of $R^n$ have Euler characteristic 1?
It is well-known that the closed $n$-ball has Euler characteristic $1$. Is it true that every closed (i.e., compact), connected $n$-dimensional submanifold (with boundary) of $\mathbb R^n$ having ...
7
votes
1answer
406 views
Homotopy type of continuous/smooth/analytic loop spaces?
Apologies in advance if this is well-known; a google search did not produce anything useful.
Let $(M,p)$ be a pointed real analytic manifold. Are the (free or pointed) loop spaces of continuous, ...
5
votes
0answers
103 views
Refinement of hypercovers by ordinary covers
I am asking for references and discussions of statements of the form
Every bounded hypercover can be refined by an ordinary cover
E.g., are there conditions for a site making this statement true?
My ...
6
votes
1answer
218 views
Orientation reversal and restriction to submanifold of lower dimension
Let $M$ be a connected closed oriented manifold with at least one orientation-reversing homeomorphism $M\to M$.
Let $S\subset M$ be a connected closed embedded submanifold of lower dimension. Let $f:M\...
8
votes
0answers
117 views
Cohomology algebra of the maximal nilpotent subalgebra of a semisimple Lie algebra
The answer to this question may be well-known, but I failed to locate it in any obvious source. From the results of Bott (Ann. Math. 66 (1957), 203-248) and Kostant (Ann. Math. 74 (1961), 329-387), it ...
9
votes
0answers
185 views
What is the operad for homotopy associative, homotopy commutative objects?
There is an operad whose algebras are objects with a homotopy unital multiplication -- the $A_2$ operad.
There is an operad whose algebras are objects with a homotopy unital, homotopy associative ...
6
votes
1answer
356 views
Comparison of two monodromies
Let us consider a smooth projective curve $\Sigma_b$ of genus $b$, and assume that there is a surjective group homomorphism $$\varphi \colon \pi_1 (\Sigma_b \times \Sigma_b - \Delta) \to G,$$ where $\...
9
votes
2answers
284 views
Proofs that the classifying space of Connes' cycle category $\Lambda$ is $\mathbb C \mathbb P^\infty$
Connes showed in Cohomologie cyclique et foncteurs $Ext^n$ (1983) that the classifying space of his cycle category $\Lambda$ is $\mathbb C \mathbb P^\infty = B(S^1) = K(\mathbb Z,2)$.
Connes' proof is ...
1
vote
0answers
119 views
Aut/Inn/Out Automorphism Groups of the unitary group $𝑈(𝑁)$
Given a group $G$, we denote the center Z$(G)$, we like to know the
automorphism group Aut($G$), the outer automorphism Out($G$) and the inner automorphism Inn($G$). They form short exact sequences:
$$...
0
votes
0answers
131 views
Adjunction formula for non compact surfaces
Let $M$ be a non compact complex surface and S an embedded compact Riemann surface in $M$.
I already know how to show the following equality of fiber bundle:
$$\Omega^2_{M}|S =\Omega^1_S \otimes N^*_S$...
4
votes
0answers
124 views
Cyclic sets are to $S^1$-spaces as ? are to $G$-spaces?
From the article "W. G. Dwyer, M. J. Hopkins, D. M. Kan, The Homotopy Theory of Cyclic Sets" we know that, quoting from the abstract: "the homotopy theory of the cyclic sets of Connes [...
4
votes
0answers
212 views
Spectral sequence from a stratification by closed subvarieties
I am looking for a reference for the following result: If $X$ is an algebraic variety and
$$X = T_n \supset T_{n-1} \supset \cdots \supset T_{-1} = \varnothing$$
is a stratification (edit: filtration) ...
4
votes
1answer
112 views
Hadamard-like product on orientable surfaces
Denote by $C$ the category of connected closed orientable surfaces.
Is there a functor $F:C\times C\to C$ such that $b_1(F(S\times S'))=b_1(S)b_1(S')$?
7
votes
1answer
211 views
Isotopies, Fiber Bundles and Selection Theorems
The following problem is a culmination of a few questions I've asked the last two months, and it's still giving me some issues. I think I know the right way to solve it, but I'm having trouble with ...
16
votes
0answers
188 views
Reference request: Milnor rank of spheres
Milnor defines the rank of a smooth manifold $M$ as the maximum cardinality of a linearly independent set of vector fields on $M$ whose elements are pair wise commuting. In other words, the rank of $M$...
