# Questions tagged [at.algebraic-topology]

Homotopy theory, homological algebra, algebraic treatments of manifolds.

6,875
questions

**6**

votes

**1**answer

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

**1**answer

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

**0**answers

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

**0**answers

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

**1**answer

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

**2**answers

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

**0**answers

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

**0**answers

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

**0**answers

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

**0**answers

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

**0**answers

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

**0**answers

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

**1**answer

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

**1**answer

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

**0**answers

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

**1**answer

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

**1**answer

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

**0**answers

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

**1**answer

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

**0**answers

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

**0**answers

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

**2**answers

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

**0**answers

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

**1**answer

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

**1**answer

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

**0**answers

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

**2**answers

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

**1**answer

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

**1**answer

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

**1**answer

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

**1**answer

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

**2**answers

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

**0**answers

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

**1**answer

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

**0**answers

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

**1**answer

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

**1**answer

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

**0**answers

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

**1**answer

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

**0**answers

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

**0**answers

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

**1**answer

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

**2**answers

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

**0**answers

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

**0**answers

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

**0**answers

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

**0**answers

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

**1**answer

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

**1**answer

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

**0**answers

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 $M$...