Homotopy, stable homotopy, homology and cohomology, homotopical algebra.

**2**

votes

**0**answers

116 views

### Reference request for a “truncated version” of the de Rham algebra

Let's start on the $n$-torus for sake of simplicity.$\newcommand{\T}{\mathbb T}$
If I understand the relevant definitions correctly, the usual de Rham algebra of smooth differential forms on $\T^n$ is ...

**6**

votes

**0**answers

753 views

### Ordinary cohomology groups of $(\mathbb{C}^3\times T^2)/\mathbb{Z}_k$ that I need for my string theory research

Let $X=(B^6\times T^2)/\mathbb{Z}_k\subset (\mathbb{C}^3\times T^2)/\mathbb{Z}_k$ where $k=2,3,4,6$, where the generator of $\mathbb{Z}_k$ acts on $\mathbb{C}^3$ by the multiplication by a primitive ...

**0**

votes

**0**answers

130 views

### Is there a t-structure on the homotopy category of spectra that has the sphere spectrum in its heart?

Maybe the heart of such a t-structure should be the category of abelian groups, and the t-homology functor should be given by taking usual homology groups of the spectrum. Is it impossible?

**10**

votes

**2**answers

535 views

### A homology theory which satisfies Milnor's additivity axiom but not the direct limit axiom?

Let us agree on the following: a "homology theory" means a functor $h_*$ from the category of pointed CW complexes to the category of graded abelian groups, together with natural isomorphisms ...

**4**

votes

**0**answers

61 views

### The metric gives the optimal element in a class

In geometry there is plenty of examples in which the following happens:
Some elements are considered equivalent, in some topological or algebraic sense
We take the quotient
The metric is usually not ...

**0**

votes

**1**answer

167 views

### group actions on fibre bundles

Suppose that we have a group $G$ acting on the spaces $E$ and $B$. Suppose moreover that we have fibre bundles $\xi$ and $\eta$ in the following commutative diagram
If $\xi$ is a trivial bundle, ...

**7**

votes

**1**answer

158 views

### (Non)-equivariant equivalence in $G$-spectra

In HHR, an important part is the periodicity theorem. For proving the theorem, they invert a carefully defined class $D \in \pi^{C_8}_{19\rho_8}(N^8_2MU_{\mathbb{R}})$ and they can find an element in ...

**8**

votes

**1**answer

311 views

### Non-Cartesian Monoidal Model Structure on a Slice Category

Given a monoidal model category $(M,\otimes, 1)$, and a monoid therein $A$, one can take the slice model category $M_{/A}$. This category has a natural monoidal structure induced by taking fibered ...

**0**

votes

**1**answer

61 views

### Conjugation Cells [equivariant cohomology]

I'm studying conjugation spaces, I have found in many sources that a conjugation cell is a conjugation space (without a proof). The widest approach that I have found so far is this paper (example 3.5)
...

**3**

votes

**0**answers

168 views

### some terminologies on limiting mixed hodge structures or rather Derived categories

$f: X\rightarrow S$ is proper surjective homomorphism map from connected complex manifold to unite disk. $Y=f^{-1}(0)$ is algebraic and normal crossing in X, f is smooth away from 0, $X^*=X\setminus ...

**10**

votes

**1**answer

331 views

### Wild half-line in a Euclidean space

Is there an $m$-dimensional simplicial complex $S$ with the following properties:
The cone over $S$ is homeomorphic to $\mathbb{E}^{m+1}$. Here $\mathbb{E}^{m+1}$ denoes the $(m+1)$-dimensional ...

**34**

votes

**1**answer

793 views

### A dictionary of Characteristic classes and obstructions

I apologize in advance as this is not a research level question but rather one which could benefit from expert attention but is potentially useful mainly to novice mathematicians.
In an effort to ...

**16**

votes

**2**answers

1k views

### Why do people say DG-algebras behave badly in positive characteristic?

It seems to be a common wisdom in derived algebraic geometry that commutative DG-algebras are not, in general, a good model for derived rings, with the stated reason that they behave badly in positive ...

**6**

votes

**0**answers

107 views

### Detection tools for (reduced) suspension

I'm learning about loop spaces and the work of Stasheff on $A_{\infty}$-spaces. The broad idea that I'm getting is the following. Given a space $Y$, we want to know under which conditions there exists ...

**6**

votes

**3**answers

772 views

### Does anyone know the classification of fourth order surfaces?

Does anyone know the classification of fourth order surfaces? By "fourth order surface" I mean a surface defined by an equation of the form $$f(x, \, y, \, z)=0,$$
where $f$ is a polynomial of degree ...

**4**

votes

**0**answers

108 views

### What structure of a monoidal simplicial model category is preserved by taking the opposite category?

Suppose we have $(M,\otimes,1)$, a monoidal simplicial model category. Then we can consider the opposite model category $M^{op}$ with the opposite model structure (fibrations become cofibrations, ...

**4**

votes

**1**answer

136 views

### The principal bundle of embeddings

In a paper of P. Michor, it was shown that Emb(M,N) is a smooth principal diff(M)-bundle, M and N are smooth locally compact manifolds provided dim M < dim N. My question is why there is a ...

**3**

votes

**0**answers

98 views

### Are there necessary and sufficient conditions for a chain complex $0 \to C_0 \to C_1 \to C_2 \to 0$ to be Poincare?

I am looking for necessary and/or sufficient conditions for the chain complex $0 \to C_0 \to C_1 \to C_2 \to 0$ over a principal ideal domain to be Poincare in the sense that $H_0 \cong H^2$, $H_1 ...

**7**

votes

**1**answer

363 views

### higher algebraic homotopy groups for schemes?

I think I understand how to define the algebraic fundamental group $\pi^{alg}_{1}(X)$ of a scheme and I think I understand the relation between $\pi^{alg}_{1}(X)$ and $\pi_{1}(X(\mathbb{C}))$, where ...

**6**

votes

**0**answers

211 views

### Two different Thom diagonals in recent literature?

Taking the point of view that a Thom spectrum functor should be a map $Top_{/BGL_1(R)}\to LMod_R$, for $R$ some $\mathbb{E}_n$-ring spectrum, there seem to be two morphisms (in $Top_{/BGL_1(R)}$) that ...

**1**

vote

**0**answers

211 views

### Can one prove the poincare duality for projective scheme by proving it for projective space?

It's well known the relationship between Poincare duality and Thom isomorphism（I mean cohomology purity $R^q i^! F=0$ if $q\neq c $ ) $\quad $
$Rf_!Ri_!=R(f|_Z)_!$ where f is $P_k^n\rightarrow k$ ...

**13**

votes

**0**answers

187 views

### Uniqueness of connected cover of Morava K-theory

Let $k(n)$ denote the connected cover of Morava $K$-theory $K(n)$ at the prime $2$ and in particular $n=2$. It is known that $$ H^*(k(n)) = A//E(Q_n), $$
where $A$ is the Steenrod algebra and $Q_n$ is ...

**10**

votes

**1**answer

297 views

### Dimension in CW-approximation

The following question was something that came to my mind during my (unsuccessful) attempt at answering this MO-question.
Let $X$ be a topological space, and let $\tilde{X}\to X$ be a ...

**21**

votes

**6**answers

1k views

### Book recommendation for cobordism theory

I am planning to organize a seminar on cobordism theory and I'm looking for a reference. Such a reference is preferably a book, but I'm open to other ideas.
The audience is familiar with ...

**10**

votes

**1**answer

456 views

### What is the Status of Borel conjecture today?

Let me recall the conjecture: $M$ and $N$ two aspherical closed $n$-manifolds with isomorphic fundamental groups, then $M$ and $N$ are homeomorphic.

**4**

votes

**0**answers

133 views

### References for bilinear forms on chain complexes?

I am looking for references that include general results and theorems for bilinear forms defined on chain complexes. That is, bilinear forms $\langle \cdot , \cdot \rangle_i : C_i \times C_i \to ...

**6**

votes

**1**answer

137 views

### Rational cohomology of the Rosenfeld projective planes

The bioctonionic plane $(\mathbb{C} \otimes \mathbb{O})\mathbb{P}^2$, the quarteroctonionic plane $(\mathbb{H} \otimes \mathbb{O})\mathbb{P}^2$ and the octooctonionic plane $(\mathbb{O} \otimes ...

**6**

votes

**2**answers

444 views

### Genuine equivariant ambidexterity

A particular case of Lurie and Hopkins' ambidexterity theory is that if $G$ is a finite group acting on a $K(n)$-local spectrum $X$ then the norm map
$$ X_{hG} \to X^{hG} $$
is a $K(n)$-local ...

**3**

votes

**1**answer

137 views

### cohomology module of unit tangent vector bundles over spheres

Let $S^m$ be the $m$-sphere and $\tau (S^m)$ the sphere bundle consisting of unit tangent vectors in the tangent bundle $TS^m$. Then we have a fibration
$$
S^{m-1}\longrightarrow ...

**9**

votes

**0**answers

277 views

### What does convergence of a Bousfield-Kan spectral sequence say about the homotopy type of the totalization?

Given a cosimplicial space or spectrum $X^\bullet$, there is an associated Bousfield-Kan spectral sequence. This starts out as the bigraded object obtained by taking homotopy groups of each $X^n$ and ...

**29**

votes

**5**answers

3k views

### What is modern algebraic topology(homotopy theory) about?

At a basic level, algebraic topology is the study of topological spaces by means of algebraic invariants. The key word here is "topological spaces". (Basic) algebraic topology is very useful in other ...

**2**

votes

**1**answer

163 views

### $n$-th cohomology of locally compact subsets in R^n

Where can I find a reference that for any locally compact (or just open) subset $U$ of $\mathbb{R}^n$, $H^n(U;\mathbb{Z})$ (the n-th Cech integral cohomology) is trivial?

**6**

votes

**0**answers

163 views

### What is the paper “Structure and homology of configuration spaces” by F. Cohen and L. Taylor?

Cohen and Taylor, in their paper Computations of Gelʹfand-Fuks cohomology, the cohomology of function spaces, and the cohomology of configuration spaces (1978), cite this numerous times:
[9] F. R. ...

**6**

votes

**2**answers

498 views

### quotient space of Eilenberg-MacLane space

Let $\pi$ be a group and $K(\pi,1)$ the Eilenberg-MacLane space. Let $G$ be a finite group acting on $K(\pi,1)$ such that the following is a covering map
$$
K(\pi,1)\longrightarrow K(\pi,1)/G.
$$
...

**1**

vote

**1**answer

326 views

### Reference request for Godement's “Topologie algébrique et théorie des faisceaux”

Does anybody know if an english translation of this paper exists please?

**3**

votes

**0**answers

190 views

### Are all 4-manifolds $Pin^{\tilde{c}}$?

It's known that all oriented 4-manifolds admit a $Spin^c$ structure, ie. a spin structure on $TX\oplus\mathcal{L}$ for some complex line bundle $\mathcal{L}$.
A usual generalization of this ...

**26**

votes

**1**answer

673 views

### What is, really, the stable homotopy category?

When you try to understand the fuss behind the new good categories of spectra that arose on the 90's, you read things such as the following paragraph written by Peter May (from "The Hare and the ...

**3**

votes

**0**answers

59 views

### Is the functor of PA forms known to be equivalent to the functor of PL forms for noncompact spaces?

In the following paper:
Robert Hardt, Pascal Lambrechts, Victor Turchin, and Ismar Volić, Real homotopy theory of semi-algebraic sets, Algebr. Geom. Topol. 11 (2011), no. 5, 2477–2545.
the authors ...

**16**

votes

**1**answer

574 views

### (really) basic intuition for $\mathbb A^1$-homotopy theory

Apologies in advance if this question is inappropriate for MO.
I'm trying to read here and there about $\mathbb A^1$-homotopy theory in algebraic geometry. I understand some abstract machinery is ...

**30**

votes

**2**answers

446 views

### If $A$ is the ring of continuous functions on a genus $g$ surface, can the genus of $X$ be seen by simple algebra in $A$?

I was describing to a friend the result that a compact Hausdorff space is determined up to homeomorphism up to by its ring of continuous functions, and he asked how one could see the genus of a ...

**1**

vote

**0**answers

50 views

### Semicovering and homotopy lifting property

Has a semicovering map ( local homeomorphism + unique path lifting property ), the homotopy lifting property? Clearly it has the homotopy path lifting property.

**9**

votes

**1**answer

375 views

### Spectra as functors from Spaces to Spaces

I will use the notation of this question. So, if $X$ is a (nice) topological space and $G$ is an abelian group, we can form its $G$-linearization $G[X]$. In McCord's article, this was denoted ...

**6**

votes

**0**answers

163 views

### Homotopy classes of maps and cohomology classes (Hatcher, AT, Thm 4.57) [duplicate]

Hatcher's AT Theorem 4.57 is used in both the algebraic topology construction of Seifert surfaces, and the (similarly flavored) proof that given a compact 3-manifold (with or without boundary), we can ...

**4**

votes

**2**answers

251 views

### homotopy equivalence between configuration spaces

Let $M$ be a $m$-dimensional compact manifold without boundary and $W(M)$ the non-compact $CW$-complex obtained by glueing $[0,1)\times (0,1)^{m-1}$ to $M$, identifying the boundary ...

**2**

votes

**1**answer

201 views

### isotopy equivalence (topological meaning) between $CW$-complexes

Let $M$ and $N$ be $CW$-complexes.
Definition. (different from the isotopy notion in geometry of submanifolds). A (topological) isotopy is a fibre-wise continuous map
$$
F: M\times ...

**9**

votes

**1**answer

269 views

### embedding of quaternionic projective spaces

Let $\mathbb{H}P^m$ be the $m$-th quaternionic projective space. What is the smallest integer $N$ such that there exists an embedding
$$
\mathbb{H}P^2\longrightarrow \mathbb{R}^N?
$$
Are there any ...

**7**

votes

**1**answer

178 views

### homological 2 dimensional groups

In a Commentarii Mathematici Helvetici paper by Benno Eckman and Heinz Müller in 1980 (volume 50, pages 510-520) proved that poincaré Duality Groups of dimension 2 with positive first ...

**1**

vote

**0**answers

54 views

### simplicial approximation of an open subset

I am interested in the following problem :
Let $K$ be an abstract simplicial complex, let $\vert K \vert$ be its geometrical realization, let $U\subset \vert K \vert$ be an open subspace (not a ...

**2**

votes

**1**answer

179 views

### Smallest homotopy equivalent space inside a manifold with boundary

It is well known that any compact manifold with boundary is homotopy equivalent to its interior. Is there a notion of some smallest space in the interior of the manifold that is homotopy equivalent to ...

**1**

vote

**0**answers

107 views

### cohomology ring of compact submanifolds of Euclidean spaces

Suppose we have a compact $m$-dimensional submanifold $M$ of $\mathbb{R}^N$ and we want to know the cohomology ring $H^*(M;\mathbb{Z})$.
Let $\epsilon>0$ and a $m$-dimensional finite simplicial ...