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

**11**

votes

**2**answers

277 views

### What's an example of 2 elliptic curves with the same ground ring s.t. their associated cohomology theories detect different things?

My understanding is that a complex-oriented spectrum is a ring spectrum $E$ with a map $MU \to E$.
Analogously, a ring with a formal group law is a ring $R$ with a map $L \to R$ (where $L$ is the ...

**2**

votes

**1**answer

80 views

### Curvature of a principal bundle and the exterior covariant derivative

I am sorry if this is too elementary; I had posted it on math.stack but no one answered.
Let $P\to M$ a principal fibre bundle with fibre $G$, and let $A\in \Omega^{1}(P)\otimes\mathfrak{g}$ be a ...

**0**

votes

**0**answers

7 views

### dual basis of cohomology algebra [migrated]

Let $H^*(M)$ be the cohomology algebra of oriented manifold $M$ with rational coefficients.
Let $\{b_i\}$ be a basis of $H^*(M)$ as a vector space over $\mathbb{Q}$.
Let the dual basis be
...

**20**

votes

**1**answer

373 views

### Interplay between Loop Quantum Gravity and Mathematics

It is known that there are many interesting connections between String Theory and modern Mathematics, with a rich feedback going on in both directions: there have been advances in mathematics thanks ...

**2**

votes

**0**answers

111 views

### Classifying spaces of topological groups whose underlying spaces are homotopy equivalent

Let $G$, $H$ be topological groups and $f:G\rightarrow H$ a continuous group homomorphism which happens to be a homotopy equivalence of the underlying topological spaces. Let us assume that $G$, $H$ ...

**7**

votes

**1**answer

257 views

### Must we know $MU^*(X)$ in order to compute $Ell^*(X)$?

Let $Ell^*(X)$ be the elliptic cohomology theory (associated to a given elliptic curve $E$) of a nice space $X$.
Recall the Landweber-Ravenel-Stong construction:
$MU^*(X) \otimes_{MU^*} R \simeq ...

**3**

votes

**1**answer

132 views

### Toral rank conjecture

In 1985, S.~Halperin conjectured in the topological context of maximal free torus actions on topological manifolds, that:
If $X$ is a topological space, then $$\dim H^*(X;\mathbb Q)\geq 2^{rk(X)}.$$
...

**0**

votes

**3**answers

194 views

### orbit space of $\mathbb{Z}_p$ action over complex projective space by permuting the homogeneous coordinates

$Z_p$:=cyclic group of order $p$.
I want to understnd $H_\ast(\mathbb{C} P^{n}/Z_{n+1};Z)$ with $(n+1)$ being a prime number,and the action is given by permuting the homogeneous coordinates.
For ...

**4**

votes

**1**answer

146 views

### Are all quotients of a weakly contractible space via a free group action classifying spaces of the group?

I asked this question on math.stackexchange a week ago, but did not get an answer.
First of all, I don't want to restrict to any kind of "nice spaces" since I am interested in the most general ...

**2**

votes

**1**answer

150 views

### Loop defects in Walker-Wang model

My question is about the description of general defects (specially loop defects) in the Walker-Wang (WW) model.
Elementary excitations in the WW model can be point particles, loop defects and more ...

**1**

vote

**0**answers

71 views

### Poincare-Lefschetz duality, universal coefficients, and middle cohomology [migrated]

Sorry if the question is too simple, algebraic topology is not my strong suit. Let $(M,\partial M)$ be a $2n$-dimensional manifold with boundary, with one-dimensional middle cohomology. By ...

**5**

votes

**1**answer

212 views

### When does the Borel construction have the homotopy type of a CW-complex?

Suppose that $G$ is a Lie group acting smoothly on a manifold $M,$ does the Borel $M \times_G EG$ construction have the homotopy type of a CW-complex? If not, under what conditions would this be true? ...

**-5**

votes

**1**answer

142 views

### Stiefel-Whitney class of complex projective spaces [on hold]

Let $T\mathbb{C}P^m$ be the tangent bundle of complex projective space. What is the total Stiefel-Whitney class $w(T\mathbb{C}P^m)$?
Let $a_m$ be the maximal integer such that the $a_m$-th dual ...

**1**

vote

**1**answer

156 views

### Moduli space of flat connections over a torus

Let us fix a principal bundle $G\hookrightarrow P\to T^{2}$, where $T^{2}$ is a torus. Is the moduli space of flat connections on $P$ known? At least, it is known for some particular gauge groups, ...

**1**

vote

**0**answers

73 views

### Why is the oriented $G$-homotopy type of a $G$-complex uniquely determined by the periodicity generator?

Say we have a periodicity generator $e \in H^k(BG)$. I can show that we then have a $(k-1)$-dimensional $G$-complex $X$ with free $G$-action. It's also not that difficult to see that it has trivial ...

**7**

votes

**1**answer

202 views

### Homotopy of orthogonal groups in the unstable range

We fix an integer $n$ and consider the stabilization map $O(n)\to O$.
Using rational methods one can easily check that the map
$\pi_{4i-1}(O(n))\to \pi_{4i-1}(O)\cong\mathbb{Z}$ vanishes for ...

**8**

votes

**3**answers

409 views

### When are (weak) homotopy equivalence testable on open covers?

I asked this question on math.stackexchange, but did not get an answer.
Let $f\colon X\rightarrow X'$ be a continuous map between two spaces $X,X'$, which might be arbitrary wild, especially I don't ...

**2**

votes

**1**answer

105 views

### rational cohomology of finite real grassmannian

Let $p_j$ to be the $j$-th Pontryagin class of the universal $n$-plane bundle $E_n(\mathbb{R}^\infty)\to G_n(\mathbb{R}^\infty)$. Then according to Theorem 1.6, The Cohomology of BSO n and BO n with ...

**4**

votes

**1**answer

242 views

### $K_0$ of integral group ring of cyclic group $\mathbb{Z}/p\mathbb{Z}$

Is there a table for the computation of $K_0(\mathbb{Z}[\mathbb{Z}/p\mathbb{Z}])$?
These groups are also known as ideal class group in number theory.In topology,they are the home of some important ...

**2**

votes

**0**answers

138 views

### Continuous maps to fat geometric realizations of simplicial spaces

The nLab page on partitions of unity mentions the application of partitions of unity as a way to construct continuous maps to geometric realizations of simplicial spaces. However I often feel ...

**10**

votes

**1**answer

340 views

### Pseudomanifolds and Poincaré duality

1) A $n$-dimensional homology manifold is a topological space $X$ such that for any $x\in X$, the homology groups
$$H_p(X,X-x,\mathbb{Z})$$
are trivial unless $p=n$ where
$$H_n(X,X-x,\mathbb{Z})\cong ...

**2**

votes

**1**answer

153 views

### cohomology ring of configuration spaces

In the paper configuration spaces: applications to Gelfand-Fuks cohomology, by F. Cohen and L. Taylor, Bull. Amer. Math. Soc., 1978, theorem 1, I did not find the proof. What method did the author use ...

**5**

votes

**2**answers

203 views

### homology of a mapping spectrum

If $X$ and $Y$ are two spectra, I denote by $F(X,Y)$ their mapping spectrum. This is uniquely determined by the existence of a natural isomorphism $[X\wedge Y, Z]\cong [X,F(Y,Z)]$.
I denote by $H_*$ ...

**1**

vote

**1**answer

115 views

### cohomology of the orbit space of a group action

Let $M$ be a manifold. Let a finite group $G$ act on $M$ discretely. Let $F$ be a field.
Suppose the induced action of $G$ on the cohomology algebra $H^*(M,F)$ is known. We want to obtain ...

**22**

votes

**1**answer

557 views

### Homotopy Type Theory: What is it?

My question is, broadly, what is the main project of Homotopy Type Theory (HoTT). I asked a professor who is likely to be correct and he say the following:
There are three directions:
...

**10**

votes

**1**answer

189 views

### Two H-space structures on S^3 and [X,S^3] different as groups for each: Explicit Example?

There are twelve continuous maps $S^3\times S^3\to S^3$ up to homotopy that make the three-sphere $S^3$ into an H-space. This follows from a result of James [1], which says that if there exists one ...

**1**

vote

**1**answer

104 views

### Retractions of ENR

Let $i:X\rightarrow \mathbb{R}^N$ be an imbedding of a topological space $X$. Assume that there exists an open neighborhood $U$ containing $i(X)$ which also admits a retraction $p:U\rightarrow X$. The ...

**2**

votes

**0**answers

80 views

### project limit on $n$- simplical complex which is principal homogeneous with respect to an action

The setting:
Let G be compact locally $\Bbb{Q}_p$ analytic group. We fix a countable basis of open normal subgroups $G\supset G_1\supset ...G_r\supset...$
We suppose that we are given a system of ...

**0**

votes

**0**answers

44 views

### question about Computations of gelfand-fuks cohomology, the cohomology of function spaces, and the cohomology of configuration spaces

In the paper Computations of gelfand-fuks cohomology, the cohomology of function spaces, and the cohomology of configuration spaces, F. R. Cohen, L. R. Taylor, Geometric Applications of Homotopy ...

**3**

votes

**1**answer

251 views

### Mixed Hodge structure and cup product

I'm looking for a reference for the answer to the following questions.
Let $X$ be an algebraic variety over C. When is the cup product a morphism of Mixed Hodge structures? Does $X$ have to be ...

**-2**

votes

**1**answer

120 views

### Does there exist a fibre bundle $K(\mathbb{Z}_4,1)\rightarrow K(\mathbb{Z}_2,1)$ with fiber $K(\mathbb{Z}_2,1)$? [closed]

Does there exist a fibration $K(\mathbb{Z}_4,1)\rightarrow K(\mathbb{Z}_2,1)$, evidently with fiber $K(\mathbb{Z}_2,1)$?

**1**

vote

**0**answers

73 views

### A noncommutative vector bundle associated with a codimension one foliation

Assume that we have a codimension one foliation of a manifold $M$ which is generated by a one form $\alpha$. So the following $\phi$ satisfies $\phi \circ \phi =0$:$$\phi:\Omega^{i}(M)\to ...

**5**

votes

**0**answers

141 views

### proving the injectivity half of de Rham's theorem by construction in degrees other than $1$ and $n$

(This is a revision of a question I asked on MSE.)
Let $M$ be a smooth manifold of dimension $n$, and let $\omega$ be a differential form of degree $p$ on $M$. Then we have (I'm pretty sure) the ...

**2**

votes

**0**answers

122 views

### Kernel of the Weil homomorphism for compact symmetric spaces

Let $X = G/K$ be a Riemannian symmetric space of compact type and consider the "Weil homomorphism" $$w^\bullet: H^\bullet(BK; \mathbb R) \to H^\bullet(X; \mathbb R),$$ i.e. the map in cohomology ...

**0**

votes

**0**answers

62 views

### Dyer-Lashof algebra structures over graded modules

In Lecture Notes in Mathematics, Vol. 533, The homology of iterated loop spaces, Chapter 3, The homology of $C_{n+1}$-spaces, F. Cohen, Section 2, page 222, line 4, 5, 6:
for an arbitrary graded ...

**1**

vote

**1**answer

117 views

### Classification of $SU(2)$-bundles versus the classification of $SO(3)$-bundles

As explained in:
Classification of $SU(2)$ principal fibre bundles over four-dimensional manifolds
principal $SU(2)$ bundles $P_{SU(2)}$ over a four-dimensional manifold $M$ are classified by their ...

**2**

votes

**1**answer

229 views

### Dimension leaking in homology as opposed to homotopy

In homotopy theory we have the Seifert van-Kampen theorem, which is a clean statement about the fundamental groupoid of a pushout in $\mathsf{Top}$. There is also a 2d version of SvK in R Brown's ...

**1**

vote

**0**answers

138 views

### Geometric representatives of homology classes of manifolds

Is it true that for even dimensional differentiable manifold $M^{2n}$ all singular homology classes in dimension less than $n$ can be represented by a submanifold?

**2**

votes

**1**answer

96 views

### cohomology algebra of submanifold in euclidean space

If we write a manifold or CW-complex $X$ as a subset of $\mathbb{R}^n$, in expression of coordinates, for example, \begin{multline}
F(S^2,k+1)=\{(x_1,x_2,x_3,\cdots, ...

**4**

votes

**0**answers

176 views

### Fundamental group of the complement homogeneous variety in $\mathbb{C}P^{n-1}$

Let $f,g:\mathbb{C}^n\to \mathbb{C}$ are two irreducible homogeneous polynomials. If there is a homeomorphism $h:\mathbb{C}^n\to \mathbb{C}^n$ such that $h(X)=Y$ and $h(0)=0$, where $X=f^{-1}(0)$ and ...

**1**

vote

**0**answers

104 views

### Classification of $SU(n)$-principal bundles over a four-dimensional base

It is well-known that a principal $SU(2)$-bundle $P$ over a four-dimensional manifold $M$ is topologically classified by its second Chern-class $c_{2}(P)\in H^{4}(M,\mathbb{Z})$, as explained for ...

**6**

votes

**1**answer

256 views

### Properties of coefficients of ring spectra

This is an awkwardly backwards question, but bear with me here: Suppose I have a graded ring $R$ with unit, which has an invertible element $u$ in degree $2$. The multiplicative formal group law ...

**1**

vote

**0**answers

151 views

### Equivariant Homotopy

Let $G=\mathbb{Z}/2\mathbb{Z}$ be $\{\pm1\}$ and let there be two $G$-spaces given: $X=$ The surface of a cylinder including its boundary circles and $S^4$. That means we two G-actions $f_1:G\times ...

**1**

vote

**0**answers

135 views

### Classifying Spaces and Eilenberg-Maclane objects in the category of simplicial rings

[Skip down to the bottom for a correction] Let's work over a field k, assume it is as nice as you need it to be.. Suppose I have an ordinary (edit: commutative) affine group scheme G = Spec(A) over k, ...

**8**

votes

**4**answers

692 views

### Topological Grothendieck Construction

Let $C$ be a small category and $F\colon C^{op}\rightarrow Set$ a functor. The Grothendieck construction is the category $F\wr C$ with objects being pairs $(c,x)$ where $c$ is a object of $C$ and ...

**0**

votes

**0**answers

87 views

### permutation action on cohomology of configuration space

Let $F(M,n)$ be the $n$-th configuration (ordered) of manifold $M$.
In the paper The Integral Cohomology Algebras of Ordered Configuration Spaces of Spheres, ...

**1**

vote

**1**answer

169 views

### permutation action on cohomology of Stiefel manifolds

Let $V_k(\mathbb{R}^n)$ be Stiefel manifolds.
In the paper The cohomology rings of real Stiefel manifolds with integer coefficients, Martin Čadek, Mamoru Mimura, and Jiří Vanžura, J. Math. Kyoto ...

**0**

votes

**1**answer

58 views

### Fixed point property for the projectivization of manifold of fixed rank matrices

Let $M$ be the manifold of all matrices in $M_{n}(\mathbb{R})$ with fixed rank $0<k<n$. The projectivization of $M$ is denoted by $PM$.
Does $PM$ satisfy fixed point property?

**0**

votes

**1**answer

113 views

### $\mathbb{Z}_{2}$ -equivariant vector bundles over manifold of rank-$k$ matrices

Edit: I remove the trivial part of the first version, according to comment of Alex Degtyarev
Let $M$ be the manifold of all matrices in $M_{n}(\mathbb{R})$ with fixed rank $0<k<n$. There is ...

**13**

votes

**1**answer

191 views

### Completed and uncompleted operations for Morava $E$-theory

Let $E = E_n$ be the $n$-th Morava $E$-theory with coefficient ring
$$
E_* = \mathbb{W}(\mathbb{F}_{p^n})[\![u_1,\ldots,u_{n-1}]\!][u^{\pm 1}].
$$
It is usual to consider the completed co-operations
...