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

**4**

votes

**0**answers

195 views

### Etale Slice Theorem

I found the Luna's Slice Theorem very Technical. It will be helpful if someone illustrates the geometry involved in the theorem. Also why this theorem so useful?
This is Luna's Slice theorem from a ...

**4**

votes

**1**answer

285 views

### Free action of $\mathbb{Z}(2^{\infty})$ on a compact space

Assume that $X$ is a Hausdorff compact space such that $\forall n\in \mathbb{N}$, we have a free action of $\mathbb{Z/{2^{n}}\mathbb{Z}}$ on $X$. Must $\mathbb{Z}(2^{\infty})$ act freely on ...

**4**

votes

**0**answers

134 views

### spectral sequence of a bicomplex equipped with a group action

Let $(A_1^\bullet,\partial_1)$ and $(A_2^\bullet,\partial_2)$ be complexes of $\mathbb{C}$-vector spaces.
We suppose that $(A_1^\bullet,\partial_1)$ and $(A_2^\bullet,\partial_2)$ are equipped with ...

**15**

votes

**2**answers

610 views

### When is a topological space the homotopy colimit of an open covering?

Suppose that $X$ is a topological space and $\left(U_i \to X\right)$ is an open cover. We can associate to it the Cech diagram of this cover $$C_U:\Delta^{op} \to Top.$$ I know that for many good ...

**1**

vote

**0**answers

91 views

### Identifying a cohomology class arising from a Postnikov decomposition of BU(2)

For various reasons I'm currently interested in Principal $U(n)$-bundles $U(n)\hookrightarrow E\xrightarrow{p} S^m\times X_g$ over the Cartesian product of an $m$-sphere and a closed oriented surface ...

**2**

votes

**1**answer

185 views

### Simplicial version of the A-infinity operad

I am looking for a description of the $A_\infty$ operad in the category of simplicial sets. More specifically, I am looking for a formulation of the loop space recognition principle for simplicial ...

**1**

vote

**0**answers

264 views

### Testing the faithfulness of group homomorphisms by testing on the level of induced Lie Algebras

Let $G$ be a group and let $\Gamma_G(k)$ be the $k$th term of the
lower central series of $G$. For each $k\geq 1$, set
$\mathrm{gr}_k(G)=\Gamma_G(k)/\Gamma_G(k+1)$ and
...

**8**

votes

**2**answers

210 views

### Configuration space like subspace of sphere product

For $k \geq 2, n \geq 1$ let $$M^{n,k} = \{(x_1,\dots,x_k) \in S^n \times \dots \times S^n \ | \ x_1 + \dots + x_k = 0\}$$ This is a compact CW-complex and almost, but not quite, a manifold. I ...

**4**

votes

**0**answers

185 views

### Topology of the space of foliations on a 3-manifold

Denote by $\mathcal{P} (M)$ the space of smooth plane fields(oriented and transversely oriented) on a given closed and orientable 3-manifold $M$ with the $C^{\infty}$ topology, and by $\mathcal{F}(M)$ ...

**2**

votes

**3**answers

423 views

### Is true that $\left[\frac{\hat{A}(\mathbb HP^m)} { \hat{M}(\mathbb HP^m) }\right]_{4m} = 0$?

There are two questions:
How to prove that in general
$[\hat{A}(\mathbb HP^m)]_{4m} = 0$
It is possible to verify it for low values of $m$.
How to prove that in general
...

**9**

votes

**2**answers

272 views

### Is there any relationship between the topologies of the clique complex and the independence complex?

Let $G$ be a simple graph on a finite vertex set. The clique complex $X(G)$ is the simplicial complex whose faces are complete subgraphs of $G$, and the independence complex $I(G)$ is the simplicial ...

**7**

votes

**1**answer

495 views

### Relation between $BG$ in topology and in algebraic geometry

This could as well have been asked in the comments to this question, but I prefer to open a new one for the sake of clarity.
Say $G$ is a reductive group over the complex numbers, with compact real ...

**1**

vote

**0**answers

101 views

### Quantization of Chern number $c_1^n$ on 2n dimensional spin manifold [closed]

All orientable 2-manifolds are spin manifolds, and we know that the quantization of the first Chern number $c_1$ of a complex line bundle on 2-manifold is $\mathbb{Z}$.
For 4-manifolds, the second ...

**2**

votes

**1**answer

310 views

### fixed point and homotopy fixed points

Let $G$ be a group and $X$ be a $G$-space (finite G-CW-complexe when needed).
Let $p$ a prime number and $G= \mathbf{Z}/p\mathbf{Z}$,
If I'm not wrong Miller-Lannes,... theory provides tools and ...

**2**

votes

**0**answers

89 views

### Computing $\text{Tor}_*^{R_G} (\mathbb{Z}, \mathbb{Z}) $ for a compact Lie group $G$

Let $R_G$ be the representation ring of $G$ a connected, simply connected Lie group, $I_G$ the augmentation ideal and $\mathbb{Z}=R_G/I_G$. $R_G$ acts on $\mathbb{Z}$ via $V \cdot n = (\dim V ) n$. I ...

**0**

votes

**0**answers

107 views

### An integrality theorem for immersions of quaternion projective spaces in the euclidean space

There are three questions:
Please let me know your proof of the following theorem:
If $\mathbb HP^2$ can be immersed in $\mathbb R^{12}$ with an Euler class $W_{4}(\nu)$ for the normal bundle of ...

**0**

votes

**1**answer

164 views

### maps $\mathbb{S}^{n} \to \mathbb{S}^{n}$ [closed]

I'm trying to prove that if two (continuous) maps $f, g : \mathbb{S}^{n} \to \mathbb{S}^{n}$ are such that $f(x) \neq -g(x)$ for any $x \in \mathbb{S}^{n}$, then $f$ and $g$ are homotopic. But I ...

**15**

votes

**1**answer

487 views

### Soft and hard part of geometry [duplicate]

While listening to some lecture of Alain Connes about noncommutative geometry, he spoke about various generalizations of the classical concepts from geometry and divided it into "soft" and "hard" ...

**0**

votes

**1**answer

161 views

### An integrality theorem for immersions of complex projective spaces in the euclidean space

There are three questions:
Please let me know your proof of the following theorem:
If $CP^3$ can be immersed in $R^8$ with an Euler class $W_{2}(\nu)$ for the normal bundle of $CP^3$ respect to ...

**10**

votes

**1**answer

432 views

### Is there a category whose isomorphisms are precisely the simple homotopy equivalences?

If we start with the category of finite complexes and continuous maps, and then identify two morphisms iff they are homotopic, we get the homotopy category of finite complexes, and it is trivial to ...

**3**

votes

**2**answers

416 views

### Reference request for cohomology of coverings

Let $B$ be a compact manifold, and $\hat{B}\to B$ be the maximal abelian covering of $B$; i.e. $\hat{B}$ is the quotient of the universal cover with respect to the commutator subgroup of $\pi_1(B)$. ...

**5**

votes

**0**answers

197 views

### Gompf's invariant of $2$-plane fields

I am interested in low dimensional contact topology. These days I read "Handlebody construction of Stein surfaces" written by R. E. Gompf, and study an invariant $\theta (\xi)$. This invariant is ...

**11**

votes

**2**answers

671 views

### from a circle to higher spheres

Question: Is there a group $G$ and a CW-complex $X$ such that
1) $X$ is homotopy equivalent to the circle $S^{1}$.
2) $G$ acts on $X$
3) the space of fixed points $X^{G}$ is weakly equivalent to ...

**3**

votes

**1**answer

368 views

### Hopf-algebras in associative ring spectra

I'm interested in a definition of cocommutative Hopf-algebra objects in the $\infty$-category of associative (read: $A_\infty$) ring spectra. One thought I had was to think of cocommutative ...

**1**

vote

**1**answer

237 views

### A Comparison between $\pi_{1}$ of cohomology and cohomology of $\pi_{1}$

Let we have a complex of abelian topological or lie groups $$\ldots \to G_{n}\to G_{n+1}\to \ldots$$ such that the image of $G_{n}$ is a closed subgroup of $G_{n+1}$. Then we have a complex of ...

**4**

votes

**1**answer

182 views

### Are homotopy braid groups residually nilpotent?

A group is called residually nilpotent if given any non-identity element, there is a normal subgroup not containing that element, such that the quotient group is nilpotent. It is known that pure braid ...

**5**

votes

**1**answer

273 views

### Framed version of braided monoidal category

The operad $\mathcal{D}_2$ of little $2$-disks is an operad whose $n$-th space is a $K(PB_n,1)$ where $PB_n$ denotes the pure braid group on $n$-strands. Algebras over $\mathcal{D}_2$ have a ...

**2**

votes

**0**answers

188 views

### What is the right categorical framework for diagonal approximations, cup and cap products and identities between them?

I wanted to know what is the right categorical analog for the diagonal approximation for the singular chain complex $\Delta: C\rightarrow C\otimes C$ as a morphism in the homotopy category of chain ...

**3**

votes

**1**answer

233 views

### Realization of second Stiefel-Whitney class

I hope this is not too trivial.
Let $M$ be a compact oriented manifold and $x\in H^2(M, \mathbb{Z}/2\mathbb{Z})$. My question is that if there exists a real oriented vector bundle $V$ over $M$ such ...

**0**

votes

**0**answers

92 views

### Degree of Map between Pseudomanifold

There are two different ways to define a degree of map.
Let $M$, $N$ be smooth, and $M$ is compact, $N$ is connected. If $f\in C(M,N)$, we define $\deg f$ by smooth approximation. [Milnor/Topology ...

**6**

votes

**2**answers

172 views

### Are all unstable homotopy groups of $U(n)$ torsion?

The first few unstable homotopy groups of the unitary groups $U(n)$ were calculated by Borel-Hirzebruch, Toda, and Kervaire, and they are all torsion. There is a paper by Matsunaga (details below) in ...

**1**

vote

**0**answers

104 views

### (The Homotopy type of the) lifting of homeomorphism of Grassmanian

For $k<n$ put $FM_{k\times n}$ for the space of all $k\times n$ full rank matrices with real or complex entries. Note that the permutation group $S_{n}$ has an obvious action on this space ...

**6**

votes

**1**answer

332 views

### K-theory of the h-cobordism category

I was reading through Kervaire and Milnor's "Groups of Homotopy Spheres", in which the authors begin to compute the groups $\Theta_n$ of h-cobordism classes of homotopy $n$-spheres (with group ...

**1**

vote

**1**answer

116 views

### What is the relationship between $BU$ and $\textrm{Fred}_0(H)$?

Let $U=\cup_{n=1}^{\infty} U(n)$ be endowed with the weak topology, let $BU$ be its classifying space. Then for any compact CW complex $X$, $[X,BU]$ classifies all the vector bundles on $X$ up to ...

**-1**

votes

**1**answer

365 views

### loop homology product for oriented compact manifolds with boundary

This is my first steeps in string topology and please forgive the basic level of my questions: I reformulate my question
Chas and Sullivan define the loop homology product for closed (=compact with ...

**1**

vote

**0**answers

115 views

### Explicitly showing that a free group is LERF [closed]

Let $F$ be a free group on a finite set $X$, and let $M$ be a finitely generated subgroup.
Marshall Hall's theorem states that $M$ is closed in the profinite topology on $F$. That is, $M$ is the ...

**1**

vote

**0**answers

88 views

### Role of determinant of the matrix corresponding to $i$-th Homology group.

I was thinking about the proof of the Lefschetz's Fixed point theorem and the ingeniuty of the Hopf's Trace formula, i.e. associating the trace of the matrix for deciding about the fixed points. Now ...

**0**

votes

**1**answer

111 views

### Unseparability of two linked rings in higher dimensions [closed]

I am not familiar with topology. We know that in $R^3$, we cannot separate two "rings": two copies of $S^1$, if they are "linked".
I wonder that is there any similar results for two copies of ...

**1**

vote

**2**answers

182 views

### Sheaf cohomology on non paracompact topological spaces

I have some confusion on the subject of sheaf cohomology on non-paracompact topological spaces, i hope you can help me.
My reference is Godement's book "Topologie algebrique et theorie dex ...

**-1**

votes

**1**answer

207 views

### Creating topological spaces with portals [closed]

I'm trying to rigorously describe an object that I'm calling a "portal". The situation is easiest to describe in two dimension.
I start with a line segment $pq$ in $\mathbb{R}^2$. I want to remove ...

**1**

vote

**1**answer

108 views

### Differential of homological atiyah-Hirzebruch Spectral sequence for K-homology

The first non vanishing differential $d_3$ of the cohomological Atiyah-Hirzebruch spectral sequence for computing (Complex) Topological $K$-theory out of ordinary cohomology has a ...

**10**

votes

**0**answers

166 views

### Homotopy groups of $MO(2)$

Have there been any computations of the higher homotopy groups of $MO(2)$, the Thom space of the universal $O(2)$-bundle? Thom himself noted in his landmark 1954 paper that
$$
\pi_1(MO(2))=0,\quad ...

**22**

votes

**2**answers

2k views

### Why can't we take three loops?

Apologies for the vague title and soft question. According to Etingof, Igor Frenkel once suggested that there are three "levels" to Lie theory, which I guess could be given the following names:
No ...

**12**

votes

**2**answers

606 views

### Eilenberg-Mac lane spaces and a generalization

Let $G$ and $H$ be two abelian groups and let $n>1, m>1$ be two different integers. How many different spaces $X$ (up to homotopy) do we have with the property $\pi_{n} X=G$ , $\pi_{m} X=H$ and ...

**5**

votes

**1**answer

303 views

### Localized J homomorphism

Let $X$ be a simply connected finite CW complex, $\xi$ and $\eta$ vector bundles over $X$ of the same dimensions and their dimension is big enough, so they are stable bundles. Let $p$ be a prime.
Are ...

**1**

vote

**2**answers

388 views

### Does a graded vector space isomorphism between the homology of two loop spaces imply the existence of an algebra isomorphism?

Consider ordinary homology with coefficients in a field. For $X$ a path-connected pointed space, the graded vector space $\bigoplus_{q\ge 0} H_q(\Omega X)$ has the structure of an algebra with the ...

**7**

votes

**0**answers

156 views

### Does the de Rham version of Cohen's theorem hold in the $\infty$-setting?

One of the first results that one needs to prove in the theory of chiral algebras is a de Rham version of Cohen's theorem on the homology of $C_n$ spaces. This is achieved in Beilinson-Drinfeld's book ...

**3**

votes

**1**answer

119 views

### Associated graded Lie algebra of braid groups

Let $G$ be a group and let $\Gamma_G(k)$ be the $k$th term of the lower central series of $G$. For each $k\geq 1$, set $\mathcal{L}_G(k)=\Gamma_G(k)/\Gamma_G(k+1)$ and ...

**10**

votes

**0**answers

200 views

### What is the symmetric monoidal functor from Clifford algebras to invertible K-module spectra?

There ought to be a symmetric monoidal functor from the symmetric monoidal $2$-groupoid whose objects are Morita-invertible real superalgebras (precisely the Clifford algebras), morphisms are ...

**4**

votes

**0**answers

101 views

### Nowhere vanishing section of vector bundles over varieties as connectivity of morphism of stacks

The following is, amongst others, a Hartshorne exercise:
Let $V$ be a $k$-variety of dimension $n$ and $\mathcal{E}$ a vector bundle of rank greater than $n$, then, generically, a generating section ...