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

**2**

votes

**0**answers

99 views

### Generalization of Jordan Curve Theorem

Jordan Curve Theorem says that any plane continuum homeomorphic to $\mathbb{S}^1$ separates the plane into exactly two components.
Now
"Let $\alpha$ and $\beta$ be two homeomorphic plane continua. ...

**7**

votes

**1**answer

292 views

### Thom isomorphism from the ABGHR perspective

In ABGHR Thom spectra are described in the following way: we start with a morphism of Kan complexes $X\to \mathbb{S}\text{-line}$, where $\mathbb{S}\text{-line}$ is an $\infty$-groupoid which is ...

**6**

votes

**1**answer

215 views

### Action of the homotopy braid groups on reduced free groups

Firstly some definitions:
$B_n$ is the braid group with $n$ strands.
$\widetilde{B_n}$ is "homotopy braid group", which is a factor group of $B_n$ by adding the relation that $A_{j,k}$ ...

**0**

votes

**0**answers

20 views

### Action as specified in the extension given by the holomorph

I am reading Braid Groups, Free Groups, and The Loop Space of the 2-sphere
by F.R. Cohen and J. Wu, which is available here.
Here is an extract of the paper:
(Please refer to the paper for ...

**7**

votes

**0**answers

381 views

### Failure of “equivariant triangulation” for finite complexes equipped with a $G$-action

Let $\mathcal{S}$ be the $\infty$-category of spaces, and let $G$ be a finite group, and let $BG$ be the groupoid with one object and automorphisms given by $G$.
Consider the $\infty$-category ...

**2**

votes

**0**answers

129 views

### Which homology classes from loop space?

Fix a closed connected manifold $Q$ and let $LQ$ denote its free loop space. We can get second homology classes on $Q$ by "doing things" to loops in $Q$.
For instance, if we have a loop of loops, it ...

**2**

votes

**2**answers

275 views

### When do zero-simplices of a simplicial diagram determine its homotopy colimit?

Suppose that I have a diagram of simplicial sets $X_\bullet:\mathscr{C} \to Set^{\Delta^{op}},$ with $\mathscr{C}$ a small category such that for each $C \in \mathscr{C},$ $X_\bullet(C)$ is a Kan ...

**2**

votes

**1**answer

112 views

### Does $A_{j,k}$ commute with all its conjugates in homotopy braid groups?

Let $\tilde{B_n}$ be the homotopy braid group; namely, in the deformation of braids, a braid string is allowed to intersect itself. Similarly let $\tilde{P_n}$ be the homotopy pure braid group.
I am ...

**5**

votes

**2**answers

508 views

### Mayer-Vietoris sequence for topological K-theory

I'm reading the paper Loop groups and twisted K-theory I by Freed, Hopkins, and Teleman. They give some examples of computing (twisted) K groups using the Mayer-Vietoris sequence.
I'm a bit ...

**0**

votes

**0**answers

141 views

### Self homotopy equivalence

Given $X$, a simply connected CW-complex of finite type, ${\rm aut}(X)$ denotes the set of its self homotopy equivalences, that are maps $f: X\rightarrow X$ which admits a homotopy inverse (i.e., ...

**2**

votes

**1**answer

166 views

### Axiomatization of Degree Theory

I am reading Ruiz's Mapping Degree Theory, and find an axiomatization of degree theory of $\mathbb R^n$ in P38. It says that there exists a unique map $d(f,D,y)\in\mathbb Z$ satisfies
Normality
...

**8**

votes

**2**answers

441 views

### Are there some tables or handbooks of homology and homotopy groups of every manifold which has been calculated?

Are there some tables or handbooks of homology and homotopy groups of every manifold which has been calculated? Or are there some tables or handbooks which list some common calculated results of ...

**4**

votes

**0**answers

223 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

287 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

146 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

647 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

94 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

192 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

265 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

217 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

192 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

425 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

296 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

510 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

105 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

333 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

94 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

109 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

165 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

502 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

165 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

455 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

418 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

203 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

675 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

383 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

244 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

192 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

282 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

231 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

239 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

98 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

184 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

106 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

348 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

125 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

369 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

90 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

114 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 ...