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

**-3**

votes

**0**answers

72 views

### spin structures on knot complements [migrated]

Let $K$ be a knot in $S^3$, and let $M=S^3/N(K)$ be its knot complement, where $N(K)$ is a tubular neighborhood of $K$. $K$ is given for example by a its projection onto the plane.
The question is ...

**1**

vote

**0**answers

123 views

### Bockstein cohomology

There is a notion of Bockstein homomorphism $\beta$. I am interested precisly in the case of sequence $$0 \rightarrow \mathbb{Z}/p \mathbb{Z} \rightarrow \mathbb{Z}/p^2 \mathbb{Z} \rightarrow ...

**0**

votes

**0**answers

48 views

### Simplicial approximation diagram [on hold]

Let $K$ and $L$ be simplicial complexes as given, and let $\phi:|K|\to|L|$ be the continuous map, where $A=\phi(a)$, $B=\phi(b)$, and so on.
Check whether the map $\phi$ has a simplicial ...

**3**

votes

**1**answer

257 views

### K theory long exact sequence

(1) Suppose that $Z\subset X$ is a closed embedding, $U = X\setminus Z$ is the complement. If relevant, suppose that both $X, Z$ are smooth and even (if relevant) that the normal bundle of $Z\subset ...

**9**

votes

**0**answers

172 views

### Homological stability for orthogonal groups

In Vogtmann's paper "Spherical posets and homological stability for $O_{n,n}$" it is shown that for all fields different than the field $F_2$ with two elements the homology groups of the orthogonal ...

**6**

votes

**2**answers

298 views

### Is this almost-cosimplicial object familiar?

I have a sequence $X_0,X_1,\ldots$ of Abelian groups, along with face maps $d_0,\ldots, d_n: X_{n-1}\to X_n$ which satisfy the standard cosimplicial identities EXCEPT at the very bottom, where in ...

**1**

vote

**0**answers

124 views

### cohomology ring of second unordered configuration space

For a topological space $X$, let the second unordered configuration space $B(X,2)$ be the collection of
$
(a,b)\in X\times X
$
with $a\neq b$ modulo the equivalent relation $(a,b)\sim (b,a)$.
Let ...

**12**

votes

**0**answers

190 views

### Koszul complex for non-Koszul algebras

Let $A$ be a graded, connected, locally finite, quadratic algebra over a field $k$; that is, $A$ may be presented as $T(V)/I$, where $V = A_1$ is a finite dimensional $k$ vector space, and the ideal ...

**6**

votes

**1**answer

176 views

### Suspension of the third Hopf map

There are Hopf maps in three dimensions, I denote their homotopy classes by $h_1 \in \pi_3(S^2)$, $h_2 \in \pi_7(S^4)$ and $h_3 \in \pi_{15}(S^8)$ respectively. For $h_1$ and $h_2$ it is true that ...

**5**

votes

**1**answer

210 views

### abelian and nonabelian parts of Aut($\widehat{F_2}$)

Let $F$ be the free profinite group on two generators. Let $\text{IA}(F) := \ker\left(\text{Aut}(F)\rightarrow GL_2(\widehat{\mathbb{Z}})\right)$, the group of "IA automorphisms" of $F$. (I'm also ...

**2**

votes

**1**answer

301 views

### embeddings of product of spheres in Euclidean spaces [closed]

I notice that $T^2=S^1\times S^1$ can be embedded in $\mathbb{R}^3$ as a hypersurface (submnaifolds of codimension 1).
In general,
(1). could the product of spheres $S^{m_1}\times\cdots\times ...

**3**

votes

**0**answers

85 views

### configuration space of product of spheres

Let the $2$-nd ordered configuration space of a space $X$ be
$$
F(X,2)=\{(x,y)\mid x\neq y, x,y\in X\}
$$
and the $2$-nd unordered configuration space of a space $X$ be
$$
B(X,2)=\{(x,y)\mid x\neq y, ...

**3**

votes

**1**answer

118 views

### covering map from spheres to projective spaces and the associated vector bundle

Let $S^n$ be the $n$-sphere and consider a $2$-sheeted covering
$$
S^n\longrightarrow\mathbb{R}P^n.
$$
We have an associated vector bundle
$$
\xi: \mathbb{R}^2\longrightarrow ...

**12**

votes

**2**answers

299 views

### Cohomological obstructions to lift $\pi_0$ of a topological group

Let $G$ be a topological group. Denote the same group with the discrete topology by $G^\delta$ and denote the group of connected components of $G$ by $\pi_0G$. I am interested in the question when we ...

**3**

votes

**0**answers

156 views

### Operadic Lift of Lurie's Relative Tensor Product

In Section 4.4 of his book Higher Algebra, Lurie introduces, for a monoid object $A$ of a monoidal quasicategory $C$, and right and left $A$-modules $M,N$, the relative tensor product $M\otimes_AN$. ...

**6**

votes

**0**answers

115 views

### Two proofs of the Cheeger-Müller theorem

In the late 1970's, Cheeger and Müller independently proved the equality of analytic torsion and Reidemeister torsion for orthogonal representations, which had been conjectured by Ray-Singer. Their ...

**4**

votes

**3**answers

296 views

### classifying space of orthogonal groups

Let $O(n)$ be the $n$-th orthogonal group and $O$ be the direct limit of $O(n)$ with respect to $n$. Let $BO(n)$ and $BO$ be the classifying spaces.
Question:
Why $BO$ is an $H$-space? My supervisor ...

**3**

votes

**2**answers

218 views

### rational cohomology of symmetric groups

Let $\Sigma_k$ be the $k$-th symmetric group and $B\Sigma_k$ be its classifying space. How to prove:
for any $n\geq 1$ and the $n$-skeleton $sk_n (B\Sigma_k)$, there exists a finite dimensional ...

**6**

votes

**1**answer

186 views

### Under what conditions is the induced map of etale fundamental groups surjective?

Let $f:X \to Y$ be a morphism of schemes. I am interested in sufficient conditions on $f$ which would ensure that the induced map $\pi_1^{et}(X) \to \pi_1^{et}(Y)$ of etale fundamental groups is ...

**14**

votes

**1**answer

449 views

### Is the moduli space of graphs simply connected?

The moduli space of graphs $MG_n$ is the quotient of Culler-Vogtmann's outer space $X_n$ by the action of $\mathrm{Out}(F_n)$. It can be thought of as the space of metric graphs homotopy equivalent to ...

**3**

votes

**1**answer

170 views

### “Ambient homotopy” between preimages under a fiber bundle?

Choose a notion of an "ambient homotopy" between maps of topological spaces. For example, say that two embeddings $Y \rightarrow X$ are ambiently homotopic if there is a path between them in the space ...

**0**

votes

**0**answers

36 views

### Finite topological space [migrated]

At http://math.stackexchange.com/questions/1528995/finite-topological-space the user asked "Let $X$ be a Hausdorff topological space such that every closed subset has finitely many connected ...

**3**

votes

**1**answer

192 views

### What is kernel $\phi:G\rightarrow \pi_1(X/G,p(x_0))$?

Let $G$ be a discontinuous group (this means that it acts discontinuously with finite stabilizers) of homeomorphisms of a simply connected, locally compact metric space $X$. Let $p:X\rightarrow X/G$ ...

**4**

votes

**1**answer

142 views

### Transgression in terms of k-invariant for chain complexes

I am looking for a reference for the following. Say we have a $G$-space $X$ whose homology groups (in field coefficients $k$) are non-zero only in dimension zero and for a fixed $n>0$. Let $M$ ...

**2**

votes

**0**answers

84 views

### Chern character of finite $CW$-complexes and rational Pontrjagin class of vector bundles

Let $K$ be a finite $CW$-complex. Could you give any references or explanations for the following two items? I do not understand. Thanks!
(1). The Chern character from $\tilde{KO}^0(K)$ to the ...

**4**

votes

**1**answer

127 views

### The cooperations algebras Johnson-Wilson theory and truncated BP-theory

Given a prime $p$, there is a well known homology theory $BP$, known as Brown-Peterson homology. Has several related theories, namely the Johnson-Wilson theories $E(n)$ and the truncated ...

**5**

votes

**1**answer

186 views

### how to prove the $n$-times self-product of a map is null-homotopic

Let $k$ be a fixed positive integer and $\Sigma_k$ the $k$-th symmetric group. By letting $\Sigma_k$ permuting an orthonormal basis of a $k$-dimensional Euclidean space, there is a "regular ...

**7**

votes

**1**answer

142 views

### classifying maps of Whitney sums of vector bundles

For an $n$-dimensional vector bundle $\xi$ with structure group $G\leq O(n)$ over a $CW$-complex $B$, we have a classifying map up to homotopy
$$
f(\xi): B\longrightarrow BG,
$$
$f(\xi)\in [B;BG]$, ...

**8**

votes

**0**answers

139 views

### Which map realizes the isomorphism $KO_n(X)\otimes \mathbb{Q}\to \bigoplus_{i\in\mathbb{Z}}H_{n-4i}(X;\mathbb{Q})$?

The description of the real $KO$-homology groups $KO_n(X)$ can be given abstractly as maps to the real K-theory spectrum $KO$ smash $X$, or via triples $(M,x,\phi)$ where $M$ is a closed manifold, ...

**1**

vote

**0**answers

67 views

### Alexander Duality in the complex plane

Thanks to Alexander duality, we know that for each compact subset $K$ of $\mathbb{C}$ there is an isomorphism $$H_1(\mathbb{C} \backslash K) \simeq \prod_{i \in CC(K)} \mathbb{Z},$$ where $CC(K)$ is ...

**3**

votes

**1**answer

152 views

### Definition of Left Operadic Kan Extension for $\infty$-operads

In Lurie's book Higher Algebra, he makes the following definition:
Definition 3.1.2.2: Let $M^\otimes\to N(Fin_\ast)\times\Delta^1$ be a correspondence from an $\infty$-operad $A^\otimes$ to another ...

**2**

votes

**0**answers

83 views

### order of elements in a mapping space

Let $B$ be a finite CW-complex and $\xi$ be a vector bundle over $B$ with structure group $\Sigma_n$, the $n$-th symmetric group.
Then corresponding to $\xi$, we have a classifying map
$$
g\in \tilde ...

**0**

votes

**0**answers

92 views

### a paper by D. Kahn published in 1975

I want to find the reference:
D. Kahn, Homology of the Barratt-Eccles decomposition maps, Reunion Sobre Teoria De Homotopia, Universidad de Northwestern, Agosto, 1974. Sociedad Matemtica Mexicana, ...

**3**

votes

**2**answers

103 views

### equivariant embeddings from the k-th configuration space to the k+1-th configuration space

Let $S$ be a closed, orientable surface in $\mathbb{R}^3$ and $S'$ be the manifold $S\setminus\text{one point}$. Let $F(S',k)$ be the $k$-th (ordered) configuration space on $S'$. It is claimed in ...

**3**

votes

**1**answer

169 views

### Homology of solvable Lie groups made discrete

In what follows "homology" will mean group homology, i.e. $H_*(BG^\delta;{\mathbf R})$ for the group $G$ with the discrete topology.
It is well-known how to compute the homology of abelian groups, ...

**1**

vote

**0**answers

87 views

### Augmentation of the sphere spectrum

I am wondering if it is sensible to talk about the augmentation ideal of the sphere spectrum in the category of spectra, as well as the `submodule of decomposables', whose construction comes from the ...

**3**

votes

**0**answers

110 views

### Topologized category of bounded chain complex

I am reading Segal's paper 'categories and cohomology theories' [1], but there is one claim (in the last example in sec.2) I don't quite understand:
Let $\mathcal{C}$ be the category of bounded chain ...

**5**

votes

**2**answers

244 views

### 3-manifolds homotopy equivalent to a surface

I have heard that an open, orientable 3-manifold $X$ (non-compact, without-boundary) that is homotopy equivalent to an orientable surface $S_g$ must itself already be homemorphic to $S_g \times ...

**3**

votes

**1**answer

190 views

### the “Kahn-Priddy map” and “multiplicative $p$-local equivalence”

The following is a part of a paper that I need to understand
I totally do not know the argument. Could you explain? Thanks.
Let $\Sigma_n$ be the $n$-th symmetric group and $\Sigma_\infty$ be the ...

**4**

votes

**0**answers

147 views

### Cofiltered diagram of path connected spaces with empty homotopy limit?

Is it possible to have a filtered category $J$, a functor $F: J^\mathrm{op} \to \mathrm{Spaces}$ such that $F(i)$ is path connected for all $i$ and such that $\mathrm{holim} F = \emptyset$?
If $J$ is ...

**4**

votes

**1**answer

213 views

### Triviality of a fiber bundle

Is the principal fiber bundle $GL^+(6,\mathbb R)$ over $GL^+(6,\mathbb R)/SL(3,\mathbb C)$ trivial ?

**0**

votes

**0**answers

112 views

### when is “fibering” preserved under homotopy equivalence

Suppose I have an oriented $F$ bundle over $B$ with total space $E$ (all of the three are closed manifolds) and i have a closed manifold $E'$ which is homotopy equivalent to $E$.Is there any condition ...

**1**

vote

**0**answers

68 views

### Homology of spherical braid groups

By the spherical braid group, I mean the fundamental group of the configuration space of distinct unordered points in $S^2$. I am wondering what is known about the group homology of the spherical ...

**1**

vote

**1**answer

169 views

### Different model structures on Top

There is at least 3 model structures on the category of topological spaces, the Quillen Model structure, the Storm model structure and the Mixed model structure.
In the Mixed model structure ...

**2**

votes

**0**answers

128 views

### vector bundles over finite CW-complex with its structure group the symmetric group

Let $S_n$ be the $n$-th permutation group and $X$ be a topological space with a free and properly discontinuous $S_n$-action. We have a covering space
$$
X\to X/S_n
$$
and an associated vector bundle
...

**1**

vote

**0**answers

137 views

### Homotopy equivalence of Lens spaces

I find the following statement about the homotopy equivalence of Lens spaces in Wikipedia. The three-dimensional spaces $L(p,q_1)$ and $L(p,q_2)$ are homotopy equivalent if and only if $q_1 q_2\equiv ...

**12**

votes

**1**answer

421 views

### Automorphisms of Eilenberg-Mac Lane spaces and semidirect products (and the odd line)

If $A$ is an abelian group, we have
$Aut\left(K\left(A,n\right)\right)=Aut(A) \ltimes K\left(A,n\right),$
where the left hand side is the space of self-homotopy equivalences. Is there an easy way to ...

**4**

votes

**2**answers

244 views

### Gluing two 3 manifolds along their boundary

Let $X,Y$ be two compact, smooth, orientable 3 manifolds, each with an incompressible boundary component diffeomorphic to some genus $g $ surface $S_g$. Under an orientation-reversig diffeomorphism ...

**0**

votes

**0**answers

190 views

### triviality of vector bundles with the reduced homology of base space entirely torsion [migrated]

Let $\xi$ be an $n$-dimensional vector bundle over a manifold $M$ such that the reduced cohomology
$\tilde H^*(M;\mathbb{Z})$ is entirely torsion (every element has finite order under addition).
...

**0**

votes

**1**answer

127 views

### Equivalence relations of topological spaces not comparable with homotopy [closed]

The question is pretty much contained in the title:
What are examples of equivalence relations of topological spaces which are neither stronger nor weaker than homotopy equivalence?
Something that ...