# Tagged Questions

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

**5**

votes

**2**answers

174 views

### What is this construction using iterated face maps of semisimplicial sets?

Let $X$ be a semisimplicial set (face maps but no degeneracy maps). Fix a positive integer $k$. Let $Y_n$ be $X_{(n+1)k}$ and then define $\partial^Y_i:Y_n\to Y_{n-1}$ by
$$\partial^Y_i = ...

**4**

votes

**1**answer

165 views

### Localization at the Johnson-Wilson spectrum and rationalization

Is there a clean proof that the $L_n$, localization at $E(n)$, is simply rationalization (i.e. $L_0$) on Eilenberg-MacLane spectra? Eric Peterson asked this here, but I haven't seen an answer.

**0**

votes

**1**answer

188 views

### fiber, homotopy fiber of spaces

Suppose we have a pullback of topological spaces (CW-complexes) $B\rightarrow A\leftarrow C$ which I will denote by $D$.
Assumptions
The induced map $D\rightarrow C$ is a trivial fibration
The map ...

**3**

votes

**0**answers

126 views

### What's the story with the Hopf fibration and the Jacobi identity?

I like the Hopf fibration of the 3-sphere $S^3$ enough that I found a nice way to make a physical model of it. All you need is to combine a bunch of key rings in such a way that (ii) every pair of ...

**7**

votes

**1**answer

266 views

### What are some examples of total derived functors that can't be computed from a functorial replacement?

(Or more generally, what are some examples of Kan extensions which are not pointwise?)
By a total derived functor of a functor $F: C \to D$, I simply mean a (left or right) Kan extension of $F$ along ...

**1**

vote

**0**answers

223 views

### Generalization for Leray Hirsch theorem for Principal $G$-bundle [closed]

This is a general question:
Is there a generalized Leray Hirsch theorem for Principal $G$-bundle? with $G$ finite group with discrete topology. I know it does not make sense to compare with original ...

**2**

votes

**1**answer

106 views

### positions of regular cubes in Euclidean space with all its vertices without distinction

Let $P$ be the standard regular cube, centered at the origin of $\mathbb{R}^3$ with all its vertices on the unit sphere.
If the vertices are labelled by $1,2,3,4,5,6,7,8$, then the collection of ...

**5**

votes

**1**answer

181 views

### Existence of a non-null-homotopic simple closed curve

Assume that $X$ is a path-wise connected Hausdorff space, and assume that its fundamental group is non-trivial. Does it always exist a simple curve in $X$ which is non-null-homotopic?
Such curve does ...

**8**

votes

**1**answer

200 views

### references for Stiefel-Whitney class of Stiefel manifolds and Grassmannians

Let $M$ be a manifold. The total Stiefel-Whitney class of $M$ is defined to be the Stiefel-Whitney class of the tangent bundle $TM$
$$
w(M)=1+w_1(TM)+w_2(TM)+\cdots
$$
I want to find references for
$$
...

**8**

votes

**0**answers

243 views

### Descent theorems for fundamental groups and groupoids?

Grothendieck in his 1984 "Esquisse d'un programme" (Section 2) wrote (English translation):
" ..,people still obstinately persist, when calculating with fundamental groups, in fixing a single base ...

**3**

votes

**1**answer

209 views

### Polynomial differential forms on $BG$

Let $\Omega^{*}_{\text{poly}}\: : \: sSet\to dg_{\geq 0}Comm_{+}$ be the polynomial De Rahm functor on simplicial sets, where the codomain is the category of commutative differential graded algebras ...

**3**

votes

**1**answer

131 views

### unwinding the definition of $H_i(KU)$ as a map of spectra $\mathbb{S}^i \to HZ \wedge KU$

I asked this on mathstackexchange but didn't get any response (or many views) so I'm asking it here, although clearly it belongs over there.
In the answer to this question on mathoverflow, it says:
...

**4**

votes

**2**answers

247 views

### triviality of Whitney sums of a vector bundle

Let a $3$-dimensional subspace $V$ of $\mathbb{R}^4$ be $$V=\{(x_1,x_2,x_3,x_4)\in\mathbb{R}^4\mid\sum_{i=1}^4x_i=0\}.$$ The alternating group $A_4$ acts on $V$ by
...

**5**

votes

**1**answer

317 views

### What is the algebraic fundamental groups of $SO(n)$ and $Sp(2n)$?

Let $k$ be an algebraically closed field of characteristic zero. and let $$\sigma: SL_n(k)\rightarrow SL_n(k)$$
be an involution.
My questions are:
How could one calculate the fundamental group of ...

**2**

votes

**0**answers

71 views

### good choice of extension of equivariant map

Let $K$ be a compact Lie group. Let $C_K$ denote the category whose objects are the compact lie groups containing $K$ and whose morphisms are inclusion of the groups. Let $Y$ be a $K-$space such that ...

**4**

votes

**1**answer

111 views

### Is the signature of inverse images of diffeomorphic submanifolds (along a homotopy equivalence) the same?

Suppose it is given an orientation preserving homotopy equivalence $h:N→M$ between closed oriented connected manifolds. Let $X,Y\subset M$ be diffeomorphic submanifolds, and assume $h$ to be ...

**7**

votes

**1**answer

188 views

### Operads and the Stable Module Category

I am seeking references to places where operads and their algebras have been studied for the stable module category. Colored operads are fine too.
Let $k$ be a field and $R$ a $k$-algebra. The stable ...

**6**

votes

**1**answer

240 views

### vector bundles associated to a covering space

Let $M$ be a $m$-dimensional manifold whose cohomology ring and cell structure are well-understood, such that there is a free action of the symmetric group $S_n$ on $M$. Then we have a $n!$-sheeted ...

**1**

vote

**1**answer

92 views

### Cancellation of 2-component links

Consider a two-component tame link in 3-space, consisting of an arc from $(-1,1,0)$ to $(1,1,0)$ and an arc from $(-1,-1,0)$ to $(1,-1,0)$, confined to the slab $-1 \leq x \leq 1$. Call such a link ...

**1**

vote

**1**answer

169 views

### How to extend an equivariant map from a compact Lie group

Let $G$ be a compact Lie group and let $H$ be a closed subgroup of it. Let $g$ be a torsion element of $G$ and $C_G(g)$ the centralizer of it. Let $Y$ be a $C_G(g)-$space. I'm working on the space ...

**6**

votes

**1**answer

187 views

### Fundamental class in $KO[1/2]$

Let $M^m$ be an oriented Riemannian manifold.
The signature operator associates to $M$ a class $\Delta_M\in KO_m(M)[1/2]$.
I have two questions about this class $\Delta_M$:
Rationally, $\Delta_M$ is ...

**18**

votes

**4**answers

2k views

### How to get convinced that there are a lot of 3-manifolds?

My question is rather philosophical : without using advanced tools as Perlman-Thurston's geometrisation, how can we get convinced that the class of closed oriented $3$-manifolds is large and that ...

**6**

votes

**4**answers

886 views

### Topological structure of SO(n) as a product

I’m interested in the question for which $n$ the special orthogonal group is homeomorphic to the product
$$ \mathrm{SO}(n) \approx S^{n-1} \times \mathrm{SO}(n-1). $$
Allen Hatcher [1, p. 293 f.] ...

**4**

votes

**0**answers

85 views

### characteristic classes of a covering space with symmetric group action

Let $S_n$ be the $n$-th symmetric group. Suppose we have a $n!$-sheeted covering space
$$
S_n\to M\to M/S_n
$$
where $M$ is a manifold. Let $\mathbb{K}$ be the real numbers $\mathbb{R}$, complex ...

**8**

votes

**1**answer

232 views

### Why does strong convergence of the EMSS imply that Tot commutes with suspension spectrum?

Given a fiber square of simplicial sets
$$\begin{array}{cc}
& \hspace{-7mm} E \\
&\hspace{-7mm}\downarrow \\
\ast\longrightarrow &\hspace{-7mm} B
\end{array}$$
...

**7**

votes

**2**answers

342 views

### first Chern class of complex vector bundles and first Pontrjagin class of quaternionic vector bundles

Let $\xi$ be a (real) vector bundle of dimension $n$. Then the first Stiefel-Whitney class
$$
w_1(\xi)=0
$$
if and only if $\xi$ is orientable, i.e. the structure group of $\xi$ can be reduced to ...

**8**

votes

**0**answers

99 views

### Overview and/or reference of theory of pro-universal covers?

This question will contain very little in the way of concrete information, because I don't have much to go on. I've heard whispers of something called a "pro-universal cover," which is the inverse ...

**3**

votes

**1**answer

481 views

### What is the Euler characteristic of a mapping space?

Suppose that $A$ and $B$ are topological spaces homotopy equivalent to finite cell complexes, and let $B^A = \mathrm{maps}(A,B)$ denote the space of maps from $A$ to $B$. Is it there a formula for ...

**3**

votes

**2**answers

269 views

### mod p cohomology ring of alternating groups

Let $A_n$ be the alternating group of $\{1,2,\cdots,n\}$.
(1). What is the cohomology ring
$$
H^*(A_4;\mathbb{Z}/3)
$$
and its Steenrod operation $P^i$'s?
(2). Are there general results about the ...

**11**

votes

**1**answer

182 views

### Equivariant Fredholm operators classify equivariant K-theory

Let $\mathcal{F}$ be the space of Fredholm operators on a separable Hilbert space $H$ with the topology induced by the operator norm.
If $X$ is compact,
Atiyah-Jänich proved that
...

**4**

votes

**0**answers

107 views

### Exotic 2-adic lifts of mod $2$ Steinberg idempotent

Denote $B_n$ the Borel subgroup of $Gl_n(Z/2)$, i.e., the subgroup of
upper triangular matrices, $\Sigma _n$ the subgroup of permutation matrices.
The (conjugate) Steinberg idempotent is defined to be ...

**3**

votes

**1**answer

193 views

### Geometric interpretation of the conjugation operation in the dual Steenrod algebra

As the dual mod 2 Steenrod algebra, $A$, is a Hopf algebra, it has the conjugation operation, $\chi:A\to A$. Milnor also gives a formula for this.
I wonder if there is any source telling about a ...

**3**

votes

**1**answer

120 views

### What's the topology on the mapping space $Map_H(G, Y)$ when $G$ is not finite

When $G$ is a finite group and $H$ a closed subgroup of it, the sets of right cosets $H\backslash G$ has the discrete topology on it. Let $Y$ be a $H-$space. We have the $G-$homeomorphism ...

**10**

votes

**1**answer

480 views

### positions of a methane molecule with carbon atom at the origin

Let $\text{CH}_4$ be the molecule of Methane:
The four hydrogen atoms form vertices of a regular tetrahedron with the carbon atom in the center of the regular tetrahedron.
Here we regard all atoms ...

**5**

votes

**1**answer

479 views

### Intuition behind salient numbers in number of h-cobordism classes of smooth homotopy n-spheres

The Wikipedia article on Exotic Sphere displays this sequence of numbers (see also OEIS A001676 and the Milnor link therein) for the order of the classses as
$$1, \;1, \;1,\; 1,\; 1, \;1, \;28,\; ...

**11**

votes

**2**answers

594 views

### Does every (co)homology functor (in particular, stable homotopy) factor through chain complexes?

Ordinary homology and cohomology factor through chain complexes via singular homology and cohomology. What about other (co)homology theories?
That is, for each spectrum $E$, do we have a lift in the ...

**10**

votes

**0**answers

180 views

### Mapping class groups in high dimension

Let $M$ be a $1$-connected, closed, smooth manifold with $dim(M)>4$ and let us set $MCG(M)=\pi_0(Diff(M))$. Dennis Sullivan proved that $MCG(M)$ is commensurable to an arithmetic group.
I was ...

**11**

votes

**1**answer

250 views

### Reference request: sheaves on the site of d-manifolds

I believe I know how to prove the following results. I also know to whom to cite fancy-shmancy results that have these as a very special case. My question is: what are the correct citations for ...

**10**

votes

**0**answers

144 views

### How does the HHR Norm functor interact with the cotensor over $G$-spaces?

Let $N_H^G$ be the norm functor from orthogonal $H$-spectra to orthogonal $G$-spectra. We know the category of orthogonal $G$-spectra $\mathcal{S}_G$ is enriched over the category of based $G$-spaces ...

**10**

votes

**3**answers

785 views

### Is there a notion of “flat vector bundle over a topological space”?

I am reading this paper and at the top of page 5 the author makes reference to categories consisting of flat complex vector bundles over $X$ where $X$ is an arbitrary topological space. However, the ...

**5**

votes

**0**answers

181 views

### Is $D(n)$ a Thom spectrum?

Fix $p=2$ and let $D(n)$ be a spectrum which filters the Eilenberg-Moore spectrum $H\mathbb{Z}/2$, i.e. $\mathrm{colim}\ D(n)=H\mathbb{Z}/2$. This spectrum can be considered as the cofibre of ...

**7**

votes

**2**answers

599 views

### Atiyah Singer index theorem and Hodge de Rham operator

When I read about Atiyah Singer index theorem I met the following example: let $M$ is (orientable closed smooth) Riemannian manifold and consider Hodge-de Rham Dirac operator defined by $d+d^*$ ...

**11**

votes

**2**answers

306 views

### Gerstenhaber conjecture for free loop space

I- Is the following statement still a conjecture see this article ?
Conjecture (?)
Let $M$ be a simply connected compact oriented $d$-manifold (smooth), then $HH^{\ast}(C^{\ast}(M))$ the Hochschild ...

**9**

votes

**1**answer

176 views

### Inducing up the group homomorphism between mapping class groups

There are many ways to embed the braid group into the mapping class group of a surface. To describe one of them, let ${C}_{2g+2}(\mathbb{D}^2)$ be the configuration of unordered $2g+2$ points in the ...

**3**

votes

**1**answer

134 views

### very basic question on p-typical group law and MU and BP

I am a beginner trying to understand Dan Quillen's idempotent map $\xi:MU_{(p)} \to MU_{(p)}$ described in his classic paper 5-page paper On formal group laws...
Let $F$ be a formal group law over a ...

**3**

votes

**0**answers

162 views

### What is the formula for the homology class represented by the diagonal?

Let $M$ be a compact oriented manifold and $\{\mu_{i}\}$ a basis
for the homology $H_*(M, \mathbb{Z})$ (we are ignoring any torsion).
Now consider the diagonal $\Delta_{M}$ inside $M\times M$. ...

**7**

votes

**0**answers

170 views

### Is there a definition of an $\infty$-groupoid in HoTT whose terms are $n$-manifolds and whose higher morphisms are diffeomorphisms/isotopies/etc?

Suppose you want to work with TQFTs in homotopy type theory (HoTT). Working with $(\infty,n)$-categories, or even $(\infty,1)$-categories, is something that I gather is too difficult for HoTT at the ...

**7**

votes

**0**answers

193 views

### Mapping class group action on fundamental group of punctured elliptic curves

Let $(\mathcal{M}_{1,1})_{\overline{\mathbb{Q}}}$ be the moduli stack of elliptic curves over $\overline{\mathbb{Q}}$. By Oda, we know that its etale fundamental group is $\widehat{SL_2(\mathbb{Z})}$.
...

**3**

votes

**2**answers

356 views

### Cup product of cohomology in a Serre spectral sequence

How to use Serre spectral sequence to compute cup product structures?
Let $F\to E\to B$ be a fibration. Suppose all the differentials of the corresponding Serre spectral sequence of cohomology are ...

**8**

votes

**1**answer

248 views

### Stiefel-Whitney class of fibre bundles

Let $F,E,B$ be manifolds (we may assume them to be compact, without boundary if necessary) and $$F\to E\to B$$ be a fibre bundle. Suppose $$w(F), w(B),$$ the Stiefel-Whitney class of $F$ and $B$, are ...