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

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39 views

### Non-flat fibration - 1. fibres still homotopic? 2. references/examples?

I stumbled over fibrations $\pi: E\rightarrow B$ that are not flat, i.e. where the dimension of the fibre $\pi^{-1}(b)$ jumps over certain points $b \in B$. Are these still 'ordinary' fibrations in ...

**4**

votes

**2**answers

160 views

### Charts needed for an atlas

I just read this question link and asked myself, if there is any easy way to decide how many charts you actually need to cover a given compact manifold in $\mathbb{R}^3$, maybe at least in this ...

**2**

votes

**1**answer

124 views

### Example s.t. the unbased loop-space is not $\Omega X \times X$

For a connected pointed CW-complex $X$, let us write (as usual) $\Omega X$ for the space of based loops at $X$. I am looking for an example where the space $\Omega' X$ of all (unbased) loops in $X$ is ...

**18**

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**4**answers

3k views

### Integrals from a non-analytic point of view

I've mentioned before that I'm using this forum to expand my knowledge on things I know very little about. I've learnt integrals like everyone else: there is the Riemann integral, then the Lebesgue ...

**-2**

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**4**answers

315 views

### Studying topology: which first, algebraic or differential? [on hold]

I have recently studying the basics of topology (ideas in point set, connectedness compactness) and I want to continue my studies but i'm interested in both differential and algebraic topology. which ...

**1**

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**0**answers

96 views

### manifold branched covering space for orbifolds

An orbifold structure on some topological space $X$ is a covering of $X$ with local quotient charts $V/G$, where $V$ is some connected manifold and $G$ effectively acts on $V$ via a finite group of ...

**3**

votes

**1**answer

63 views

### Polygons with centroid at origin and vertices on compact codimension one submanifold of $\mathbb{R}^{n}-\{0\}$

Let $M$ be a compact codimension one submanifold of $\mathbb{R}^{n}$ which does not contaion $0$ and the origin lies in the bounded component of$\mathbb{R}^{n}-\{0\}$.
Is it true to say that:
...

**13**

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**1**answer

239 views

### What are explicit obstructions to realizability of formal group laws as complex-oriented ring spectra?

Recall that a complex-oriented spectrum is a ring spectrum E with a map $MU \to E$.
Analogously, a ring with a (1-d commutative) formal group law is (represented by) a ring $R$ with a map $L \to R$ ...

**3**

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**1**answer

143 views

### Parallelizable nearly-Kahler manifolds

In this question, we have discussed how the following bundle:
$E_{d} = TS^{d}\oplus \Lambda^2 T^{\ast}S^{d}$
is always trivial, where $S^{d}$ is the $d$-dimensional standard sphere. Now, let us take ...

**23**

votes

**4**answers

2k views

### What is a simplicial commutative ring from the point of view of homotopy theory?

Let $k$ be a field. There are two natural categories to consider:
The category of simplicial commutative $k$-algebras.
The category of connective $E_\infty$ $k$-algebras (i.e., chain complexes of ...

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**0**answers

134 views

### Fiber bundle in smooth category and topological category

Let $M$ be a smooth manifold and $G$ be a Lie group. Denote by $Bun(M,G)$ the set of all equivalent smooth Principal bundles on $M$ with structural group $G$ in smooth category. And denote by ...

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89 views

### TQFT characterization of braiding statistics

In the TQFT language, quasiparticles correspond to Wilson loop operators. It is well-known that quasiparticles can have non-trivial braiding statistics.
Take $2+1$ dimensional Abelian Chern-Simons ...

**3**

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**0**answers

132 views

### Using $\mathcal{U(H)}$ as a model for $EG$ and working with the Fredholm Operators

Let $\mathcal{H}$ be a unitary universe for some group $G$. As $\mathcal{H}$ is a faithful representation the representation map is an injection $G \to \mathcal{U(H)}$, so there's a free $G$ action on ...

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votes

**3**answers

279 views

### A conjecture about parallelizable generalized spheres

Let $S^{d}$ denote the standard $d$-dimensional sphere. I heard from a physicist that from physical arguments they have been able to show that the vector bundle:
$E_{d} = TS^{d}\oplus \Lambda ...

**11**

votes

**1**answer

393 views

### Condition on a Hopf operad for tensor product in the base category to be a (categorical) coproduct for algebras

A Hopf operad will be an operad endowed with a coproduct $P(n) \longrightarrow P(n) \otimes P(n)$ which is compatible in the obvious sens with operad laws (no more structure is assumed a priori. ...

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votes

**1**answer

422 views

### How do small changes in a filtered complex affect the associated spectral sequence

I have recently been learning about spectral sequences, not in the context of any problem, but mostly out of curiosity. There are two questions that have occurred to me which I've been unable to ...

**2**

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**1**answer

70 views

### Generators of the colored braid group (two colors), reference

I consider the group $B_{n,n}$, the braids, colored in two colors, say all odd strings are black and all even strings are white.
It is easy to find a set of generators for $B_{n,n}$:
$$
\begin{cases}
...

**5**

votes

**2**answers

278 views

### Interpretation of the monomorphism $H^2(\pi_1(X),\mathbb{Z}) \rightarrow H^2(X,\mathbb{Z})$

Let $X$ be a nice topological space and denote by $\pi_1(X)$ its fundamental group.
It is well-known that there is a well-defined map
$$
0 \rightarrow H^2(\pi_1(X),A) \rightarrow H^2(X,A),$$
where ...

**-5**

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**0**answers

58 views

### vector bundle and characteristic classes [closed]

why every orient able line bundle is trivial? I give an orientation s(x) to each fib re such that s(x).s(x)=1 using the euclidean metric but I can not understand why this section becomes continuous

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**1**answer

105 views

### Coverings/Cech cohomology of totally disconnected spaces

For any topological space $X$ we have a natural functor
$\text{Cov}_X \rightarrow \text{Fun}(\pi_1(X),\text{Set})$
from the category of coverings of $X$ to the category of functors $\pi_1(X) ...

**5**

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**0**answers

129 views

### Is there a more general obstruction to the existence of moduli spaces than the existence of automorphisms?

We are taught that, in general:
A type of objects that has nontrivial automorphisms cannot have a fine moduli space.
The proof generally goes along the lines of:
Take an object $X$ with a ...

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votes

**1**answer

177 views

### Given a map of classifying spaces, can the target be described as a groupoid quotient of the source mod some action of some (co)kernel?

Let $H \to G$ be a homomorphism of affine algebraic groups (over characteristic $0$, if it matters). The case I care most about is when $H \to G$ is an inclusion. There is a corresponding map $f: ...

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votes

**2**answers

268 views

### Maps to the group completion

Let $M$ be an H-space, topological monoid (homotopy-commutative if necessary):
What does the group comletion $\Omega BM$ represent in homotopy category? Is $[X,\Omega B M]$ always equal to the ...

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votes

**1**answer

208 views

### is this map a closed inclusion?

I apologize in advance if this question is too technical. I haven't found a reference in the literature yet, and it seems difficult enough that perhaps it has not been answered.
Let $A$, $B$, and $C$ ...

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votes

**1**answer

165 views

### Difference between coherent nerve of simplical model category and simplicial category

Suppose I have a simplicial model category $M$. Then I can take the homotopy coherent nerve of $M$ to obtain a quasicategory. This, however, only depends on the fact that $M$ is a category enriched in ...

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votes

**2**answers

3k views

### Group completion theorem

Let $M$ be a topological monoid. How does the homology-formulation of the group completion theorem, namely (see McDuff, Segal: Homology Fibrations and the "Group-Completion" Theorem)
If $\pi_0$ is ...

**8**

votes

**1**answer

243 views

### When is a quasicategory over $N(\Delta)^{op}$ a planar $\infty$-operad?

In Lurie's DAG II, a notion of monoidal $\infty$-category is given that differs from the notion given in his later book Higher Algebra. In the former, the relevant structure is a cocartesian ...

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114 views

### Does the diffeomorphism group of an open manifold act naturally on the compactification of the manifold? [closed]

I need help regarding this question please:
Does the diffeomorphism group of an open manifold act trnsitively/freely on the compactification of the manifold?
any recommended references!

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75 views

### Global dimension of graded Lie algebra

The rational global dimension of a graded algebra $A=(A_k)_{k\geq 0}$, with $A_0=\mathbb Q$, denoted here ${\rm gl}\dim A$ is defined to be the greatest integer $k$ (or $\infty$) such that ${\rm ...

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votes

**1**answer

258 views

### Cohomology of the Image of J spectrum

Let $J$ denote the image of $J$-homomorphism spectrum and let $j$ denote its connective cover. I am interested in knowing the cohomology of $j$ i.e.
$$ [j, HZ/p]_*$$
as a module over Steenrod algebra. ...

**33**

votes

**9**answers

4k views

### understanding Steenrod squares

There is a function on $\mathbb{Z}/2\mathbb{Z}$-cohomology called Steenrod squaring: $Sq^i:H^k(X,\mathbb{Z}/2\mathbb{Z}) \to H^{k+i}(X,\mathbb{Z}/2\mathbb{Z})$. (Coefficient group suppressed from ...

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votes

**1**answer

457 views

### Non-unique splittings of homotopy idempotents

By a homotopy idempotent I mean a map $f:X\to X$, where $X$ is a space, equipped with a homotopy $f\circ f \sim f$. In contrast to the situation in stable homotopy theory (where $X$ would be a ...

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votes

**0**answers

76 views

### A suitable (transfer) map of Thom spectra $BT(n)^{-ad_{O(n)}|_{T(n)}-\gamma_1^{\times n}}\to BT(n)^{-\rho_n}$

For a Lie group $G$, write $ad_G=EG\times_G g\to BG$ for the adjoint bundle, $g$ is the Lie algebra of $G$ on which $G$ acts through its adjoint representation. Let $T(n)=O(1)^{\times n}$. I am ...

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votes

**1**answer

217 views

### What is an example of a formal group law that is Landweber-exact but not flat?

Quick Background: The $p$-series of $F$ (where $F$ is a formal group law over a graded ring $R$) will be of the form $[p](x) = px + v_1x^{p^1} + ... + v_nx^{p^n} + ...$ ; $(F, R)$ is Landweber-exact ...

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**0**answers

101 views

### Are there any known ``topological" invariants for branched coverings?

My question is the following: let $f:\Omega\to \mathbb{R}^n$ be a branched covering, namely $f$ is continuous, discrete (each fiber is a discrete subset of $\Omega$) and open (open sets get mapped ...

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votes

**1**answer

250 views

### A lower-dimensional algebraic topology problem between homology group and fundamental group

Let
\begin{equation}
A\stackrel{\alpha}{\longrightarrow}B\stackrel{\beta}{\longrightarrow}C\quad\quad (1)
\end{equation}
be a short sequence of (abelian or nonabelian) groups and homomorphisms. We say ...

**1**

vote

**1**answer

116 views

### Chern classes of three (two) dimensional complex vector bundles

Let $M$ be a manifold.
Let $F(M,3)=\{(m_1,m_2,m_3)\mid m_1, m_2, m_3\in M, m_i\neq m_j, \text{ for any } i\neq j\}$.
Let $S_3$ be the symmetric group of order $3$.
Let $S_3$ act on $F(M,3)$ by ...

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votes

**0**answers

196 views

### Reference for a proof of the fiberwise Stokes theorem

The fiberwise Stokes theorem says that given a differential form on a smooth fiber bundle whose fibers have boundary,
the difference between the fiberwise integral of the differential and the ...

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votes

**0**answers

72 views

### Reference request: linearly independent cycles in a manifold

The following seems to be well known to experts, but I would be happy if there is a paper or textbook that I can cite.
Note: all of the manifolds are assumed to be without boundary.
Suppose that $C$ ...

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votes

**0**answers

43 views

### Product structure on manifolds via lifting classifying maps

Let's say you want to study $d$-dimensional manifolds $M$ which decompose functorially into $M\cong N\times P$ for a fixed $P$. Can this structure be expressed by a lift of the stable normal bundle?
...

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votes

**1**answer

151 views

### How to write the Thom spectrum representing cobordism as an $\Omega$-spectrum?

It is often said [e.g. Atiyah, "Bordism and Cobordism" (1961)] that the Thom spectrum $MSO(i)$ represents oriented cobordism, in the following sense:
\begin{eqnarray}
MSO^n(X,Y) &:=& \lim_{i ...

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votes

**0**answers

134 views

### Finitness of the connected components of a stack

Let $X$ be an algebaic stack over a scheme $S$, for any $S$-scheme $Y$ we can consider the groupoid $X(Y)$ of $Y$-points. Denote by $\pi_0(X(Y))$ the set of isomorphism classes of the groupoid.
Are ...

**22**

votes

**9**answers

2k views

### What are some interesting problems in the intersection of Algebraic Number Theory and Algebraic Topology?

I'm a beginning graduate student and while my background is primarily in algebraic number theory, I've found myself a bit smitten with the subject of algebraic topology recently after only having read ...

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**1**answer

152 views

### Reference Request: Grouplike Algebras over the little $n$-cubes operad are $n$-fold loop spaces

In Geometry of the iterated loop space, Peter May proved his famous recognition theorem, which is, in a simple form, stated on page 3 as the following.
There exist $\Sigma$-free operads ...

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vote

**1**answer

340 views

### Existence of $n$-connected topological groups with $m$-dimensional action extending that of $GL(m)$ on $\mathbb{R}^m$

I'll first state the question as concisely as I can and then provide some motivation.
Consider two positive integers $m$ and $n$ such that $m < n+2$. Does there exist a topological group $G$ ...

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votes

**1**answer

942 views

### Classical and Quantum Chern-Simons Theory

Please excuse a sloppy question from an old user who hasn't been here in a long time. I think the expertise here is such that it can be answered anyway.
Let $\Sigma$ be a two-manifold and $M$ a ...

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vote

**1**answer

156 views

### Homotopy type of certain maps on complex grassmanian

$G(k,n)$ is the complex grassmanian which is homeomorphic to the space of projections in $M_{n}(\mathbb{C})$ with trace $k$. So we can Identify $G(k,n)$ with $$\{A\in M_{n}(\mathbb{C})\mid ...

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111 views

### Pro-p topology on free group

Let $H$ be a finitely generated subgroup of the free group $F(A)$ and $G_P$ the pseudovariety of all finite $p$-group with $p$ fixed prime number. We endow $F(A)$ with the pro-$G_p$ topology. Suppose ...

**1**

vote

**0**answers

84 views

### Explicit calculation of G-CW(V) structure of a G-space

I know explicitly the $Z/6$-CW($ξ^2$)-complex structure of $D(ξ^2)$, where $ξ$ is the non-trivial irreducible representation of $Z/6$ without fixed points. I am looking for an explicit calculation of ...

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votes

**1**answer

280 views

### semi flat connections

Let $L\to V$ be complex line bundle and $F_{t}:V\to V$, $t\in [0,1]$, be a loop of diffeomorphisms, $F_0=F_1=$ identity.
For every $x\in V$, we get a loop $\gamma_x(t)=\{F_t(x)\}$ whose class in ...