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

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90 views

### Decomposable elements in cohomology ring [on hold]

I feel this must be well known, but I don't have enough knowledge on the matter. So, if there is a literature on this, then please provide a reference.
Let $E$ be a ring spectrum, and $X$ a pointed ...

**12**

votes

**1**answer

249 views

### Measuring the failure of pushforward to commute with Steenrod squares

Let $f \colon X \rightarrow Y$ be a map of topological spaces. Let's say that they are (closed) manifolds (not necessarily orientable), though to be honest I'm really interested in the more general ...

**7**

votes

**1**answer

697 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 ...

**25**

votes

**3**answers

1k views

### Examples for non-naturality of universal coefficients theorem

Does anyone have good examples of a space $X$ and a map $f: X \to X$ so that $f_*: H_*(X) \to H_*(X)$ is the identity but (e.g.) $f_*: H_*(X; \mathbb{F}_2) \to H_*(X; \mathbb{F}_2)$ is not the ...

**6**

votes

**2**answers

765 views

### A simple minded Poincare duality for orbifolds?

Suppose $X^n$ is an orientable compact orbifold (without boundary) with stabilisers in codimension 2, and $\bar X^n$ is the underlying topological space. We can assume, moreover, that $X^n$ is a ...

**1**

vote

**1**answer

176 views

### Free Symmetric Operads and $\mathbb{S}$-modules

In the definition of operads, if we restrict our attention to $\mathbb{S}$-modules where the action by the symmetric groups is free, then the free operads
have still an underling free ...

**13**

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**2**answers

234 views

### Schur multiplier of $Sp(2g, \mathbb{Z}/2)$ for $g \geq 3$

This question is about the computation of $H_2(Sp(2g, \mathbb{Z}/2), \mathbb{Z})$, where $Sp(2g, \mathbb{Z}/2)$ is the group of symplectic $2g \times 2g$ matrices over $\mathbb{Z}/2$.
With respect to ...

**0**

votes

**1**answer

368 views

### Can monodromy be described by the same matrix for chosen generators in case of the same singularity type?

Let $X$ be a surface in $\mathbb{P}^3$. We have a fibration $f: X \longrightarrow \mathbb{P}^1$, and $f^{-1}(s_1)$ and $f^{-1}(s_2)$ have the same singularity type. Let $\gamma_1$ and $\gamma_2$ be ...

**12**

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**10**answers

2k views

### Orbifold fundamental group in terms of loops?

In chapter 13 in his notes on 3-manifolds, Thurston defines the orbifold fundamental group to be the group of deck transformations of the universal cover of the orbifold. He also makes a statement ...

**9**

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**2**answers

303 views

### Homotopy groups of Moore spaces

Is there anything known about the homotopy groups of the Moore spaces $M(\mathbb Z_m,n)$ if $m\neq 2$ and $n \geq 2$?

**0**

votes

**0**answers

47 views

### How to compute a signature of n-dimensional torus [closed]

I can use cup product on cohomology, but what exactly means fundamental class?
Should I use Chern-Weil correspondence to compute its Pontrjagin class (actually what does Pontrjagin class for ...

**10**

votes

**2**answers

349 views

### Cohomology of $T^n/W$ for compact Lie groups

Let $G$ be a compact, connected and simply connected Lie group.
Let $T\subset G$ be a maximal torus and let $W$ be the corresponding
Weyl group.
Then we have the diagonal action of $W$ on $T^{n}$ ...

**13**

votes

**2**answers

477 views

### Integral cohomology of $G/N(T)$

Let $G$ be a compact connected simple Lie group, $T$ a maximal torus, $N(T)$ the normalizer of $T$, and $W=N(T)/T$ the Weyl group. It is well-known that $H^*(G/T,\mathbb{Q})$ is the regular ...

**56**

votes

**8**answers

5k views

### equivalence of Grothendieck-style versus Cech-style sheaf cohomology

Given a topological space $X$, we can define the sheaf cohomology of $X$ in
I. the Grothendieck style (as the right derived functor of the global sections functor $\Gamma(X,-)$)
or
II. the Čech ...

**12**

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**3**answers

487 views

### Nice things that can be proved easily with characteristic classes

A bit of context for this question: as a project for my master's degree my supervisor asked me to understand the construction of Milnor's exotic spheres. After learning the heavy material (I knew very ...

**8**

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**0**answers

85 views

### Are the unwound thin realization and fat realization homotopy equivalent?

This is a question about a theorem (proposition 2) in the article--On the homotopy type of classifying spaces
Recall some definitions first:
Given a category $\mathcal{C}$ internal in ...

**13**

votes

**1**answer

174 views

### Vector bundles with exactly one nonzero SW-class

I am interested in seeing examples of a space $X$ (preferably a closed smooth manifold, but any finite-dimensional CW-complex would also be of interest) with a vector bundle $\xi\colon E \to X$ on it, ...

**2**

votes

**1**answer

159 views

### The compatibility of the Gysin sequence with mixed Hodge structures

Let $X$ be a compact complex $n$-manifold and $D$ be a smooth comdimension $1$ submanifold. Also let $U:= X\setminus D$ and $j$ be the inclusion of $U$ in $X$.
Then it is well known that the ...

**19**

votes

**2**answers

839 views

### The homotopy category is not complete nor cocomplete

I understand that the homotopy category of (pointed) topological spaces and continuous maps is not complete. Nor is it cocomplete. In particular it neither has all pullbacks nor all pushouts.
What ...

**2**

votes

**2**answers

259 views

### Is it true that all sphere bundles are some double of disk bundle?

Let's consider a smooth sphere bundle over a smooth manifold with structure group is equal to the diffeomorphism group of sphere. Then, can we say that this is a double of some disk bundle? Thank you ...

**1**

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**1**answer

279 views

### Torsion of $H_{n-1}$

Suppose $X$ is a non-orientable manifold. Using Universal Coefficient Theorem (UCT) for homology, we can get that the torsion of $H_{n-1}$ is a cyclic group of order $2$. I am looking for a proof of ...

**12**

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**1**answer

308 views

### Characteristic classes for odd $K$-theory

There are different models of odd $K$-theory. In one case,
one takes the group $U=\lim\limits_{\longrightarrow}U(n)$ as classifying space. Similarly, if $\mathcal U$ denotes the unitary group of a ...

**10**

votes

**3**answers

490 views

### Maps with Hopf invariant zero are suspensions

Let $h:\pi_{2n-1}(S^n) \rightarrow \mathbb{Z}$ be the Hopf invariant. I believe that in the same paper that proves his suspension theorem, Freudenthal proved that if $x \in \pi_{2n-1}(S^n)$ satisfies ...

**17**

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**5**answers

880 views

### Computation of homotopy groups of spheres via Pontryagin-Thom

The Pontryagin-Thom construction identifies $\pi_{n+k}(S^n)$ with the group of bordism classes of framed $k$-dimensional submanifolds of $S^n$. Before Serre's work introduced algebraic tools into the ...

**1**

vote

**1**answer

257 views

### cohomology of the orbit space of a group action

Let $M$ be a manifold. Let a finite group $G$ act on $M$ discretely. Let $F$ be a field.
Suppose the induced action of $G$ on the cohomology algebra $H^*(M,F)$ is known. We want to obtain ...

**7**

votes

**4**answers

826 views

### What is the “right” definition of the homology(cohomology) of an orbifold?

What is the "right" analog in the orbifold case of a singular homology of a topological space?
We can not just take the homology of the underlying space, because it does not contain much information.
...

**1**

vote

**0**answers

83 views

### Mayer-Vietoris sequence for orbifolds

Is there a version of the Mayer-Vietoris long exact sequence for orbifolds? I am interested in orbifold homology as opposed to the homology of the underlying topological space.

**7**

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**1**answer

218 views

### Thorough reference on regular homotopy

I would like to learn this topic of algebraic topology but I cannot find a relevant reference to answer my basic questions on the subject (for example, is there a Hurewicz theorem for regular homotopy ...

**19**

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**2**answers

2k views

### First Chern class of a flat line bundle

A referee asked me to include a reference or proof for the following classical fact. It's not hard to prove, but I'd prefer to just give a reference -- does anyone know one?
Let $X$ be a nice space ...

**5**

votes

**1**answer

333 views

### Are two equivariant maps between aspherical topological spaces homotopic?

Let $f: X \rightarrow Y$ be continuous, X,Y pathwise connected and aspherical (i.e. trivial higher homotopy groups). Then $\pi_1(X)$ acts on the universal cover of $X$ via deck transformations, and on ...

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**0**answers

65 views

### Trefoil Knot Seifert Minimal Surface Equation

I am not very familiar with knot theory nor with minimal surfaces, so I already apologize if my question appears too naive or simple :). I am trying to do the following:
Starting from a real ...

**12**

votes

**2**answers

516 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 ...

**6**

votes

**2**answers

218 views

### Flat connections on 3-manifold with boundary

Suppose $Y$ is a 3-manifold and the boundary $\Sigma:=\partial Y$ is non-empty. Let $G$ be a Lie group with trivial center. Let $\overline {\mathcal A}_{flat}(\Sigma)$ and $\overline {\mathcal ...

**5**

votes

**3**answers

641 views

### Is there a “by hand” proof on the symmetry of the Atiyah class of $TX$?

Let X be a complex manifold and $TX$ its tangent bundle. The Atiyah class $\alpha(E)\in \text{Ext}^1(E\otimes TX, E)$ for a vector bundle $E$ is defined to be the obstruction of the global existence ...

**15**

votes

**1**answer

299 views

### Classification of fake (quaternionic, octonionic) projective spaces

If $X$ is a closed $n$-manifold, a fake $X$ is another closed manifold homotopy equivalent to $X$. There is some interest in classifying manifolds (up to, say, homeomorphism) homotopy equivalent to a ...

**1**

vote

**1**answer

177 views

### Deformation Quantization

I am a beginner and I want to learn about deformation quantization. Please suggest me with which book or notes, I should start?

**3**

votes

**1**answer

206 views

### Poincaré Duality for non-compact manifolds without Zorn's Lemma

Does exists a proof of the Poincaré Duality version for non-compact manifolds without using the Zorn's Lemma? I know that there is a proof using the Whitney embedding theorem, but I don't know this ...

**9**

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**0**answers

273 views

### Finite groups inside an infinite group with the same homology

Suppose we have a triple of groups $G,H,K$ verifying the follwing conditions
$G$ and $H$ are finite groups and $K$ an infinite group.
there exists two monomorphisms $G\rightarrow K\leftarrow H$ ...

**4**

votes

**2**answers

861 views

### Tensor product of spectral sequences?

I'm wondering about a cross product for spectral sequences. I've got an idea, and I wonder if it is written up anywhere, or if it even holds water.
Let's start with three spectral sequences, $E, F$ ...

**3**

votes

**2**answers

85 views

### Fixed point set for a subcircle of torus actions

Let $T=S^{1}\times S^{1}\times ...\times S^{1}$ ($n$ times) be $n$
dimensional torus and $X$ be a $T$-space.
Lemma: If $X$ has finitely many connected orbit type, then there is a
subcircle ...

**16**

votes

**1**answer

418 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 ...

**0**

votes

**1**answer

143 views

### Atiyah-sequence-like definition of connection on vector bundles?

For a principal bundle $\pi: P \to M$ we have the following Atiyah sequence that can be used to define a connection on it
\begin{equation}
0 \to V{P} \to T{P} \to \pi^*{T{M}} \to 0
\end{equation}
A ...

**0**

votes

**0**answers

154 views

### Every vector bundle can be induced from a principal bundle? its frame bundle? [migrated]

If it is a theorem could somebody tell me the name? If it is wrong could somebody give a counterexample to illustrate what the obstruction is?
I am wondering this because in Clifford Taubes' book on ...

**10**

votes

**2**answers

465 views

### Reference request: Goodwillie tower of the identity

The Taylor (Goodwillie) tower of the identity functor on based spaces has as its $j$-th layer the infinite loop space-valued functor
$$
X\mapsto \Omega^\infty (W_j \wedge_{h\Sigma_j} X^{[j]})
$$
in ...

**0**

votes

**0**answers

90 views

### de Rham type cohomology for covariant derivative?

We know that in general for a covariant derivative $D$ on a vector bundle $\xi$ over $M$ we don't have $D \circ D = 0$. This prevents us from having the following cochain complex
\begin{equation}
...

**15**

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**0**answers

259 views

### Groups whose finite index subgroups of fixed index are isomorphic

I am interested in finitely generated groups $G$ that are residually finite and have the following property: For each $d \geq 1$, $G$ has subgroups of finite index $d$, and all such subgroups are ...

**13**

votes

**1**answer

467 views

### $p$-adic Bott periodicity?

The Bott periodicity theorem can be formulated as the existence of homotopy equivalences $\Omega^2(KU)\equiv KU$ and $\Omega^8(KO)=KO$. I always wondered whether this theorem could also be transferred ...

**29**

votes

**4**answers

3k views

### Construction of the Stiefel-Whitney and Chern Classes

I've seen two constructions of these characteristic classes. The first comes from Milnor and Stasheff's book and involves the Thom isomorphism and (at least for me) the rather mysterious Steenrod ...

**19**

votes

**4**answers

1k views

### Natural setting for characteristic classes?

In my mind, algebraic topology is comprised of two components:
Chain complex information, which is intrinsic information concerning how your object may be built up out of simple "lego blocks".
...

**8**

votes

**2**answers

342 views

### Simplest explicit counterexample for $Vect(BG) \ne Rep(G)$ as monoids

Let $G$ be a topological group, $Vect(BG)$ the monoid of complex vector bundles over its classifying space (not the stack!) and $Rep(G)$ its monoid of complex representations.
Generally $Vect(BG) \ne ...