16
votes

Accepted

### What is the relationship between spectral sequences and obstruction theory?

This is a partial answer, but every obstruction theory (in some precise sense) provides you with a spectral sequence (in fact several). Let me clarify what do I mean with obstruction theory. All this ...

16
votes

### Topological obstruction for the existence of spin$^c$ structure

The obstruction to spin$^c$ is often denoted by $W_3(M) \in H^3(M,\mathbb{Z})$. It is obtained as follows: Let
$$
\beta \colon H^2(M,\mathbb{Z}/2\mathbb{Z}) \to H^3(M,\mathbb{Z})
$$
be the Bockstein ...

14
votes

Accepted

### Possible mistake in Cohen notes "Immersions of manifolds and homotopy theory" (version 27 March 2022)

The statement is false in two ways. First, two immersions might not even be homotopic even though their normal bundles are both, say, trivial. Second, even if $M=\mathbb R^m$, regular homotopy classes ...

12
votes

Accepted

### Are open orientable 3-manifolds parallelizable via obstruction theory?

I think you are saying: for a closed $3$-manifold, vanishing $w_1$ implies vanishing $w_2$ by Wu's relations, but is this still true if the manifold is not closed? The answer is yes.
For a compact ...

11
votes

Accepted

### Finite domination and compact ENRs

It was originally conjectured by Borsuk that every compact ANR should be homotopy equivalent to a finite CW complex. While it was known that every separable ANR has the homotopy type of a countable ...

11
votes

Accepted

### Obstructions for the lifting problem after a pull-back

First, note that since you are assuming $F$ is $d-1$-connected, the primary obstruction lies in $H^{d+1}$, not $H^d$.
Now, consider the diagram $\require{AMScd}$
\begin{CD}
& & & & S^...

10
votes

### Classification of bundles, Postnikov towers, obstruction theory, local coefficients

I'll try to answer question 1. Unfortunately, I know of no especially convenient reference for the case of understanding general Postnikov towers; however, everything I say below is "well known".
...

10
votes

Accepted

### Is the Gödel universe Wick rotatable?

$\DeclareMathOperator\SL{SL}$Clearly, as Robert Bryant indicates, it is Wick-rotatable to a different Lorentzian space. However, it is also Wick-rotatable to a Riemannian space, albeit negative ...

9
votes

### Is the Gödel universe Wick rotatable?

I may be misreading the sources that you list for the definition of Wick-rotatable, but, I believe that the following construction does fit that definition: According to the Wikipedia page that the ...

9
votes

Accepted

### Obstruction to homotopy, cohomology operations and Dold-Whitney theorem

Let me explain why the condition is not the triviality of the difference cocycle. Maybe an important point to note is that the condition with the difference cocycle is not about the vanishing of an ...

6
votes

Accepted

### Non-triviality of a Postnikov class in $H^3\left(B \operatorname{PSU}(N) ; \mathbb{Z}_q\right)$

Consider the map of short exact sequences
$$\begin{array}{ccccccccc} 0 & \rightarrow & \mathbb{Z} & \xrightarrow{\cdot N} & \mathbb{Z} & \xrightarrow{mod N} & \mathbb{Z}_N &...

6
votes

### Multiplicative structures on truncated Moore spectra

EDIT:
In case you missed it, there's been a huge breakthrough in this area by Robert Burklund.
I believe that Prasit Bhattacharya's methods, as linked to in the question, can be used to show that for ...

5
votes

Accepted

### Framed version of the "copants bordism"?

The key point is the identification $\tau(S^n)+\mathbb{R}$ with the restriction of tangent bundle of the bordism. I will read bordism from bottom to top.
Even to obtain pants bordism between standard ...

5
votes

### Multiplicative structures on truncated Moore spectra

The $1$-type of $M\mathbb{Z}/(2^r)$ does have an $E_\infty$-ring structure for $r> 1$. I'm going to show it by using the algebraic models for $1$-truncated connective commutative ring spectra from:
...

5
votes

Accepted

### Injectivity of the cohomology map induced by some projection map

Ok, I will follow Fernando's advice and post an answer. I learned the computation below from the beginning of Pin(2)-equivariant Seiberg--Witten Floer homology and the triangulation conjecture.
The ...

4
votes

### Obstruction to a cohomology class on total space being a pullback of a class on the base space is the restriction to the fiber

I don't know if this is explicit enough for your purpose, but the existence of this map follows from the (dual) Blakers Massey theorem, at least with mild assumptions on $\pi$.
Let's start with the ...

4
votes

### Injectivity of the cohomology map induced by some projection map

OK, just noticed mme's comment, which considers the very same counterexample below with a clean one-line argument. I'm leaving this here in case someone finds something of any value, but I think mme's ...

4
votes

Accepted

### Homotopy class of maps into Stiefel manifolds

Maybe what you looking for is known under the name generalized curvatura integra (for the case $N> k+1$). I will formulate it not for $S^n$ but more generally for a $m$-dimensional framed manifold $...

4
votes

### What is obstructing two stably-isomorphic vector bundles from being isomorphic?

I know, I am late to the party (as always) but I could provide some more information. In this article I show that two $5$-dimensional vector bundles with $w_2=w_4=0$ over a spin $5$-manifold which ...

3
votes

### Nowhere vanishing section implies reduction of structure group

Informally, say a type of structure $S$ which can form "bundles" over a manifold is a "symmetry structure" if all fibers are isomorphic (in an appropriate, say categorical, sense). Let $G(S)$ be the ...

3
votes

### Some questions about the definition of Chern classes in Cheeger--Simons differential characters

There is a general construction of a transgression form on $P/H$ when $P$ is a principal $G$ bundle with connection, $H \subset G$ is a reductive subgroup, and $f\in \text{Sym}^n(\mathfrak{g}^*)$ is ...

3
votes

### Definition of 1st degree obstruction class

When $i=1$, the Stiefel manifold is $V_n({\mathbb R})$ which is the bundle of $n$-frames in an $n$-dimensional vector space. As such it is homeomorphic to $GL(n,{\mathbb R})$ and hence $\pi_0$ has a ...

2
votes

### Vector bundles over a homotopy-equivalent fibration

As indicated in the comments, this question ended up being accidentally rather trivial.
Specifically, the following three facts are fairly well-known and rather easy to establish:
For homotopic ...

2
votes

### Measuring failure of a setup to preserve some structure giving interesting notions

Here is a huge family of examples. Let $F:\mathcal{C}\to \mathcal{D}$ be, let's say, an additive functor between abelian categories. Let's say, $F$ is right or left exact. Its failure to be exact ...

Community wiki

1
vote

Accepted

### Obstruction to the existence of an invariant symplectic connection

Let $\nabla$ be any symplectic connection. It is easy to check that the map $\mathfrak{g} \to \Omega^1 (M, \operatorname{End}_{\omega} (TM))$ given by $X \to \mathcal{L}_X (\nabla)$ is a Chevalley-...

1
vote

### Measuring failure of a setup to preserve some structure giving interesting notions

Let $p:P\rightarrow M$ be a $G$-principal bundle, consider an extension of Lie group $1\rightarrow H\rightarrow K\rightarrow G\rightarrow 1$ ($H$ is commutative). Suppose that $K\rightarrow G$ has ...

Community wiki

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