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 ...
- 16.4k
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 ...
- 7,627
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 ...
- 55.4k
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 ...
- 55.4k
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 ...
- 5,296
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^...
- 1,463
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".
...
- 26.9k
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 ...
- 156
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 ...
- 108k
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 ...
- 17.4k
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 &...
- 1,463
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 ...
- 63.2k
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 ...
- 789
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:
...
- 15k
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 ...
- 9,553
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 ...
- 15k
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 $...
- 2,820
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 ...
- 2,820
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 ...
- 7,922
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 ...
- 31
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 ...
- 19.3k
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 ...
- 2,712
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 ...
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-...
- 243
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 ...
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