All Questions
Tagged with obstruction-theory at.algebraic-topology
31 questions
9
votes
0
answers
120
views
Reference Request: Moore--Postnikov tower of the rationalization of a fibration
Two spaces $X$ and $Y$ are said to be rationally homotopy equivalent, written $X \sim_{\mathbb{Q}} Y$, if their rationalizations $X_{\mathbb{Q}}$ and $Y_{\mathbb{Q}}$
are homotopy equivalent. Moreover,...
- 371
2
votes
0
answers
111
views
lifting a family of curves to a family of sections of a vector bundle?
This is a question in obstruction theory. It should be basic but I can't find a reference.
Let's stick to the $C^\infty$ category, so all objects mentioned are smooth. Let $\pi: E \to M$ be a vector ...
- 51
4
votes
0
answers
182
views
Obstruction to finding a Whitney disk
Let $p,q\geq 3$ be integers. Let $M$ be a compact oriented smooth $(p+q)$-manifold and $P$ and $Q$ compact submanifolds of dimensions $p$ and $q$ intersecting transversely. Assume that $M,P$ and $Q$ ...
0
votes
1
answer
155
views
Vector bundles over a homotopy-equivalent fibration
I think this question is related to what is known as "obstruction theory", but I'm not very familiar with this field of mathematics, so I am asking here.
Let $\pi:N\rightarrow M$ be a smooth ...
- 2,792
2
votes
1
answer
160
views
Obstruction to a cohomology class on total space being a pullback of a class on the base space is the restriction to the fiber
Let $\pi \colon E \to X$ be a fiber bundle with fiber $F$ and suppose that $\tilde H^i(F) = 0$ for $0 \leq i \leq k-1$.
Using the Leray-Serre spectral sequence, we get an exact sequence
$$
0 \to H^k(...
- 293
9
votes
1
answer
330
views
Non-triviality of a Postnikov class in $H^3\left(B \operatorname{PSU}(N) ; \mathbb{Z}_q\right)$
Let $\alpha\in H^2(B\operatorname{PSU}(N) ; \mathbb{Z}_N)$ be the obstruction class for lifting a $\operatorname{PSU}(N)$-bundle to an $\mathrm{SU}(N)$-bundle. Note that $\operatorname{PSU}(N)\cong \...
- 2,273
7
votes
2
answers
538
views
Injectivity of the cohomology map induced by some projection map
Given a (compact) Lie group $G$, persumably disconnected, there exists a short exact sequence
$$1\rightarrow G_c\rightarrow G\rightarrow G/G_c\rightarrow 1$$
where $G_c$ is the normal subgroup which ...
- 125
15
votes
1
answer
1k
views
Possible mistake in Cohen notes "Immersions of manifolds and homotopy theory" (version 27 March 2022)
In Theorem 2 of these notes, Ralph Cohen reformulates the main theorem of Hirsch-Smale theory merely in terms of normal bundles.
In particular, he says that if $N, M$ are two manifolds, $\dim N< ...
- 2,533
8
votes
1
answer
468
views
Finite domination and compact ENRs
Edit: In the comments, Tyrone points out that West's positive answer to Borsuk's conjecture implies that every compact ENR is homotopy equivalent to a finite CW complex. It follows that the only ...
- 18.8k
10
votes
2
answers
404
views
Multiplicative structures on truncated Moore spectra
As discussed for example in this paper (and this MO thread), there are obstructions for the existence of $A_n$-structures on Moore spectra $M(p^r):=\mathbb{S}/p^r$, which in general don't vanish. In ...
- 2,300
9
votes
1
answer
316
views
Framed version of the "copants bordism"?
The "pants" bordism in dimension n is a bordism which goes from $S^n \sqcup S^n$ to $S^n$ witnessing the connected sum operation - equivalently by attaching 1-handle to the trivial bordism, ...
- 27.5k
3
votes
1
answer
217
views
Definition of 1st degree obstruction class
Recently I go through obstruction class illustrated by Milnor.
He defined $\mathfrak{o}_i$by an element in $H^i(M; \pi_{i-1}(V_{n-i+1}(F))$, which is cohomology with local coefficients.
But the 0th ...
- 1,168
4
votes
0
answers
302
views
Where can I read about non-principal obstruction theory?
Most treatments of obstruction theory assume a principal Postnikov tower. I have the vague sense that if one uses cohomology with local coefficients, one does not need to make any assumptions on one's ...
- 63.9k
2
votes
0
answers
182
views
obstruction cocycle for nonsimple spaces using local coefficients
This question is similar to here but I was hoping for a concrete theorem statement surrounding the obstruction cocycle for non-simple spaces.
I'm hoping for a theorem like the following:
Let $A \...
- 131
3
votes
1
answer
363
views
Homotopy class of maps into Stiefel manifolds
Motivation
Hopf theorem, asserts that $C^0$-maps $f:M^n\to \mathbb{S}^n$ from an orientable, closed n-manifold into an n-sphere are classified up to homotopy by their degree $deg(f)$.
The theorem ...
- 2,533
2
votes
2
answers
214
views
Measuring failure of a setup to preserve some structure giving interesting notions
I am looking for some examples of failure of some structures giving interesting notions. For example, we have the following situation:
Let $P(M,G)$ be a principal bundle. Let $\Gamma\subseteq TP$ be ...
8
votes
1
answer
606
views
Obstruction to homotopy, cohomology operations and Dold-Whitney theorem
I am reading the famous paper by Dold and Whitney "Classification of Oriented Sphere Bundles Over A 4-Complex".
I'll state their theorem for the case of SO(3) bundles
Classification Theorem:Let $B_1,...
- 2,533
11
votes
1
answer
858
views
What is the relationship between spectral sequences and obstruction theory?
Let $X,Y$ be objects of some category $\mathcal C$, and suppose I want to study homotopy classes of maps from $X$ to $Y$ (almost everything one does in algebraic topology can be viewed this way). It ...
- 63.9k
11
votes
1
answer
1k
views
Classification of bundles, Postnikov towers, obstruction theory, local coefficients
RECAP on classification of bundles
We want to classify $G$-principal bundles over $X$ (smooth manifold, G compact Lie). These are in 1-1 correspondence with homotopy classes of maps $[X,BG]$ (where $...
- 2,533
5
votes
0
answers
745
views
Questions about obstruction theory (Hatcher's book)
I'm actually studying obstruction theory as presented in the last section of chapter $4$ of the book Algebraic Topology by Allen Hatcher. He first finds condition so that a space $X$ admits a ...
- 521
2
votes
0
answers
158
views
Extension of a given section and obstruction cocyles
Let $p:E \to X$ be a fibration with the fiber $F$ where $X$ is a CW-complex. Denote by $U$ the set $U:=D^n \times \{0\} \cup S^{n-1} \times I$ (part of a cylinder) and let $\tilde{f}:U \to E$ be a ...
- 9,330
8
votes
1
answer
651
views
Obstructions for the lifting problem after a pull-back
This is a cross-post from a MSE question which received no answers. Beware that the notation here is a little different.
Consider the following lifting problem(s):
$\require{AMScd}$
\begin{CD}
&...
- 1,726
5
votes
1
answer
507
views
Are open orientable 3-manifolds parallelizable via obstruction theory?
In the case of orientable closed $3$-manifolds, we have 4 ingredients that ensure parallelizability:
1a) Closed smooth $n$-manifolds have homotopy type of a $CW$-complex
1b) Closed smooth $n$-...
- 441
6
votes
0
answers
163
views
Reference to the theorem about linear bundles
The following fact looks like it is well-known. I can prove it myself, but I would like to know of a reference which has a proof?
Let $M$ be an $n$-manifold, $C\subset M$ a codimension-1 closed ...
- 1,095
9
votes
0
answers
417
views
Equivariant obstruction theory done wrong
Let $G$ be a finite group, and let $p:E\to B$ be a $G$-fibration with fibre $F$. The correct framework for studying equivariant sections of $p$ is Bredon cohomology. With the right definitions, most ...
- 35.9k
7
votes
1
answer
542
views
Topological obstruction for the existence of spin$^c$ structure
Recently I asked on stack exchange the following question: https://math.stackexchange.com/questions/2088888/vanishing-of-certain-cohomology-class-and-existence-of-spin-structure
I would like to know ...
- 9,330
4
votes
0
answers
182
views
Naturality of primary obstruction under fiber-preserving maps
Let $B$ be a path-connected CW complex, and let $p:E\to B$ and $p':E'\to B$ be fibrations. Let $f:E'\to E$ be a fiber-preserving map, which therefore induces a map of fibers $\bar{f}:F'\to F$.
Let us ...
- 35.9k
2
votes
0
answers
222
views
obstructions to embeddings of manifolds into Grassmannians
Let $G_k(\mathbb{R}^n)$ be the Grassmannian consisting of $k$-dimensional subspaces in $\mathbb{R}^n$ and $AG_k(\mathbb{R}^n)$ the "affine Grassmannian" consisting of $k$-dimensional planes in $\...
- 543
19
votes
2
answers
1k
views
What is obstructing two stably-isomorphic vector bundles from being isomorphic?
The specific situation is the following:
Let $n>0$ be a natural number, let $X$ be a finite CW complex of dimension $n$, and let $\xi_0,\xi_1$ be oriented real vector bundles of rank $n$ over $X$ ...
- 521
5
votes
1
answer
630
views
What's a good reference for the following obstruction theory yoga?
Fix a colored operad, which I will leave implicit, and a field $\mathbb K$ of characteristic $0$. By algebra in this post I will mean a dg algebra over $\mathbb K$ for the given colored operad. I ...
- 54.6k
0
votes
0
answers
148
views
Extending a 2-frame field - manifolds with boundary
If $M$ is an orientable compact 3-manifold with boundary such that there is defined $s\in\Gamma(\partial M,V_2(TM))$ a section of the 2-frame bundle of $TM$, then $s$ extends to $\tilde s\in\Gamma(M,...
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