Questions tagged [obstruction-theory]

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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< ...
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8 votes
1 answer
261 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 ...
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10 votes
2 answers
322 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 ...
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3 votes
0 answers
143 views

Virtual fundamental class of punctual Hilbert scheme of points

$\DeclareMathOperator\Hilb{Hilb}$It is well known that the Hilbert scheme $\Hilb^n(\mathbb C^3)$ has a (symmetric) perfect obstruction theory. Consider the punctual part at $0 \in \mathbb C^3$, which ...
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0 answers
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Obstruction to lifting homomorphism of groups

Is there a "cohomology" group that encodes obstructions to constructing a lift in a diagram of groups below? If $X\to Y$ is an extension and the bottom row is the identity map it's just $H^1(...
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9 votes
1 answer
261 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, ...
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1 vote
0 answers
58 views

Would the iterated finite abelian descent obstruction equality hold for curves?

Let $X$ be a smooth projective geometrically integral variety over a number field $k$. We begin with some established notions in the theory of descent obstruction to the local-global principle, ...
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1 vote
0 answers
78 views

Obstruction to deformation of composite morphism (Reference request + question)

Let $f_0:X_0\xrightarrow{g_0}Y_0\xrightarrow{h_0}Z_0/S_0$ be a morphism of smooth projective $S_0$-schemes such that $g_0,h_0$ are flat. Let $S_0\subset S$ be a first-order thickening, and let $X,Y,Z$ ...
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3 votes
1 answer
97 views

Obstruction to the existence of an invariant symplectic connection

Let $M$ be a symplectic manifold with a symplectic action of a Lie algebra $\mathfrak{g}$. I am interested whether there exists a $\mathfrak{g}$-invariant symplectic connection on $M$. Where does the ...
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3 votes
1 answer
175 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 ...
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4 votes
0 answers
194 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 ...
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2 votes
0 answers
110 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 \...
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3 votes
1 answer
303 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 ...
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2 votes
2 answers
207 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 ...
5 votes
1 answer
426 views

Nowhere vanishing section implies reduction of structure group

Description I noticed a repeating theme in vector bundle theory, and wonder if there's a theorem that describes this kind of phenomenon. Given a vector bundle $E$ over a manifold $X$. If there is a ...
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8 votes
1 answer
462 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,...
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11 votes
1 answer
624 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 ...
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11 votes
1 answer
812 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 $...
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5 votes
0 answers
539 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 ...
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2 votes
0 answers
109 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 ...
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8 votes
1 answer
559 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} &...
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5 votes
1 answer
439 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 $n$-...
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  • 393
6 votes
0 answers
148 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 ...
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6 votes
0 answers
114 views

Topological constraints for existing of certain differential operators on manifolds

At the beginning a word of warning: this would be rather vague question: vague as it is, I'm not requiring a precise answer, rather some intuitive explanation. In the flat case $M=\mathbb{R}^n$ ...
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1 vote
0 answers
116 views

Obstruction theory on $A_{\infty}, C_{\infty}$-algebras

Let $\mathcal{P}_{\infty}$ be $A_{\infty}$ or $C_{\infty}$. Let $A=A^{1}\oplus A^{2}$ be a graded vector space concentrated in degree 1 and 2. Let $m_{n}\: : \:{A^{1}}^{\otimes n}\to A^{2}$ be a ...
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4 votes
0 answers
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Obstruction to the existence of lifting of the classifying map

Let $E$ be an $n$-plane bundle over CW complex $X$. Then $E$ is a pullback of tautological bundle $\gamma_n$ over $BO(n)$ i.e. $E=f^*(\gamma_n)$. This $f$ is called classyfing map. One can show that ...
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9 votes
0 answers
342 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 ...
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5 votes
1 answer
440 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 ...
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  • 8,680
4 votes
0 answers
147 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 ...
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3 votes
2 answers
230 views

Asymptotics of list size in Robertson-Seymour theorem

A planar graph cannot have $K_5$ and $K_{3,3}$ as minors. Robertson-Seymour theorem generalizes this by stating for every genus $g$ there is a finite list of forbidden minor graphs that are ...
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2 votes
0 answers
183 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 $\...
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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 $X$ ...
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  • 521
6 votes
2 answers
570 views

Obstructions for $E_n$-algebras

In Alan Robinson's paper, Classical Obstructions and S-algebras, he provides conditions for a ring spectrum to have an $A_n$ and $\mathbb{E}_\infty$-structure. Have the obstructions for an object ...
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3 votes
2 answers
459 views

Recommended textbooks for Hamiltonian group actions?

I am doing a project on Hamiltonian group actions on symplectic manifolds, and my supervisor was able to list several good books on Riemannian geometry to start me off, but he didn't know of any ...
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5 votes
1 answer
520 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 ...
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0 votes
0 answers
143 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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2 votes
0 answers
113 views

Lazard's $\Gamma_n(f)$ as cocycle

In Michel Lazard's "Commutative Formal Groups" Springer Lecture Notes, he defines an operator on a polynomial 3-cochain $f$ denoted $\Gamma_n(f)$, which defines as the $n^{th}$ homogeneous piece of ...
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