Questions tagged [obstruction-theory]

Obstruction theory is a name given to two different mathematical theories, both of which yield cohomological invariants.

43 questions
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If $S^1 \hookrightarrow X$ induces an injection of $H_1$, then $X$ retracts onto $S^1$

This is an exercise in Hatcher, section 4.3, exercise 3, page 419, on which I'm struggling. Suppose that a CW complex X contains a subcomplex $S^1$ such that the inclusion $S^1 \hookrightarrow X$ ...
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When does a continuous map $f:X\rightarrow \mathbb{H}P^n$ lift to $S^{4n+3}$?

Consider the Hopf bundles $$S^1\rightarrow S^{2n+1}\rightarrow \mathbb{C}P^n$$ and $$S^3\rightarrow S^{4n+3}\rightarrow \mathbb{H}P^n.$$ In this question (and also here), it is shown that for any ...
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Manifolds with non-vanishing vector field and vast homology

Let $n \ge 3$. Is there n-fold $M^n$ with both $\chi(M)=0$ and $\dim H_*(M,\mathbb{R}) \ge$ given number?
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Is this inclusion a cofibration?

Let $(X,A)$ be a relative CW-complex. Consider the inclusion $$i:\ \ X\times \{0,1\} \cup A\times I \rightarrow X\times I.$$ I was wondering if this is a cofibration. I guess it is, for there is a ...
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Obstruction theory for fibrations.

I'm currently reading Kirk's & Davis's Lecture notes in Algebraic Topology. On the page 190 they discuss obstruction for lifting $f:X \rightarrow B$ to $f':X \rightarrow E$ where $E \rightarrow B$...
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Do deformations of a vector bundle form a vector space?

I wonder whether deformations of a vector bundle $F$ over a local Artinian ring $(A, m)$ form a vector space $\operatorname{Ext}^1(F, F) \otimes m$, or what happens in a sequence (*) below? I read ...
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Extending a section to a basis of sections in a trivial bundle.

Suppose we have a manifold $M$ of dimension $m$ and let $E$ be the rank $n$ trivial vector bundle on $M$. Let $\Gamma$ be a section of the trivial bundle $E$. My question is, are there some conditions ...
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a question on oriented bundles and Euler class

In Characteristic classes, J. Milnor, J. Stasheff, Prop. 9.7, it is proved that: if the oriented vector bundle $\xi$ possesses a nowhere zero cross section, then the Euler class $e(\xi)=0$. I want ...
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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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Lifting problem of a map $g:M\to B^2\mathbb{Z}_n$ to $f:M\to BPSU(n)$

Since there is a short exact sequence of groups: $$1\to\mathbb{Z}_n\to SU(n)\to PSU(n)\to1,$$ we have a fiber sequence: B\mathbb{Z}_n\to BSU(n)\to BPSU(n)\stackrel{\iota}{\to} B^2\mathbb{Z}_n\to\...
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Finding explicit form of the first characteristic class of a fiber bundle over $S^2$

These two problems are from an exam I've taken some time ago that I still didn't solve: (1) Does there exist a fiber bundle over $S^2$ with fiber $S^1$, whose characteristic class is equal to ...
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Nontrivial obstruction cocycle whose class is trivial

Are there any interesting examples of relative CW complexes $(X,A)$ where a map $f \colon X_n \rightarrow Y$ into an abelian space has the property that the obstruction cocycle $c_f$ is nonzero, ...
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Extending a 2-frame field - manifolds with boundary

It is known that every smooth manifold can be homotoped to a cell complex. In particular this is true for manifolds with boundary. My question: Under the homotopy to a cell complex, is the boundary ...
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Extending map into tangent bundle

Let M be an orientable smooth n-manifold and $p: TM \rightarrow M$ be its tangent bundle. In addition, suppose $f: E \rightarrow M$ is a Hurewicz fibration. Write $e(M) \in H^n(M)$ for the Euler class....
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Showing that the Mobius strip is non-orientable using obstruction theory

In the notes available here, the first example (p. 2) says that the Mobius band is non-orientable because there is an obstruction to a map sending a disk to the band. I don't really understand what's ...
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spin structures on knot complements

Let $K$ be a knot in $S^3$, and let $M=S^3/N(K)$ be its knot complement, where $N(K)$ is a tubular neighborhood of $K$. $K$ is given for example by a its projection onto the plane. The question is ...
Let $1 \to K \to G \to H \to 1$ be a short exact sequence in the category of groups(interested in non-Abelian groups). My question is the following: Where does the obstruction to the splitting of the ...