Questions tagged [obstruction-theory]

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

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Basic obstruction theory : where does the obstruction to uniqueness of lifting lie?

This is a question about a remark someone said to me without giving much precision. Suppose you have two nice spaces $X,Y$ and are trying to build a map $X\to Y$ with certain nice properties. ...
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First nonzero Stiefel-Whitney class

In Milnor-Stasheff's Characteristic Classes, one of the problems says that the smallest nonzero $w_i(E)$ happens when $i$ is a power of 2 in the case where $w(E) = 1$. I read here (Prop 7): https://...
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Classification of circle bundles over $\mathbb{RP}^2$

I am trying to understand isomorphism classes of bundles $$\mathbb{S}^1\hookrightarrow E\to \mathbb{RP}^2.$$ These are classified by homotopy classes $[\mathbb{RP}^2,BAut(\mathbb{S}^1)]$. ATTEMPT 1. ...
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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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Is a principal $\mathbb{Z}_2\ltimes PSU(4)$-bundle over a 3-manifold $M$ equivalent to an element in $H^1(M,\mathbb{Z}_2)\times H^2(M,\mathbb{Z}_4)$?

Given a 3-manifold $M$ and a principal $\mathbb{Z}_2\ltimes PSU(4)$-bundle $P$ over $M$ whose isomorphism class is represented by the homotopy class of a map $f:M\to B(\mathbb{Z}_2\ltimes PSU(4))$ ...
192 views

Euler class as obstruction to have a never vanishing cross section

We know that (see Hatcher's vector bundles and K-theory Prop. 3.22) the Euler class of an orientable vector bundle or rank $r$, $E\to M$ is the first obstruction to the existence of a never vanishing ...
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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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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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Obstruction to extending $G$-bundle to 4-dimensions in Chern-Simons theory

I am reading Dijkgraaf and Witten's paper on Chern-Simons and finite gauge groups and something they have written about the obstruction to extending the bundle to the 4-manifold confuses me. My ...
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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 ...
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 $S^1 \hookrightarrow X$ ...
Let $K$ denote a simplicial complex and $Y$ some topological space. Let us also denote by $K^n$ the $n$-skeleton of $K$. I would like to have an example for the following situation: There is a map \$...