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In Stefan Metzger's thesis (https://arxiv.org/abs/hep-th/0512285) on page 9, the following statement appears in connection with the compactifications type-IIB string theory on two Calabi-Yau manifolds that are related by a geometric transition. [The geometric transition is described on the previous page in the thesis.]

It is now interesting to see what happens if we compactify Type IIB string theory on two Calabi-Yau manifolds that are related by such a transition. Since one is interested in $\mathcal{N} = 1$ effective theories it is suitable to add either fluxes or branes in order to further break supersymmetry. It is then very natural to introduced D5-branes wrapping the two-spheres in the case of the small resolution of the singularity (Edit: small resolution refers to blowing up the singularity using an $S^2$). The manifold with a deformed singularity (Edit: deformed singularity refers to blowing up the singularity using an $S^3$) has no suitable cycles around which D-branes might wrap, so we are forced to switch on flux in order to break supersymmety.

So first of all, when can we wrap a D$p$-brane around a $k$-cycle (i.e. for what $p$ and $k$)? I thought $p \geq k$ is all we needed?

More specifically, what are the internal cycles of an $S^2$ and an $S^3$ and why can't these D-branes be wrapped around them? Is the reasoning obvious?

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