We study necessary and sufficient conditions for a 4-dimensional Lefschetz fibration over the 2-disk to admit a ${\text{Pin}}^{\pm}$-structure, extending the work of A. Stipsicz in the orientable setting. As a corollary, we get existence results of ${\text{Pin}}^{+}$ and ${\text{Pin}}^-$-structures on closed non-orientable 4-manifolds and on Lefschetz fibrations over the 2-sphere. In particular, we show via three explicit examples how to read-off ${\text{Pin}}^{\pm}$-structures from the Kirby diagram of a 4-manifold. We also provide a proof of the well-known fact that any closed 3-manifold M admits a ${\text{Pin}}^-$-structure and we find a criterion to check whether or not it admits a ${\text{Pin}}^+$-structure in terms of a handlebody decomposition. We conclude the paper with a characterization of ${\text{Pin}}^+$-structures on vector bundles.

For a manifold with boundary, the restriction of Chern's transgression form of the Euler curvature form over the boundary is closed. Its cohomology class is called the secondary Chern– Euler class and was used by Sha to formulate a relative Poincaré–Hopf theorem under the condition that the metric on the manifold is locally product near the boundary. We show that the secondary Chern–Euler form is exact away from the outward and inward unit normal vectors of the boundary by explicitly constructing a transgression form. Using Stokes’ theorem, this evaluates the boundary term in Sha's relative Poincaré–Hopf theorem in terms of more classical indices of the tangential projection of a vector field. This evaluation in particular shows that Sha's relative Poincaré–Hopf theorem is equivalent to the more classical law of vector fields.

Let d be the degree of an algebraic one-dimensional foliation $\mathcal F$ on the complex projective space ${\mathbb P}_n$ (i.e. the degree of the variety of tangencies of the foliation with a generic hyperplane). Let $\Gamma$ be an algebraic solution of degree $\delta$, and geometrical genus g. We prove, in particular, the inequality $(d-1)\delta+2-2g\geq {\mathcal B}(\Gamma)$, where ${\mathcal B}(\Gamma)$ denotes the total number of locally irreducible branches through singular points of $\Gamma$ when $\Gamma$ has singularities, and ${\mathcal B}(\Gamma)=1$ (instead of 0) when $\Gamma$ is smooth. Equivalently, when $\Gamma=\bigcap_{\lambda=1}^{n-1} S_\lambda$ is the complete intersection of n - 1 algebraic hypersurfaces $S_\lambda$, we get $(d+n-\sum_{\lambda=1}^{n-1}\delta_\lambda)\delta \geq {\mathcal B}(\Gamma)-{\mathcal E}(\Gamma)$, where $\delta_\lambda$ denotes the degree of $S_\lambda$ and ${\mathcal E}(\Gamma)=2-2g+(\sum_\lambda\delta_\lambda-(n+1))\delta$ the correction term in the genus formula. These results are also refined when $\Gamma$ is reducible.

an index of a collection of 1-forms on a complex isolated complete intersection singularity corresponding to a chern number is defined and – in the case when the 1-forms are complex analytic – expressed as the dimension of a certain algebra.

Let X be a compact connected Riemann surface and ξ a square root of the holomorphic contangent bundle of X. Sending any line bundle L over X of order two to the image of dim H0(X, ξ ⊗ L) − dim H0(X, ξ) in Z/2Z defines a quadratic form on the space of all order two line bundles. We give a topological interpretation of this quadratic form in terms of index of vector fields on X.

In the first paper with the same title the authors were able to determine all partially oriented flag manifolds that are stably parallelizable or parallelizable, apart from four infinite families that were undecided. Here, using more delicate techniques (mainly K-theory),we settle these previously undecided families and show that none of the manifolds in them is stably parallelizable, apart from one 30-dimensional manifold which still remains undecided.

The existence of metastable immersion or span for space forms and homogeneous spaces is shown to depend only on the two-type of the space.