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For a germ of a variety V,0⊂CN,0$\mathcal {V}, 0 \subset \mathbb {C}^N, 0$, a singularity V0$\mathcal {V}_0$ of "type V$\mathcal {V}$" is given by a germ f0:Cn,0→CN,0$f_0: \mathbb {C}^n, 0 \rightarrow \mathbb {C}^N, 0$, which is transverse to V∖{0}$\mathcal {V}\setminus \lbrace 0\rbrace$ in an appropriate sense, such that V0=f0−1(V)$\mathcal {V}_0 = f_0^{-1}(\mathcal {V})$. In part I of this paper, we introduced for such singularities the Characteristic Cohomology for the Milnor fiber (for V$\mathcal {V}$ a hypersurface), and complement and link (for the general case). It captures the cohomology of V0$\mathcal {V}_0$ inherited from V$\mathcal {V}$ and is given by subalgebras of the cohomology for V0$\mathcal {V}_0$ for the Milnor fiber and complements, and is a subgroup for the cohomology of the link. We showed these cohomologies are functorial and invariant under diffeomorphism groups of equivalences KH$\mathcal {K}_{H}$ for Milnor fibers and KV$\mathcal {K}_{\mathcal {V}}$ for complements and links. We also gave geometric criteria for detecting the nonvanishing of the characteristic cohomology. In this paper, we apply these methods in the case V$\mathcal {V}$ denotes any of the varieties of singular m×m$m \times m$ complex matrices, which may be either general, symmetric, or skew‐symmetric (with m$m$ even). For these varieties, we have shown in another paper that their Milnor fibers and complements have compact "model submanifolds" for their homotopy types, which are classical symmetric spaces in the sense of Cartan. As a result, we first give the structure of the characteristic cohomology subalgebras for the Milnor fibers and complements as images of exterior algebras (or in one case a module on two generators over an exterior algebra). For links, the characteristic cohomology group is the image of a shifted upper truncated exterior algebra. In addition, we extend these results for the complement and link to the case of general m×p$m \times p$ complex matrices. Second, we then apply the geometric detection methods introduced in Part I to detect when specific characteristic cohomology classes for the Milnor fiber or complement are nonzero. We identify an exterior subalgebra on a specific set of generators and for the link that it contains an appropriate shifted upper truncated exterior subalgebra. The detection criterion involves a special type of "kite map germ of size ℓ$\ell$" based on a given flag of subspaces. The general criterion that detects such nonvanishing characteristic cohomology is then given in terms of the defining germ f0$f_0$ containing such a kite map germ of size ℓ$\ell$. Furthermore, we use a restricted form of kite spaces to give a cohomological relation between the cohomology of local links and the global link for the varieties of singular matrices. [ABSTRACT FROM AUTHOR]

We define the Milnor number of a one-dimensional holomorphic foliation F as the intersection number of two holomorphic sections with respect to a compact connected component C of its singular set. Under certain conditions, we prove that the Milnor number of F on a three-dimensional manifold with respect to C is invariant by -1 topological equivalences. [ABSTRACT FROM AUTHOR]

We show that the base spaces of the semiuniversal unfoldings of some weighted homogeneous singularities can be identified with moduli spaces of A∞$A_\infty$‐structures on the trivial extension algebras of the endomorphism algebras of the tilting objects. The same algebras also appear in the Fukaya categories of their mirrors. Based on these identifications, we discuss applications to homological mirror symmetry for Milnor fibers, and give a proof of homological mirror symmetry for an n$n$‐dimensional affine hypersurface of degree n+2$n+2$ and the double cover of the n$n$‐dimensional affine space branched along a degree 2n+2$2n+2$ hypersurface. Along the way, we also give a proof of a conjecture of Seidel (Proceedings of the International Congress of Mathematicians, 2002) which may be of independent interest. [ABSTRACT FROM AUTHOR]

Let M be a Liouville 6-manifold which is the smooth fiber of a Lefschetz fibration on C4 constructed by suspending a Lefschetz fibration on C³. We prove that for many examples including stabilizations of Milnor fibers of hypersurface cusp singularities, the compact Fukaya category F(M) and the wrapped Fukaya category W(M) are related through A∞-Koszul duality, by identifying them with cyclic and Calabi-Yau completions of the same quiver algebra. This implies the split-generation of the compact Fukaya category F(M) by vanishing cycles. Moreover, new examples of Liouville manifolds which admit quasi-dilations in the sense of Seidel-Solomon are obtained. [ABSTRACT FROM AUTHOR]

We prove that Waldhausen K-theory, when extended to a very general class of quasicategories, can be described as a Goodwillie differential. In particular, K-theory spaces admit canonical (connective) deloopings, and the K-theory functor enjoys a simple universal property. Using this, we give new, higher categorical proofs of the approximation, additivity, and fibration theorems of Waldhausen in this article. As applications of this technology, we study the algebraic K-theory of associative rings in a wide range of homotopical contexts and of spectral Deligne–Mumford stacks. [ABSTRACT FROM AUTHOR]

This paper describes the relationship between the first non-vanishing Milnor invariants of a classical link and the intersection invariant of a twisted Whitney tower. This is a certain 2-complex in the 4-ball, built from immersed disks bounded by the given link in the 3-sphere together with finitely many ‘layers’ of Whitney disks.The intersection invariant is a higher-order generalization of the intersection number between two immersed disks in the 4-ball, well known to give the linking number of the link on the boundary, which measures intersections among the Whitney disks and the disks bounding the given link, together with information that measures the twists (framing obstructions) of the Whitney disks.This interpretation of Milnor invariants as higher-order intersection invariants plays a key role in our classifications [J. Conant, R. Schneiderman and P. Teichner, ‘Higher-order intersections in low-dimensional topology’, Proc. Natl Acad. Sci. USA 108 (2011) 8131–8138; J. Conant, R. Schneiderman and P. Teichner, ‘Whitney tower concordance of classical links’, Geom. Topol. 16 (2012) 1419–1479] of both the framed and twisted Whitney tower filtrations on link concordance. Here, we show how to realize the higher-order Arf invariants, which also play a role in the classifications, and derive new geometric characterizations of links with vanishing length at most 2k Milnor invariants. [ABSTRACT FROM AUTHOR]

The Milnor degree of a 3-manifold is an invariant that records the maximum simplicity, in terms of higher-order linking, of any link in the 3-sphere that can be surgered to give the manifold. This invariant is investigated in the context of torsion linking forms, nilpotent quotients of the fundamental group, Massey products and quantum invariants, and the existence of 3-manifolds with any prescribed Milnor degree and first Betti number is established. Along the way, it is shown that the number Mrk of linearly independent Milnor invariants of degree k, for r-component links in the 3-sphere whose lower degree invariants vanish, is positive except in the classically known cases (when r = 1, and when r = 2 with k = 2, 4 or 6). [ABSTRACT FROM PUBLISHER]

We introduce blow-up and blow-down operations for generalized complex 4-manifolds. Combining these with a surgery analogous to the logarithmic transform, we then construct generalized complex structures on for n odd, a family of 4-manifolds that admit neither complex nor symplectic structures unless n = 1. We also extend the notion of a symplectic elliptic Lefschetz fibration, so that it expresses a generalized complex 4-manifold as a fibration over a 2-dimensional manifold with boundary. [ABSTRACT FROM PUBLISHER]