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1.
Claus Hertling 《Compositio Mathematica》1999,116(1):1-37
Classifying spaces and moduli spaces are constructed for two invariants of isolated hypersurface singularities, for the polarized mixed Hodge structure on the middle cohomology of the Milnor fibre, and for the Brieskorn lattice as a subspace of the Gauß–Manin connection. The relations between them, period mappings for -constant families of singularities, and Torelli theorems are discussed. 相似文献
2.
Ulrich Koschorke 《manuscripta mathematica》1992,76(1):241-249
Linkings of higher dimensional disjoint singular spheres in euclidean space ℝm are compared with linkings in the sphereS
m. Somewhat surprisingly, link homotopy inS
m can be interpreted as a special case of link homotopy in ℝm. This leads to a considerable refinement of standard invariants such as the generalized Milnor invariants. 相似文献
3.
Link-homotopy and self Δ-equivalence are equivalence relations on links. It was shown by J. Milnor (resp. the last author) that Milnor invariants determine whether or not a link is link-homotopic (resp. self Δ-equivalent) to a trivial link. We study link-homotopy and self Δ-equivalence on a certain component of a link with fixing the other components, in other words, homotopy and Δ-equivalence of knots in the complement of a certain link. We show that Milnor invariants determine whether a knot in the complement of a trivial link is null-homotopic, and give a sufficient condition for such a knot to be Δ-equivalent to the trivial knot. We also give a sufficient condition for knots in the complements of the trivial knot to be equivalent up to Δ-equivalence and concordance. 相似文献
4.
Laurence R. Taylor 《Geometriae Dedicata》2010,148(1):371-389
A technique is discussed to control the indeterminacy in cohomology Massey triple products. It involves finding cohomology
classes with certain properties. Poincaré duality spaces always have such classes if the coefficients are in a field. 相似文献
5.
David B. Massey 《Proceedings of the American Mathematical Society》2006,134(7):2049-2060
We show how some of our recent results clarify the relationship between the Lê numbers and the cohomology of the Milnor fiber of a non-isolated hypersurface singularity. The Lê numbers are actually the ranks of the free Abelian groups--the Lê modules--appearing in a complex whose cohomology is that of the Milnor fiber. Moreover, the Milnor monodromy acts on the Lê module complex, and we describe the traces of these monodromy actions in terms of the topology of the critical locus.
6.
We describe a relation between two invariants which measure the complexity of a hypersurface singularity. One is the Hodge spectrum which is related to the monodromy and the Hodge filtration on the cohomology of the Milnor fiber. The other is the multiplier ideal, having to do with log resolutions.
Mathematics Subject Classification (2000):14B05, 32S35 相似文献
7.
In this paper we study Baer invariants of precrossed modules relative to the subcategory of crossed modules, following Fröhlich and Furtado-Coelho’s general theory on Baer invariants in varieties of Ω-groups and Modi’s theory on higher dimensional Baer invariants. Several homological invariants of precrossed and crossed modules were defined in the last two decades. We show how to use Baer invariants in order to connect these various homology theories. First, we express the low-dimensional Baer invariants of precrossed modules in terms of a new non-abelian tensor product of a precrossed module. This expression is used to analyze the connection between the Baer invariants and the homological invariants of precrossed modules defined by Conduché and Ellis. Specifically we prove that the second homological invariant of Conduché and Ellis is in general a quotient of the first component of the Baer invariant we consider. The definition of classical Baer invariants is generalized using homological methods. These generalized Baer invariants of precrossed modules are applied to the construction of five term exact sequences connecting the generalized Baer invariants with the cohomology theory of crossed modules considered by Carrasco, Cegarra and R.-Grandjeán and the cohomology theory of precrossed modules. 相似文献
8.
Motivic Milnor fibre of cyclic <Emphasis Type="Italic">L</Emphasis><Subscript>∞</Subscript>-algebras
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Yun Feng Jiang 《数学学报(英文版)》2017,33(7):933-950
We define the motivic Milnor fiber of cyclic L ∞-algebras of dimension three using the method of Denef and Loeser of motivic integration. It is proved by Nicaise and Sebag that the topological Euler characteristic of the motivic Milnor fiber is equal to the Euler characteristic of the étale cohomology of the analytic Milnor fiber. We prove that the value of Behrend function on the germ moduli space determined by a cyclic L ∞-algebra L is equal to the Euler characteristic of the analytic Milnor fiber. Thus we prove that the Behrend function depends only on the formal neighborhood of the moduli space. 相似文献
9.
We use techniques from homotopy theory, in particular the connection between configuration spaces and iterated loop spaces,
to give geometric explanations of stability results for the cohomology of the varieties of regular semisimple elements in
the simple complex Lie algebras of classical type A, B or C, as well as in the group . We show that the cohomology spaces of stable versions of these varieties have an algebraic stucture, which identifies them
as “free Poisson algebras” with suitable degree shifts. Using this, we are able to give explicit formulae for the corresponding
Poincaré series, which lead to power series identities by comparison with earlier work. The cases of type B and C involve ideas from equivariant homotopy theory. Our results may be interpreted in terms of the actions of a Weyl group on
its coinvariant algebra (i.e. the coordinate ring of the affine space on which it acts, modulo the invariants of positive
degree; this space coincides with the cohomology ring of the flag variety of the associated Lie group) and on the cohomology
of its associated complex discriminant variety.
Received August 31, 1998; in final form August 1, 1999 / Published online October 30, 2000 相似文献
10.
Massey Products in the Cohomology of the Moment-Angle Manifolds Corresponding to Pogorelov Polytopes
Mathematical Notes - Nontrivial Massey products in the cohomology of the moment-angle manifolds corresponding to polytopes in the Pogorelov class are constructed. This class includes the... 相似文献
11.
Darren Salven Tapp 《Journal of Pure and Applied Algebra》2008,212(10):2314-2319
Let f and g be reduced homogeneous polynomials in separate sets of variables. We establish a simple formula that relates the eigenspace decomposition of the monodromy operator on the Milnor fiber cohomology of fg to that of f and g separately. We use a relation between local systems and Milnor fiber cohomology that has been established by D. Cohen and A. Suciu. 相似文献
12.
Javier Fernández de Bobadilla 《Topology》2004,43(4):925-982
In this paper we develop a Morsification Theory for holomorphic functions defining a singularity of finite codimension with respect to an ideal, which recovers most previously known morsification results for non-isolated singularities and generalise them to a much wider context. We also show that deforming functions of finite codimension with respect to an ideal within the same ideal respects the Milnor fibration. Furthermore we present some applications of the theory: we introduce new numerical invariants for non-isolated singularities, which explain various aspects of the deformation of functions within an ideal; we define generalisations of the bifurcation variety in the versal unfolding of isolated singularities; applications of the theory to the topological study of the Milnor fibration of non-isolated singularities are presented. Using intersection theory in a generalised jet-space we show how to interpret the newly defined invariants as certain intersection multiplicities; finally, we characterise which invariants can be interpreted as intersection multiplicities in the above mentioned generalised jet space. 相似文献
13.
José Eduardo Prado Pires de Campos 《Topology and its Applications》2010,157(3):605-614
In this paper we provide a complete algebraic invariant of link-homotopy, that is, an algebraic invariant that distinguishes two links if and only if they are link-homotopic. The paper establishes a connection between the “peripheral structures” approach to link-homotopy taken by Milnor, Levine and others, and the string link action approach taken by Habegger and Lin. 相似文献
14.
The purpose of this work is to establish a link between the theory of Chern classes for singular varieties and the geometry of the varieties in question. Namely, we show that if Z is a hypersurface in a compact complex manifold, defined by the complex analytic space of zeroes of a reduced non-zero holomorphic section of a very ample line bundle, then its Milnor classes, regarded as elements in the Chow group of Z, determine the global Lê cycles of Z; and vice versa: The Lê cycles determine the Milnor classes. Morally this implies, among other things, that the Milnor classes determine the topology of the local Milnor fibres at each point of Z, and the geometry of the local Milnor fibres determines the corresponding Milnor classes. 相似文献
15.
Mikhail Katz 《Israel Journal of Mathematics》2008,164(1):381-395
We show that the existence of a nontrivial Massey product in the cohomology ring H* (X) imposes global constraints upon the Riemannian geometry of a manifold X. Namely, we exhibit a suitable systolic inequality, associated to such a product. This generalizes an inequality proved in
collaboration with Y. Rudyak, in the case when X has unit Betti numbers, and realizes the next step in M. Gromov’s program for obtaining geometric inequalities associated
with nontrivial Massey products. The inequality is a volume lower bound, and depends on the metric via a suitable isoperimetric
quotient. The proof relies upon W. Banaszczyk’s upper bound for the successive minima of a pair of dual lattices. Such an
upper bound is applied to the integral lattices in homology and cohomology of X. The possibility of applying such upper bounds to obtain volume lower bounds was first exploited in joint work with V. Bangert.
The latter work deduced systolic inequalities from nontrivial cup-product relations, whose role here is played by Massey products.
Supported by the Israel Science Foundation (grants no. 620/00, 84/03, and 1294/06) 相似文献
16.
Sadayoshi Kojima 《Topology and its Applications》1983,16(1):43-60
We present the failure of Whitney's lemma in dimension 4 from the homotopical and topological viewpoints. Those are detected by Massey products. The invariants for the examples represented by framed links are computed in terms of Milnor's -invariants. 相似文献
17.
Hirotaka Tamanoi 《Transactions of the American Mathematical Society》2000,352(11):5139-5178
On Brown-Peterson cohomology groups of a space, we introduce a natural inherent topology, BP topology, which is always complete Hausdorff for any space. We then construct a spectra map which calculates infinite BP-linear sums convergent with respect to the BP topology, and a spectrum which describes infinite sum BP-linear relations in BP cohomology. The mod cohomology of this spectrum is a cyclic module over the Steenrod algebra with relations generated by products of exactly two Milnor primitives. We show a close relationship between BP-linear relations in BP cohomology and the action of the Milnor primitives on mod cohomology. We prove main relations in the BP cohomology of Eilenberg-Mac Lane spaces. These are infinite sum BP-linear relations convergent with respect to the BP topology. Using BP fundamental classes, we define -series which are -analogues of the -series. Finally, we show that the above main relations come from the -series.
18.
A relation between the Euler characteristics of the Milnorfibres of a real analytic function is derived from a simple identity involvingcomplex monodromy and complex conjugation. A corollary is the result of Costeand Kurdyka that the Euler characteristic of the local link of an irreduciblealgebraic subset of a real algebraic set is generically constant modulo 4. Asimilar relation for iterated Milnor fibres of ordered sets of functions isused to define topological invariants of ordered collections of algebraicsubsets. 相似文献
19.
Ciprian Manolescu 《Advances in Mathematics》2007,211(1):363-416
For each positive integer n, Khovanov and Rozansky constructed an invariant of links in the form of a doubly-graded cohomology theory whose Euler characteristic is the sl(n) link polynomial. We use Lagrangian Floer cohomology on some suitable affine varieties to build a similar series of link invariants, and we conjecture them to be equal to those of Khovanov and Rozansky after a collapse of the bigrading. Our work is a generalization of that of Seidel and Smith, who treated the case n=2. 相似文献
20.
Milnor link invariants and quantum 3-manifold invariants 总被引:1,自引:0,他引:1
Let be the 3-manifold invariant of Le, Murakami and Ohtsuki. We show that , where denotes terms of degree , if M is a homology 3-sphere obtained from by surgery on an n-component Brunnian link whose Milnor -invariants of length vanish.?We prove a realization theorem which is a partial converse to the above theorem.?Using the Milnor filtration on links,
we define a new bifiltration on the vector space with basis the set of oriented diffeomorphism classes of homology 3-spheres. This includes the Milnor level
2 filtration defined by Ohtsuki. We show that the Milnor level 2 and level 3 filtrations coincide after reindexing.
Received: October 23, 1998. 相似文献