Operády v současné matematice
Atiyahova-Singerova věta o indexu
On the rational homotopy Lie algebra of spaces with finite dimensional rational cohomology and homotopy
The problem of the characterization of graded Lie algebras which admit a realization as the homotopy Lie algebra of a space of type is discussed. The central results are formulated in terms of varieties of structure constants, several criterions for concrete algebras are also deduced.
Distributive laws and Koszulness
Distributive law is a way to compose two algebraic structures, say and , into a more complex algebraic structure . The aim of this paper is to understand distributive laws in terms of operads. The central result says that if the operads corresponding respectively to and are Koszul, then the operad corresponding to is Koszul as well. An application to the cohomology of configuration spaces is given.
On the G-spaces having an -G-CW-approximation by a G-CW-complex of finite G-type
Regular functions over conformal quaternionic manifolds
A note on closed -cells in
-Invariant tensors and graphs
We describe a correspondence between -invariant tensors and graphs. We then show how this correspondence accommodates various types of symmetries and orientations.
The real K-ring of some CW-complexes of small dimension
Cohomology operations and the Deligne conjecture
The aim of this note, which raises more questions than it answers, is to study natural operations acting on the cohomology of various types of algebras. It contains a lot of very surprising partial results and examples.
Moduli spaces for fundamental groups and Link invariants derived from the lower central series.
V roce 2013 získal Abelovovu cenu Pierre Deligne za fundamentální práce svazující algebru a geometrii
Homotopy Lie algebras and fundamental groups via deformation theory
We formulate first results of our larger project based on first fixing some easily accessible invariants of topological spaces (typically the cup product structure in low dimensions) and then studying the variations of more complex invariants such as (the homotopy Lie algebra) or (the graded Lie algebra associated to the lower central series of the fundamental group). We prove basic rigidity results and give also an application in low-dimensional topology.
Abelovu cenu za rok 2011 získal John Milnor
Combinatorial differential geometry and ideal Bianchi–Ricci identities II – the torsion case
This paper is a continuation of [2], dealing with a general, not-necessarily torsion-free, connection. It characterizes all possible systems of generators for vector-field valued operators that depend naturally on a set of vector fields and a linear connection, describes the size of the space of such operators and proves the existence of an ‘ideal’ basis consisting of operators with given leading terms which satisfy the (generalized) Bianchi–Ricci identities without corrections.
Operads for -ary algebras – calculations and conjectures
In [8] we studied Koszulity of a family of operads depending on a natural number and on the degree of the generating operation. While we proved that, for , the operad is Koszul if and only if is even, and while it follows from [4] that is Koszul for even and arbitrary , the (non)Koszulity of for odd and remains an open problem. In this note we describe some related numerical experiments, and formulate a conjecture suggested by the results of these computations.
On bijectivity of the canonical transformation
Homotopy algebras via resolutions of operads
Summary: All algebraic objects in this note will be considered over a fixed field of characteristic zero. If not stated otherwise, all operads live in the category of differential graded vector spaces over . For standard terminology concerning operads, algebras over operads, etc., see either the original paper by [“The geometry of iterated loop spaces”, Lect. Notes Math. 271 (1972; Zbl 0244.55009)], or an overview [, “La renaissance des opérads”, Sémin. Bourbaki 1994/95, Exp. No. 792, Asterisque...
The integral cohomology rings of real infinite-dimensional flag manifolds
Free loop spaces and cyclohedra
It is well-known that a based space is of the weak homotopy type of a loop space iff it is a grouplike algebra over an -operad. The classical model for such an operad consists of Stasheff’s associahedra. The present paper describes a similar recognition principle for free loop spaces. Let be an operad, a -module and a -algebra. An -trace over consists of a space and a module homomorphism over the operad homomorphism given by the algebra structure on . Let be the little 1-cubes...
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