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
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.
Regular functions over conformal quaternionic manifolds
A note on closed -cells in
The real K-ring of some CW-complexes of small dimension
-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.
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
On bijectivity of the canonical transformation
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.
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.
Deformation Theory (Lecture Notes)
First three sections of this overview paper cover classical topics of deformation theory of associative algebras and necessary background material. We then analyze algebraic structures of the Hochschild cohomology and describe the relation between deformations and solutions of the corresponding Maurer-Cartan equation. In Section we generalize the Maurer-Cartan equation to strongly homotopy Lie algebras and prove the homotopy invariance of the moduli space of solutions of this equation. In the last...
Towards one conjecture on collapsing of the Serre spectral sequence
[For the entire collection see Zbl 0699.00032.] A fibration is called totally noncohomologuous to zero (TNCZ) with respect to the coefficient field k, if is surjective. This is equivalent to saying that acts trivially on and the Serre spectral sequence collapses at . S. Halperin conjectured that for and F a 1-connected rationally elliptic space (i.e., both and are finite dimensional) such that vanishes in odd degrees, every fibration is TNCZ. The author proves this being the case...
Cyclic operads and homology of graph complexes
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