ADDITIONAL OPERATIONS ON THE HOCHSCHILD COMPLEX FOR A SIMPLICIAL SET

Ladoshkin Mikhail Vladimirovich, Candidate of physical and mathematical sciences, associate professor, sub-department of mathematics and mathematics teaching technique, Mordovia State Pedagogical Institute named after M. E. Evsevyev (11a Studencheskaya street, Saransk, Russia), michldosh@gmail.com

512.662.1

Background. The process of creating analogues of algebraic structures, which are sustained during the transition to the homotopy case, has recently been a topical one in algebraic topology. Earlier, the author built a stable homotopy analogue of a simplicial object. For this object the researcher proved the existence theorem, and the proof of the theorem is constructive, as well as comparatively analyzed the obtained results with V. Smirnov’s results. The next step in the study of analogues with stable homotopy is construction of the Hochschild complex and investigation of additional operations at the complex.
Materials and methods. All major approvals, the design and proof of theorems are presented over a field of characteristic 2, i.e. over Z2. This technique is often used in algebraic topology due to simplicity of calculations and reduced complexity of designs. In addition, in most cases, the results, obtained for fields of characteristic 2, are true for the case of arbitrary fields as well.
Results. The article describes a basic set of the Hochschild complex, defines the differential, proves fulfilment of the d² = 0 condition. The work also considers additional operations, introduced on the Hochschild complex, studies their properties and relationships with the differential.
Conclusions. Additional structures, introduced on the Hochschild complex for simplicial sets, will allow to use the complex to describe a possibility of non-trivial continuation of a simplicial set to an analogue with stable homotopy.

simplicial object, homology, homotopic stability, Hochschild complex, higher simplicial operators.

1. Kadeishvili T. V. Uspekhi matematicheskikh nauk [Progress of mathematical sciences]. 1980, vol. 35, no. 3 (213), pp. 183–188.
2. Ladoshkin M. V. Izvestiya vysshikh uchebnykh zavedeniy. Povolzhskiy region. Fizikomatematicheskie nauki [University proceedings. Volga region. Physical and mathematical sciences]. 2011, no. 2 (18), pp. 80–89.
3. Smirnov V. A. Matematicheskie zametki [Mathematical notes]. 1999, vol. 66, no. 6, pp. 913–919.
4. Ladoshkin M. V. Matematicheskie zametki [Mathematical notes]. 2006, vol. 79, no. 5, pp. 717–728.
5. Ladoshkin M. V. Matematicheskie zametki [Mathematical notes]. 2007, vol. 82, no. 2, pp. 224–231
6. May J. P. Simplicial objects in algebraic topology. Van Nostred, Math. Studies. 1967, vol. 11, 162 p.