Some calculations of the homology of loop spaces of moment-angle complexes using Hall words

Some calculations of the homology of loop spaces of moment-angle complexes using Hall words

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Li Cai, Jiaotong-Liverpool University

Online Talk

Zoom link: https://princeton.zoom.us/j/92116764865

Passcode: 114700

 

Based on the previous work of Panov and Ray, Grbic, Panov, Theriault and Wu obtained a set of generators of the homology of the loop space of a moment-angle complex associated to a flag complex. It turns our that this Hopf algebra is the universal envelop algebra of its primitive elements, which is the kernel of a morphism of Lie algebras.

In this talk I will follow their approach using a more detailed calculation, with the help of Hall words. As a result, we are able to express a class of relations of the generators above, which comes from the remainders after exchanging two letters in a cycle. In the fundamental group of the corresponding real moment-angle complex (associated to the same flag complex), Panov and Veryovkin showed a similar set of generators also works. We will show that in this case, the class of relations above is sufficient, with the help of the Seifert-Van Kampen Theorem: the exchanging of letters in a cycle corresponds to the homotopy after attaching a 2-cell. If a suitable version of Seifert-Van Kampen Theorem in higher dimensions is true, then we will get all possible relations.

This work is still in progress, and several related topics will also be discussed.