Some non‐trivial cycles of the space of long embeddings detected by configuration space integral invariants using g‐loop (g=2,3) graphs

In this paper, we give some non‐trivial geometric cycles of the space of long embeddings modulo immersions. We construct a class of cycles from specific chord diagrams associated with 2‐loop or 3‐loop hairy graphs. To detect these cycles, we use cocycles obtained by the 2‐loop or 3‐loop part of modified configuration space integrals using a modified Bott–Cattaneo–Rossi graph complex. We show the non‐triviality of the cycles by a pairing argument, which is reduced to pairing of graphs with the chord diagrams. As a corollary of the 2‐loop part, we provide an alternative proof of the non‐finite generation of the ‐th rational homotopy group of the space of long embeddings of codimension two, which Budney–Gabai and Watanabe first established. We also show the non‐finite generation of the ‐th homotopy group by using the 3‐loop part.