An Invitation to Obstruction Bundle Gluing Through Morse Flow Lines

We adapt “Obstruction Bundle Gluing (OBG)” techniques from Hutchings and Taubes (arxiv: 0701300, 0705.2074) to Morse theory. We consider Morse function-metric pairs with gradient flowlines that have nontrivial yet well-controlled cokernels (i.e., the gradient flowlines are not transversely cut out). We investigate (i) whether these nontransverse gradient flowlines can be glued to other gradient flowlines and (ii) the bifurcation of gradient flowlines after we perturb the metric to be Morse-Smale. For the former, we show that certain three-component broken flowlines with total Fredholm index 2 can be glued to a one-parameter family of flowlines of the given metric if and only if an explicit (essentially combinatorial, and straightforward to verify) criterion is satisfied. For the latter, we provide a similar combinatorial criterion of when certain 2-level broken Morse flowlines of total Fredholm index 1 glue to index 1 gradient flowlines after perturbing the metric. Our primary example is the ``upright torus," which has a flowline between the two index-1 critical points.