An Invitation to Obstruction Bundle Gluing Through Morse Flow Lines

TL;DR

The paper develops Morse-theoretic Obstruction Bundle Gluing (OBG) to glue nontransverse gradient flowlines, detailing a finite-dimensional obstruction bundle and a corresponding obstruction section whose zeros parametrize gluings. It provides explicit combinatorial criteria for when a 3-component index-2 flowline with a central nontransverse piece glues without perturbation, and when a 2-level index-1 broken flowline glues after perturbing the metric to a Morse–Smale pair (t-gluing). The upright torus example illustrates both 0-gluing and t-gluing, and ties the Morse-theoretic gluing picture to boundary-obstructed phenomena described by Kronheimer–Mrowka. The work shows that the Morse complex can be defined and computed via gluings that persist under small Morse-Smale perturbations, linking Morse homology to nearby Morse–Smale data and providing a detailed, accessible template for applying OBG techniques in Floer-type settings.

Abstract

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.

Figures (11)

• Figure 1: Height function on the upright torus and flowlines for the metric induced by restricting the standard Euclidean metric on $\mathbb{R}^3$.
• Figure 2: Linearized obstruction sections depending on the asymptotic of $u_-, u_0$, and $u_+$. For a fixed parameter $R_0$ large enough, the $x$-axis plots one of the gluing parameters $R_0^-$ and the $y$-axis plots the "linearized" obstruction section $\mathfrak{s}_0$ computed with parameters $(R_0^-, R_0 - R_0^+)$.
• Figure 3: Height function on annulus in $\mathbb{R}^2$ is a Morse function with flowlines (with respect to the restriction of the standard Euclidean metric on $\mathbb{R}^2$) that are either entirely contained in the boundary or disjoint from the boundary. The flowlines $u_0^r$ and $u_0^l$ are "boundary obstructed" as in Kronheimer-Mrowka_2007. The three component flowlines $(u_-, u_0^l, u_+)$ and $(u_-, u_0^r, u_+)$ are $0$-gluable.
• Figure 4: $t$-gluing on the torus via perturbing the metric the torus. Refer to Theorem \ref{['thm: t-gluing']} for how to read off which trajectories glue and which do not.
• Figure 5: Vectors $v_l$ and $v_r$ generate $\mathrm{coker} D_{u_0^l}$ and $\mathrm{coker} D_{u_0^r}$, resp.

Theorems & Definitions (71)

• Lemma 3.1
• Proposition 3.2
• Proposition 4.1: Proposition 10.2.8 in Audin-Damian_2014
• Proposition 4.2: Proposition 10.2.8 in Audin-Damian_2014
• Example 4.3
• Remark 4.4
• Proposition 5.2
• Proposition 5.3
• Remark 5.4
• Theorem 6.1