How to Tackle the Smooth Poincare Conjecture

In principle the Ricci flow (with surgery) could also be used to prove the smooth Poincare' conjecture in dimension $4$.

There are some major problems to be overcome in this approach (problems which did not arise in dimension $3$, such as the absence of Hamilton-Ivey pinching estimates, and the new "hole-punch" singularities, as opposed to the "neck-pinch") and I know that some people are indeed working on these issues.

If you are interested, there are quite a few papers on Ricci flow on $4$-manifolds, starting with the work of Hamilton (here and here) and more recent work of Chen-Zhu (here and here) and many others.

The conjecture in question can also be thought of as the $4$-dimensional PL Poincare conjecture (because low-dimensional PL manifolds, including those of dimension $4$, carry a unique smooth structure) and this is how it is understood in most references mentioned below.

Some interesting approaches to the conjecture and its special cases can be found in several papers by Frank Quinn (some based on TQFTs and others in more classical spirit) and in some papers by Robert Craggs.

Much of the effort has been focused on the group-theoretic Andrews-Curtis conjecture, whose validity would imply that PL (or smooth) homotopy $4$-spheres given as handlebodies without $3$-handles are PL (or smoothly) standard. The latter assertion would also follow from the "Generalized Property R" conjecture. Then there's a separate industry of finding handlebody presentations of simply-connected $4$-manifolds without $3$-handles (see Problems 4.18 and 4.73 in Kirby's list, Section 6 here, Gadgil's preprint and Quinn's Corollary 3.2; note also Rasmussen's withdrawn paper arxiv.org/abs/1005.4674).

As observed by Curtis in an earlier paper (in "Topology of 3-manifolds and related topics"), every compact contractible $2$-polyhedron PL embeds in some PL homotopy $4$-sphere; so if you find one that doesn't PL embed in $S^4$, you're done with the 4D PL Poincare conjecture. This line of attack inspired some literature on PL embeddings of acyclic $2$-polyhedra in $S^4$ starting I guess with Zeeman's dunce hat paper; see this review for additional references.

There are numerous other approaches and related techniques, e.g. "Gluck twists" and "Akbulut corks". Kirby's problem list is a good source of further references prior to mid-90s; some other basic references on the Andrews-Curtis conjecture are collected here under (O1).

It seems to me your question implicitly assumes that the 4d smooth Poincaré conjecture (S4PC) is true. But if you were to take a poll of experts I think you would find that most of them suspect it is false. (See also Ryan's comment to the original question.) There are many large families of potential counterexamples to the S4PC. These are smooth 4-manifolds which are homotopy equivalent (and hence homeomorphic) to the 4-sphere, but no one knows how to prove that they are diffeomorphic to the 4-sphere.

(Akbulut and Gompf have made recent progress in showing that some, but not nearly all, of these potential counterexamples are standard 4-spheres, so I think people are more open to the idea that the S4PC might be true than they were four years ago.)

There are similarly many suspected counterexamples to the closely related Andrews-Curtis conjecture. Andrew Casson has some interesting (and unpublished) numerical evidence that the AC conjecture is false.

So if you are interested in this problem, you might want to spend as much time trying to prove that one of the potential counterexamples is an actual counterexample as you do trying to prove the conjecture.