Bordism sanity check Were framed cobordism and h-cobordism invented to use for different purposes? I have been slightly confused about all the different types of cobordism. Now I am wondering if h-cobordism was invented as an alternative to classify manifolds whereas framed cobordism was invented to help determine homotopy groups of spheres. Is this true?
 A: You are pretty much correct.
Framed bordism was introduced by Pontryagin to study homotopy groups in the following paper:

L. S. Pontryagin, A classification of continuous transformations of a complex into a sphere. Dokl. Akad. Nauk SSSR, 1938.

According to Pontryagin's autobiography, he may have started working with this idea as early as 1936. Apparently Pontryagin (as well as Rokhlin) were only able to prove what is now known as the Pontryagin-Thom theorem in a few special cases. It wasn't until Thom's involvement that the Pontryagin-Thom construction as we know it today was developed and the Pontryagin-Thom theorem was proven in full.
The concept of $h$-cobordism was reportedly introduced by Thom in

R. Thom, Les classes caractéristiques de Pontryagin des variétés triangulées. Symp. Internac. Topol. Algebr., Univ. Nac. Aut. Mexico & UNESCO, 1958.

I have never seen this paper and cannot find it anywhere, so I cannot comment on what it contains. The concept is discussed in detail in Milnor's paper:

J. Milnor, Differentiable manifolds which are homotopy spheres. 1959.

In the above paper, the term $h$-cobordism had not be introduced yet. Milnor calls it "$J$-equivalence" instead. Here Milnor utilizes $J$-equivalence/$h$-cobordism classes as a nice way to categorize manifolds and studies the group of $J$-equivalence/$h$-cobordism classes of homotopy spheres. Note that here Milnor anticipates (although without all the necessary hypotheses: he asks if $J$-equivalent/$h$-cobordant not necessarily simply connected manifolds are diffeomorphic) the $h$-cobordism theorem (Problem 5 in $\S$7). Milnor gives a counterexample to Problem 5 in

J. Milnor, Two complexes which are homeomorphic but combinatorially distinct. The Annals of Mathematics, 74(3), 1961.

Here one sees that by 1963 the term "$J$-equivalent" has been replaced by the modern "$h$-cobordant."
