What is the difference between differential topology and calculus on manifolds?

I'm trying to teach myself one and bought a book on the other. It seems to me that they both cover about the same material. This leads to the question:

What is the difference between differential topology and calculus on manifolds?

They both play off of each other nicely but, in general, differential topology is the study of the topology of a manifold given some smooth structure on that manifold. Calculus on a manifold however, is the study of the manifold and the smooth structure itself.

Perhaps the discrepency is best told by how these subjects differ in how they calculate (co)homology on the objects of their study (although this difference may appear subtle). In differential topology, the first algebraic tool you'll encounter is Morse homology which uses the differential structure (specifically the location and type of critical points) to define a sequence of groups associated to the manifold. These groups turn out to not actually depend on the differential structure on the manifold but only the topology, and so Morse homology is a topological invariant on manifolds which admit a smooth structure. This is exhibited normally by showing an isomorphism between the Morse homology and cellular homology of smooth manifolds.

In the realm of calculus on manifolds however, the first algebraic invariant you'll meet is De Rham cohomology. This sequence of groups is much more abstractly defined as a collection of equivalence classes of differential forms on the manifold and in my opinion depends more heavily on the differential structure from the beginning. Ultimately, De Rham cohomology is also a topological invariant and does not depend on the smooth structure either, so the discrepancy between how reliant on the differential structure the construction is is in their formulation. Something can also be said for their use. De Rham cohomology sheds light on much information about how integration of forms on the manifold acts. Morse homology on the other hand tells us how a smooth gradient-like vector field on our manifold will act and how critical points of various indices interact.

At the end of the day, sometimes groups of theorems don't fit in to nice boxes as we'd like them to and there can be considerable overlap between two 'subjects' (which, let's face it, is a label that humans have artificially constructed for a collection of results and 'objects'). This is definitely one case, and often it's best to study both in parallel (perhaps starting with calculus on manifolds and Riemannian geometry as they have a slightly lower entry-level).

• This is a great answer, but I have one slight issue with it. Differential geometry, in my experience, studies not just calculus but its interaction with a geometric structure imposed in addition to the smooth structure (usually a Riemannian metric).
– Neal
Aug 12, 2013 at 12:38
• @Neal Thanks, I've always had trouble with remembering the difference between calculus on manifolds and differential geometry as subjects probably because of how I was taught (alongside Lie groups, hyperbolic and Riemannian geometry with a lot of notation overlap and conflict). I've replaced 'differential geometry' in my answer with 'calculus on manifolds' which should be more accurate. Aug 12, 2013 at 12:59
• "[...] overlap between two 'subjects' (which, let's face it, is a label that humans have artificially constructed for a collection of results and 'objects')" This really is all that needed to be said, imho. +1
– Sam
Aug 12, 2013 at 13:14

Calculus on manifolds is the prerequisite to differential toplogy.
Calculus on manifolds introduces the basic notions and tools for differential topology: tangent and cotangent bundle, vector fields, differential forms, Stokes' theorem, distributions and Frobenius theorem (not the Schwartz distributions but subbundles of the tangent bundle) , DeRham cohomology, actions of Lie groups on manifolds and quotient manifolds, ...

Once these notions are reasonably mastered, the ambitious student may attack the techniques of differential topology:
$\bullet$ Vector bundles and characteristic classes
$\bullet$ Transversality and intersection theory
$\bullet$ Cobordism
...etc.

and will then arrive at the great results of differential topology:
$\bullet$ Milnor's 28 non isomorphic differential structures on the topological 7-sphere.
$\bullet$ In the opposite direction, Kervaire's topological manifolds with no differential structure at all (notice that you require differential topology to prove that a manifold has no differential structure!)
$\bullet$The uniqueness of the topology for homotopy spheres, a result due to Smale, Freedman and Perelman .
$\bullet$Moise's existence and uniqueness of a differential structure on a topological 3-manifold.

...and many, many such difficult but wonderful results which often earned their creators Fields medals .