In differential topology, the techniques of Morse theory give a very direct way of visualising and possibly computing the homology of a manifold. According to the basic insights of Marston Morse, a differentiable function on a manifold will, in a typical case, reflect the topology quite directly.
Assuming a compact manifold M, such a function f, taking real values, will have a minimum value which starts M with a bowl; and a maximum value which caps off M. For example, M might be a 2-sphere that we visualise as a globe, and we take f to depend only on latitude. The 'bowl' will be at the south pole, and the 'cap' at the north pole, for suitable f.
The major contribution to topology comes in the insight that for other, more complex cases, for example a torus, we may identify the critical points of f with 'new homology'. Thus, if instead of a sphere we have a torus with its hole in a vertical plane (think of a tyre on a car), there are tangent planes not just at the bottom and top, but also at the bottom and top of the hole. These are at saddle points, not a maximum or minimum: the tangent place touches, but in one direction the surface curves down from the plane, and in another it curves up.
What Morse did was to identify this sequence, reading from the bottom up, namely bowl-saddle point-saddle point-cap with the Betti number sequence of the torus, namely 1, 2, 1, 0, 0, ... . The four critical points each contribute a unity to the Betti numbers. To express how to map a critical point to its place, he used the Hessian matrix. This is the square matrix of all second partial derivatives of f. At a stationary point this should be positive definite (minimum), or negative definite (maximum) (see technical qualifications, below). The key revelation is this: while positive definite = bowl, when the Hessian is not definite in sign, you can still read the topological information out of it. That is, you can diagonalise the Hessian, get the number of positive and negative squares from it, and this determines the topological data. The index of the Hessian is all you need to go from some type of saddle point to the correct Betti number, to which it adds.
The theory relies on 'typical' functions f. Once it is explained precisely what this means, such f can be shown to be abundant, by means of the Baire category theorem. Intuitively this means that any given f may be rendered typical by tiny perturbations.
What we need is that the critical values of f, namely the points where the derivative Df is 0, should form a finite set on M. Further we want the Hessian to be non-singular as a matrix, at those points (so that the critical values have a definite nature as saddle points). Also it is convenient to say that f takes on different values at each critical point, so that the critical points xi can be thought of as coming in the definite order given by the f(xi).
These requirements aren't so serious, once formulated. The Morse lemma says that near such a non-degenerate critical value, co-ordinates may be taken locally on M making f into just the quadratic form one gets by diagonalising the Hessian. (In other words, such critical points fall into just one family, when you allow local diffeomorphisms.)
Given the apparatus, the basic assertion of Morse theory, that the Betti numbers can be read from f, is not technically so hard.