#### mathwonk

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Doodle Bob is the expert here but I will make some remarks subject to his review.

The concept of second derivatives which I aligned mostly with diff geom, do indeed have significant applications to algebraic topology, via morse theory.

Morse theory considers the second derivative matrix of a real valued function at a critical point, especially when that matrix of second derivatives is non singular. The subsequent identification of critical points as saddle points or maxs or mins is actually central to understanding the global topology of the manifold.

E.g. a torus is distinguished among compoact oriented 2 manifolds by having a function with a max a min and 2 saddle points.

Morse theory allows one to construct a CW complex reflecting the homotopy of the a manifold, just from knowing the critical poinmts of one non degenerate function, with its second derivatives at those points.

Moving on to Riemannian structures, it is also useful to introduce a metric to measure non topological entities like length, but which turn out to have topological implications.

E.g. in the study of homotopy groups, i.e. loop groups, it is useful to introduce a length for these loops, in order to discern "shortest" length loops or geodesics.

These give critical points for the length fucntion on the space of loops and allow one to determine the homotopy of the loop space on the manifold.

Some sphere e.g. can be realized as loops spaces of smaller spheres, via the suspension construction of freudenthal, and thus riemannian geometry yeil;ds results on the homotopy groups of spheres, a purely topological question.

The deep periodicity theorems of bott on the stable homotopy of classical groups is another consequence of these differential geometry methods.

If one tries to read about this say in milnor's book on morse theory, he/she may well wish he had followed doodle bob's advice and learned an intuitive version of curvature and differential geometry ideas from an elementary book first, like millman parker, or do carmo.

so my original comments about alg top not using diff geom, did not go far enough into alg top it seems. just my ignorance. it is always better to know something than not to.