Different approach of Poincaré Duality

Question

I am studying Algebraic Topology on my own. First I would like to tell something about me. Mainly I am focused about Hyperbolic Geometry, Geometry and Topology of 3-manifolds, knot theory, etc. I am learning Algebraic Topology as there are some sophisticated methods of algebraic topology to attack problems in knot theory.

I want to be purely differential-geometric minded people. I found that the Poincaré duality of Hatcher's Algebraic topology book purely based on category theory and Homological Algebra. But since I am thinking differential geometric way, I found a approach to Poincaré Duality using Fourier Analysis, Elliptic Regularity, Hodge-Theory etc.

My question are as follows:

1. Since all my interests are centered about geometry (diff and hyperbolic geometry), Can I omit the algebraic method (i.e. category theory and Homological Algebr)?
2. Using the analytical approach (using elliptic regularity) of Poincare duality can I do the problems of Poincaré duality of Hatcher's book?

Please help me. Thanking in advance.

Answer 1

There will be things that you will miss this way (for instance, torsion in homology/cohomology) but for many/most geometric/analytical purposes, you can work with Poincare duality over the real numbers using differential forms. My favorite reference is

Bott, Raoul; Tu, Loring W., Differential forms in algebraic topology, Graduate Texts in Mathematics, 82. New York - Heidelberg - Berlin: Springer-Verlag. XIV, 331 p., 92 figs. (1982). ZBL0496.55001.

You can also frequently use a more primitive form of duality in terms of embeddings/immersion of smooth manifolds, which you can find in

Guillemin, Victor; Pollack, Alan, Differential topology, Englewood Cliffs, N.J.: Prentice-Hall, Inc. XVI, 222 p. (1974). ZBL0361.57001.

These two references will suffice for most applications of duality that you will encounter in differential geometry. (Caveat: Suffice, until you get into forms with values in vector bundles or sheaf cohomology. The latter will be especially useful if you study complex differential geometry.)