The main ingredients of Euclid's geometry- lines, angles, circles, squares and so on- are all related to measurement.
Projective transformations can change lengths, and they can change angles, and is more flexible. Topology is a more flexible kind of geometry which can be deformed or distorted in extremely convoluted ways.
Homology studies the relations between regions in the manifold and their boundaries. Homotopy looks at what happens to closed loops in the manifold as the loops are deformed. It tackles the question: What shape is this thing?
Projective transformations can change lengths, and they can change angles, and is more flexible. Topology is a more flexible kind of geometry which can be deformed or distorted in extremely convoluted ways.
Homology studies the relations between regions in the manifold and their boundaries. Homotopy looks at what happens to closed loops in the manifold as the loops are deformed. It tackles the question: What shape is this thing?