Arguably, the value of algebraic topology is measured by the extent to which it answers questions which arise more geometrically. These classical topics were, before the development of the full algebraic tools, often treated as questions involving numbers. There are many useful examples of homology theories. Classic applications include the nonexistence of retractions of disks to their boundary and, as a consequence, the Brouwer Fixed-Point Theorem.
Homology theories are the natural settings for intersection theory.
Essentially a topic from algebraic geometry, this asks for a method of determining the greatest extent to which two subspaces of the space can avoid each other. For example, the singular cohomology rings can be further given the structure of an algebra over a key, and complicated, ring known as the Steenrod algebra. The tools for combining these data are spectral sequences. While the general theory of their development is more properly a topic in homological algebra, certain spectral sequences are very commonly used in algebraic topology. Indeed, spheres are in some sense the building blocks of topology, from the perspective of homotopy theory, and so their homotopy groups are particularly significant.