What is algebraic topology?

Algebraic topologists decompose spaces into cells (simplices) with assigned orientations that determine how boundaries are calculated. These building blocks enable systematic computation of topological invariants through linear algebra.

The circle provides the simplest non-trivial example for computing homology groups. Topologists use it to illustrate the fundamental concepts before tackling higher-dimensional spaces like the torus or sphere.

Algebraic topologists generalize the concept of holes to arbitrary dimensions using the nth homology group Hn(X). An n-dimensional hole is detected by n-cells with boundary zero that aren’t boundaries of (n+1)-cells.

Topologists define holes using the fundamental principle that closed loops in spaces with holes cannot bound any 2-dimensional region within that space. An ant on a torus surface can verify the hole’s existence through loop analysis.

Mathematicians in algebraic topology developed simplicial homology to rigorously count holes in topological spaces using group theory and linear algebra techniques. The method decomposes spaces into simple building blocks called simplices.

Mathematicians developed two equivalent definitions of homology: simplicial homology (computation-friendly) and singular homology (proof-friendly). Though defined differently, they produce isomorphic groups for any space, a non-obvious theorem.

The torus (donut surface) demonstrates how homology detects multiple independent holes in a single space. Topologists construct it by gluing opposite edges of a rectangle, creating a cell decomposition suitable for homology computation.