Combining Spaces

Algebraic Topology · Axiom Academy

Discover how complex topological spaces can be built from simpler pieces. Think about how we construct complex objects in everyday life. We often start with simple pieces and glue them together. The same principle works in topology! When we glue two spaces together, a natural question arises: what happens to their fundamental groups? Many important spaces in mathematics arise from gluing simpler pieces. Click each example to explore: Complex spaces = Simple pieces + Gluing instructions. This decomposition makes seemingly impossible problems tractable. Given and , how do we compute ? The theorem provides a systematic method: decompose the space, analyze the pieces and their intersection, then use a categorical construction to recover the whole.

This is the written version of the interactive lesson above. See the full Algebraic Topology course.