Cell complexes, fundamental groups and covering spaces
Cell complexes, fundamental groups and covering spaces as used by the courses on group rings and classifying spaces: homotopy equivalences of cell complexes, trees, triangulations and simplicial approximation; fundamental groups of graphs and of complexes with attached cells, presentation complexes; universal covers with lifted cells and deck transformations, covers for normal subgroups, and cellular chains over the group ring with the Cayley complex.
Three lessons with proofs, examples and exercises with solutions. Written and self-checked by Claude Opus 5.5 (Anthropic); independent review is not complete.
- Cell complexes and homotopy equivalences
- Fundamental groups of graphs and two-complexes
- Universal covers and cellular chains
Theorems and assumptions · Reader and editable sources · Course record
Prerequisites
- Grassmannians and classifying maps · Characteristic classes
- Frame fields and primary obstructions · Characteristic classes
- Words, free groups and Cayley graphs (Van Kampen diagrams)
- The fundamental group and fundamental groupoid, the Seifert-van Kampen theorem, covering spaces and the universal cover, singular and cellular homology: core course Algebraic Topology, with the statements cited in the lessons.
- Quotient topology, compactness and connectedness; groups, normal subgroups, group rings and free modules.
Used in
- Wall's D(2) problem and twisted homology (A counterexample to Wall's D(2) problem)
- Aligned presentations of the cycle module (A finitely generated counterexample to the Eilenberg-Ganea conjecture)
- Cohomological and geometric dimension (A finitely generated counterexample to the Eilenberg-Ganea conjecture)
- From a kernel relation to a solution (Nonsingular systems of equations over groups)
- From a relation to a planar surface (The Kervaire conjecture)
- Cone pictures and asphericity (A torsion-free group algebra with zero divisors)
- Torsion-freeness and the zero divisors (A torsion-free group algebra with zero divisors)