Elementary Applied Topology — Chapter 2: Complexes
by Robert Ghrist · University of Pennsylvania
The best entry point for the topic: it defines simplicial complexes precisely, then immediately shows the constructions (Rips, Čech, nerve, flag) that make them useful, with 268 figures across the book keeping the geometry visible. Free chapter from Ghrist's Elementary Applied Topology covering simplicial and cell complexes, nerves, Vietoris-Rips, Čech, flag and witness complexes. Heavily illustrated, with worked examples drawn from sensor networks, point clouds, phylogenetics and neuroscience.
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Wolfram MathWorld
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nLab: Simplicial Complex
Community-maintained nLab reference entry defining simplicial complexes combinatorially, with examples from graphs, nerves, posets and buildings. It contrasts them with simplicial sets, treats them as presheaves on finite sets, and covers geometric realisation and triangulable spaces.
Algebraic Topology
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Allen Hatcher's standard graduate textbook, freely downloadable from Cornell, covering fundamental groups, homology, cohomology and homotopy theory, with CW complexes introduced in Chapter 0 and used throughout. Readers learn to build spaces from cells and compute their algebraic invariants.
An Elementary Illustrated Introduction to Simplicial Sets
Expository paper explaining how simplicial sets generalise simplicial complexes, covering face and degeneracy maps, geometric realisation, and the simplicial route into homotopy theory. Illustrated throughout, written for readers who already know basic algebraic topology.
Computational Topology for Data Analysis
Cambridge University Press text whose free author-hosted pre-publication PDF treats simplicial complexes as computable data structures: Rips, Čech and Delaunay constructions, boundary matrices, matrix-reduction homology algorithms, then persistence, Reeb graphs and discrete Morse theory.