--- title: Simplicial Complexes description: Simplicial complexes are topological spaces constructed by gluing together points, line segments, triangles, and higher-dimensional tetrahedra. Learners will understand how to represent continuous spaces combinatorially and compute their simplicial homology groups. category: mathematics subcategory: algebraic-topology difficulty: beginner, intermediate, advanced url: /subject/simplicial-complexes --- # Simplicial Complexes Simplicial complexes are topological spaces constructed by gluing together points, line segments, triangles, and higher-dimensional tetrahedra. Learners will understand how to represent continuous spaces combinatorially and compute their simplicial homology groups. ## Available Resources 4 Books • 2 Courses • 5 Websites • 2 Papers ## Websites ### 1. 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. **Difficulty:** Beginner | **Price:** Free **Link:** https://ncatlab.org/nlab/show/simplicial+complex **Tags:** simplicial-complexes, simplicial-sets, geometric-realization, category-theory, algebraic-topology ### 2. Hatcher's Algebraic Topology (free PDF) **Author:** Allen Hatcher 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. **Difficulty:** Intermediate | **Price:** Free **Link:** https://pi.math.cornell.edu/~hatcher/AT/ATpage.html **Tags:** algebraic-topology, cw-complexes, homology, homotopy-theory, fundamental-group ### 3. Algebraic Topology — Cambridge Part II Lecture Notes **Author:** Oscar Randal-Williams Roughly eighty pages of Cambridge undergraduate notes whose longest chapter develops simplicial homology from chain complexes: homological algebra, explicit calculations, the Mayer-Vietoris sequence, homotopy invariance, homology of spheres and surfaces, Euler and Lefschetz numbers. **Difficulty:** Intermediate | **Language:** English | **Price:** Free **Link:** https://www.dpmms.cam.ac.uk/~or257/teaching/notes/at.pdf **Tags:** algebraic-topology, simplicial-homology, fundamental-group, mayer-vietoris, lecture-notes ### 4. Wolfram MathWorld **Author:** Eric W. Weisstein MathWorld is an online mathematics encyclopedia from Wolfram Research offering detailed, browsable articles on topics across the math spectrum, including algebra, geometry, calculus, and number theory. Each entry includes definitions, theorems, formulas, diagrams, worked examples, and links to further reading. **Difficulty:** Intermediate | **Language:** English | **Price:** Free **Link:** https://mathworld.wolfram.com **Tags:** mathematics-reference, encyclopedia, abstract-algebra, number-theory, geometry ### 5. nLab Collaborative wiki for mathematics, physics and philosophy written from a category-theoretic viewpoint, with cross-linked entries on categories, functors, adjunctions, topos theory, higher category theory and homotopy theory. Readers can look up precise definitions, examples and references for advanced structural mathematics. **Difficulty:** Advanced | **Language:** English | **Price:** Free **Link:** https://ncatlab.org **Tags:** category-theory, higher-category-theory, topos-theory, homotopy-theory, reference-wiki ## Courses ### 1. Algebraic Topology Learn simplicial complexes with N J Wildberger's Algebraic Topology course! Explore this fascinating area of math. **Difficulty:** Advanced | **Price:** Free **Link:** https://www.youtube.com/playlist?list=PLIljB45xT85D7wczwyUQdwDe2duZ7wPTf **Tags:** algebraic-topology, simplicial-complexes, simplicial-homology, euler-characteristic, lecture-series ### 2. C3.9 Computational Algebraic Topology (Oxford course notes and videos) **Author:** Vidit Nanda Oxford graduate course with eight weeks of lecture notes and recorded videos, starting from simplicial complexes, geometric realisation and simplicial maps, then nerves, carriers, homology computation, cohomology, persistent homology, cellular sheaves and discrete Morse theory. **Difficulty:** Intermediate | **Language:** English | **Price:** Free **Link:** https://people.maths.ox.ac.uk/nanda/cat/index.html **Tags:** computational-topology, persistent-homology, simplicial-complexes, cellular-sheaves, discrete-morse-theory ## Papers ### 1. An Elementary Illustrated Introduction to Simplicial Sets **Author:** Greg Friedman 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. **Difficulty:** Advanced | **Language:** English | **Price:** Free **Link:** https://arxiv.org/abs/0809.4221 **Tags:** simplicial-sets, simplicial-complexes, homotopy-theory, geometric-realization, algebraic-topology ### 2. Elementary Applied Topology — Chapter 2: Complexes **Author:** Robert Ghrist 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. **Difficulty:** Intermediate | **Language:** English | **Price:** Free **Link:** https://www2.math.upenn.edu/~ghrist/EAT/EATchapter2.pdf **Tags:** applied-topology, simplicial-complexes, vietoris-rips-complex, cech-complex, nerve-theorem ## Books ### 1. Computational Topology for Data Analysis **Author:** Tamal K. Dey, Yusu Wang 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. **Difficulty:** Advanced | **Language:** English | **Price:** Free **Link:** https://www.cs.purdue.edu/homes/tamaldey/book/CTDAbook/CTDAbook.html **Tags:** topological-data-analysis, persistent-homology, simplicial-complexes, computational-topology, discrete-morse-theory ### 2. Elements of Algebraic Topology **Author:** James R. Munkres Graduate textbook that builds algebraic topology from simplicial complexes upward, developing simplicial homology and cohomology, simplicial approximation, CW complexes, duality in manifolds, and the universal coefficient and Künneth theorems with full proofs and exercises. **Difficulty:** Advanced | **Language:** English | **Price:** Paid **Link:** https://www.amazon.com/dp/0201627280?tag=edmonddante07-20 **Tags:** algebraic-topology, simplicial-homology, cohomology, cw-complexes, simplicial-approximation ### 3. Cellular Structures in Topology **Author:** Rudolf Fritsch, Renzo A. Piccinini A book-length treatment of CW-complexes as topological spaces: closure-finiteness, the weak topology, products, subdivision, and the relationship to simplicial complexes, plus Milnor's theory of spaces of CW type and CW structures in fibration theory. Fills the exact gap Hatcher leaves: Hatcher builds CW complexes and computes with them but pushes closure-finiteness, weak-topology pathologies, products of CW complexes and the CW-versus-simplicial comparison into a short appendix. This book proves those results carefully over a full volume. **Difficulty:** Advanced | **Language:** English | **Price:** Paid **Link:** https://www.amazon.com/dp/0521063876?tag=edmonddante07-20 **Tags:** cw-complexes, cellular-structures, simplicial-complexes, point-set-topology, algebraic-topology ### 4. Elements of Algebraic Topology **Author:** James R. Munkres Textbook on homology theory built from simplicial complexes, then singular homology, cohomology, duality on manifolds, and the Künneth and universal coefficient theorems. Readers learn to compute homology groups concretely and follow the homological algebra underlying them. **Difficulty:** Advanced | **Language:** English | **Price:** Paid **Link:** https://www.amazon.com/dp/0201045869?tag=edmonddante07-20 **Tags:** algebraic-topology, homology, cohomology, simplicial-complexes, homological-algebra --- *This content is part of Dantes.io - Your Treasure Map to Knowledge* *Curated by humans at Dantes.io. Personal study use welcome; republishing this curation requires permission (team@dantes.io).* View this page online: https://dantes.io/subject/simplicial-complexes