--- title: Covering Spaces description: Covering spaces are topological spaces that locally project homeomorphically onto a base space. Learners will understand the relationship between covering spaces, fundamental groups, and deck transformations, enabling them to compute fundamental groups of complex spaces. category: mathematics subcategory: algebraic-topology difficulty: beginner, intermediate, advanced url: /subject/covering-spaces --- # Covering Spaces Covering spaces are topological spaces that locally project homeomorphically onto a base space. Learners will understand the relationship between covering spaces, fundamental groups, and deck transformations, enabling them to compute fundamental groups of complex spaces. ## Available Resources 2 Books • 2 Courses • 3 Websites • 1 Papers ## Courses ### 1. Algebraic Topology Learn about covering spaces in algebraic topology with this course by N J Wildberger. Explore topological concepts and their applications. **Difficulty:** Advanced | **Price:** Free **Link:** https://www.youtube.com/playlist?list=PL41FDABC6AA085E78 **Tags:** algebraic-topology, covering-spaces, fundamental-group, homotopy, surfaces ### 2. Topology and Groups **Author:** Jonny Evans Flipped-classroom UCL course with short pre-recorded videos and matching written notes, building from the fundamental group through van Kampen to a full section on covering spaces, covering transformations, and the Galois theory of coverings. One of the few free resources giving covering spaces its own multi-hour block plus a dedicated section on the Galois correspondence and covering transformations, in paired video-plus-notes form. **Difficulty:** Intermediate | **Language:** English | **Price:** Free **Link:** https://jde27.uk/tg/ **Tags:** algebraic-topology, fundamental-group, van-kampen-theorem, covering-spaces, galois-correspondence ## Websites ### 1. 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 ### 2. 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 ### 3. 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 ## Books ### 1. Algebraic Topology: An Introduction (GTM 56) **Author:** William S. Massey Graduate text built around the fundamental group rather than homology, covering surface classification, free products, and then covering spaces: the lifting criterion, the universal cover, the classification of coverings, and deck transformation groups. Unlike general algebraic topology texts that treat coverings as a chapter en route to homology, this book spends its whole length on the fundamental group and covering spaces, making it the deepest single-topic treatment available. **Difficulty:** Advanced | **Language:** English | **Price:** Paid **Link:** https://www.amazon.com/dp/0387902716?tag=edmonddante07-20 **Tags:** algebraic-topology, fundamental-group, covering-spaces, deck-transformations, surfaces ### 2. Topology (2nd Edition) **Author:** James R. Munkres Standard undergraduate topology text whose second part develops the fundamental group, covering spaces, lifting theorems, the classification of covering spaces, and deck transformations at a deliberate pace, with all point-set prerequisites contained in the same volume. **Difficulty:** Intermediate | **Language:** English | **Price:** Paid **Link:** https://www.amazon.com/dp/0131816292?tag=edmonddante07-20 **Tags:** topology, point-set-topology, fundamental-group, covering-spaces, lifting-theorems ## Papers ### 1. Part II Algebraic Topology (Cambridge lecture notes) **Author:** Dexter Chua (lectures by Henry Wilton) A compact, exam-scoped treatment reaching the classification theorem in about fifteen pages, so a learner can see the whole covering-space-to-subgroup dictionary in one sitting before committing to a full textbook. Cambridge Part II notes taken by Dexter Chua from Henry Wilton's Michaelmas 2015 lectures. Chapter three covers covering spaces, path and homotopy lifting, the fundamental group of the circle, universal covers, and the Galois correspondence. **Difficulty:** Intermediate | **Language:** English | **Price:** Free **Link:** https://dec41.user.srcf.net/notes/II_M/algebraic_topology.pdf **Tags:** algebraic-topology, covering-spaces, universal-cover, galois-correspondence, homotopy-lifting --- *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/covering-spaces