FROM HOMOLOGY TO DE RHAM COHOMOLOGY: An Expository Journey through the Topology of Smooth Manifolds

FROM HOMOLOGY TO DE RHAM COHOMOLOGY: An Expository Journey through the Topology of Smooth Manifolds | Authorea try { document.documentElement.classList.add('js'); } catch (e) { } var _gaq = _gaq || []; _gaq.push(['_setAccount', 'G-8VDV14Y67G']); _gaq.push(['_trackPageview']); (function() { var ga = document.createElement('script'); ga.type = 'text/javascript'; ga.async = true; ga.src = ('https:' == document.location.protocol ? 'https://ssl' : 'http://www') + '.google-analytics.com/ga.js'; var s = document.getElementsByTagName('script')[0]; s.parentNode.insertBefore(ga, s); })(); Skip to main content Preprints Collections Wiley Open Research IET Open Research Ecological Society of Japan All Collections About About Authorea FAQs Contact Us Quick Search anywhere Search for preprint articles, keywords, etc. Search Search ADVANCED SEARCH SCROLL This is a preprint and has not been peer reviewed. Data may be preliminary. 2 September 2025 V1 Latest version Share on FROM HOMOLOGY TO DE RHAM COHOMOLOGY: An Expository Journey through the Topology of Smooth Manifolds Author : Anshveer Bindra 0009-0008-3593-1450 [email protected] Authors Info & Affiliations https://doi.org/10.22541/au.175683844.48037590/v1 249 views 144 downloads Contents Abstract Supplementary Material Information & Authors Metrics & Citations View Options References Figures Tables Media Share Abstract This exposition traces a path from the geometric intuition of shapes and holes to the analytic framework of de Rham cohomology. Beginning with the classical motivations of Euler, Poincaré, and Gauss-Bonnet, we see how algebraic invariants emerged to capture global topological features invisible to local geometry. After a brief review of smooth manifolds, orientation, and Stokes' theorem, we develop the language of homology theory: simplicial complexes, chains, cycles, and boundaries. These ideas lead naturally to cohomology, where algebraic duality and the cup product reveal a richer structure that connects topology to broader areas of mathematics. The second half of the exposition introduces differential forms and the generalized Stokes' theorem, culminating in de Rham's insight that analytic invariants defined by closed and exact forms coincide with topological invariants defined by singular cohomology. The de Rham theorem, Mayer-Vietoris sequence, and functorial properties are explored in detail, alongside explicit computations on familiar manifolds such as spheres, tori, punctured Euclidean space, projective space, and surfaces of higher genus. By weaving together geometry, algebra, and analysis, this work aims to present de Rham cohomology not as an isolated construction, but as a natural culmination of the search for invariants that bridge local calculus and global topology. The result is a coherent framework in which classical results find their proper home, while new insights emerge into the structure of smooth manifolds. Supplementary Material File (de_rham_cohomology.pdf) Download 866.88 KB Information & Authors Information Version history V1 Version 1 02 September 2025 Copyright This work is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License Keywords algebraic topology cohomology differential topology exposition homology theory mathematics pure mathematics Authors Affiliations Anshveer Bindra 0009-0008-3593-1450 [email protected] View all articles by this author Metrics & Citations Metrics Article Usage 249 views 144 downloads .FvxKWukQNSOunydq8rnd { width: 100px; } Citations Download citation Anshveer Bindra. FROM HOMOLOGY TO DE RHAM COHOMOLOGY: An Expository Journey through the Topology of Smooth Manifolds. Authorea . 02 September 2025. DOI: https://doi.org/10.22541/au.175683844.48037590/v1 If you have the appropriate software installed, you can download article citation data to the citation manager of your choice. Simply select your manager software from the list below and click Download. For more information or tips please see 'Downloading to a citation manager' in the Help menu . Format Please select one from the list RIS (ProCite, Reference Manager) EndNote BibTex Medlars RefWorks Direct import Tips for downloading citations document.getElementById('citMgrHelpLink').addEventListener('click', function() { popupHelp(this.href); return false; }); $(".js__slcInclude").on("change", function(e){ if ($(this).val() == 'refworks') $('#direct').prop("checked", false); $('#direct').prop("disabled", ($(this).val() == 'refworks')); }); View Options View options PDF View PDF Figures Tables Media Share Share Share article link Copy Link Copied! Copying failed. 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