Abstract
Differential forms play a role in various realms of mathematics. Here, we work mainly from the perspective of algebraic topology, namely, we work with de Rham cohomology.
We discuss the Poincaré Lemma, the Brouwer Fixed-Point Theorem, the Jordan-Brouwer Separation Theorem, and the Stokes’s Integral Formula. There are numerous exercises concerning differential forms and cohomology.
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Notes
- 1.
Johann Friedrich Pfaff (1765–1825).
- 2.
Georges de Rham (1903–1990).
- 3.
Enrico Betti (1823–1892).
- 4.
Walther Mayer (1887–1948), Leopold Vietoris (1891–2002).
- 5.
Luitzen Egbertus Jan Brouwer (1881–1966).
- 6.
Henri Léon Lebesgue (1875–1941).
- 7.
Sir George Gabriel Stokes (1819–1903).
- 8.
Guido Fubini (1879–1943).
References
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R. Bott, L. Tu, Differential Forms in Algebraic Topology. Graduate Texts in Mathematics, vol. 82 (Springer, Berlin, 1982), xiv+331 pp.
S. Lang, Real Analysis (Addison-Wesley, Reading, 1969), xi+476 pp.
I.M. Singer, J.A. Thorpe, Lecture Notes on Elementary Topology and Geometry, Reprint of the 1967 edition. Undergraduate Texts in Mathematics (Springer, Berlin, 1976), viii+232 pp.
M. Spivak, A Comprehensive Introduction to Differential Geometry. Vol. I, 2nd edn. (Publish or Perish, Berkeley, 1979), xiv+668 pp.
H. Whitney, Geometric Integration Theory (Princeton University Press, Princeton, 1957), xv+387 pp.
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Ballmann, W. (2018). Differential Forms and Cohomology. In: Introduction to Geometry and Topology. Compact Textbooks in Mathematics. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-0983-2_3
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DOI: https://doi.org/10.1007/978-3-0348-0983-2_3
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