Cohomology of CW Complexes
Here we introduce cellular cohomology, and cellular cohomology based on finite chains. The first of these is popularly called “ordinary cohomology”. There is an intriguing double duality in this. From one point of view ordinary cohomology is considered to be the “dual” of homology as defined in Chap. 2. From another point of view, which will be made precise when we discuss Poincaré Duality in Chap. 15, cohomology based on finite chains is “dual” to homology, while ordinary cohomology is “dual” to the infinite cellular homology theory of Sect. 11.1.
As in Chap. 11, having the two cohomology theories enables us to define cohomology at the end of a CW complex.
KeywordsExact Sequence Short Exact Sequence Cohomology Theory Cochain Complex Path Component
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