Can anyone describe the meaning and concept of cohomology in a visual non technical tongue? especially when it comes to topology. E.g. is there any way we can recognize the cohomology of a surface or shape just by looking at?
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1Welp, I was given the analogy of a man walking a dog on a retractable leash. The dog can run around anywhere it wants and the man retracts the leash. If the leash can get wound around insurmountable obstacles (such as holes in a torus) the potential unretractable loops can be viewed as a group. As for seeing it. Look and how many holes the space has. Nonoriented spaces involve having an obstacle surmountable after a finite number if iterations. – fleablood Aug 23 '16 at 15:00
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It's (possibly singular) oriented submanifolds modulo (possibly singular) oriented bordism. You can drop oriented if you work over $\Bbb Z/2$. This idea lets me calculate the homology (and thus cohomology ring, via Poincare duality) from a picture of a surface or 3-manifold. – Aug 23 '16 at 18:03
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Possible duplicate of So what Cohomology is? – MJD Oct 16 '17 at 15:49