Hello everyone,
As you know, there are some geometrical meaning of the ranks of the homology groups ( the betti numbers). For example:
b0 is the number of connected components
b1 is the number of one-dimensional or "circular" holes
b2 is the number of two-dimensional "voids" or "cavities" and so on.
Are there similar geometric meanings for the cohomology groups?
Thank you in advance for your interest?
This is a great question.
Let K be a simplicial complex. Recall that we can construct a cohomology group by transforming chain groups into groups of homomorphisms and boundary maps into their dual homomorphisms. This is an observation from H. Edelsbrunner and J.L. Harer, p.104 im
Book Computational Topology: An Introduction
More to the point, see
SCHUBERT DECOMPOSITION FOR MILNOR FIBERS OF THE VARIETIES OF SINGULAR MATRICES
https://arxiv.org/pdf/1801.06893.pdf
In this paper, Milnor fibres are used to explore cohomology classes (see, e.g., p. 32).
Dear Dr. Peters,
Thank you for your polite answer, I believe it will be very useful for me. But I'm looking for more concrete meanings.
- Badges
- Science topic
- Similar topics
- Biological Science
- Bioecology
- Systems Ecology
- Landscape Ecology
- Connectivity