https://github.com/ammedmar/cpc_conjecture
Searching for counterexamples to a conjecture about local cycles in the tensor product of a simplicial complex
https://github.com/ammedmar/cpc_conjecture
Last synced: 10 months ago
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Searching for counterexamples to a conjecture about local cycles in the tensor product of a simplicial complex
- Host: GitHub
- URL: https://github.com/ammedmar/cpc_conjecture
- Owner: ammedmar
- Created: 2018-07-16T20:04:36.000Z (almost 8 years ago)
- Default Branch: master
- Last Pushed: 2019-08-16T20:47:35.000Z (almost 7 years ago)
- Last Synced: 2025-04-03T03:28:45.772Z (about 1 year ago)
- Language: Python
- Homepage:
- Size: 20.5 KB
- Stars: 0
- Watchers: 0
- Forks: 0
- Open Issues: 0
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Metadata Files:
- Readme: README.md
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README
# Constructing a counterexample for the CPC-conjecture
The script presented here constructs a counterexample for a conjecture arising in connection to the main theorem of "An Axiomatic Characterization of Steenrod cup-i products" arXiv:1810.06505
The following is a self-contained reformulation of the conjecture:
## Definitions
#### Chains
For any non-negative integer m let C_n[m] be the vector space generated by all sequences [v_0,...,v_n] with 0 <= v_0 <= ... <= v_n <= m. We take the quotient of these by the sequences with a repeated value. For simplicity we work with the field with two elements. We call such sequences n-faces. Let d : C_n[m] --> C_{n-1}[m] be the linear map that assignes to a n-face [v_0,...,v_n] the sum of all (n-1)-faces obtained by deleting one of the v_i at the time. We call C_n[m] the n-chains of the m-simplex. Putting the n-chains together we get a pair (C[m],d), the so called chains of the m-simplex.
#### Tensor product
Let (C[m] x C[m])n be the vector space generated by a pair of an i-face and and a j-face such that i+j=n subject to bilinearity relations. We call this the elements of degree n. Define (C[m] x C[m]) --> (C[m] x C[m]) by d = (d x id) + (id x d) where the d appearing on the right hand side was defined above. Putting them together we get a pair (C[m] x C[m], d), the so called tensor product of the chains of the m-simplex. An element in the kernel of d is called a cycle.
#### Unhamperedness
Let T : (C[m] x C[m]) --> (C[m] x C[m]) be the transposition map, i.e., T(a x b) = (b x a). An element
(a_1 x b_1) + ... + (a_r x b_r) in (C[m] x C[m]), with each a_i and b_i a face, is called unhampered if (a_i x b_i) is not equal to (a_j x b_j) for all i and j.
#### CPC condition
Let s_i : C_n[m] --> C_n[m-1] be the linear map induced by the only surjective order preserving function {1,...,m} --> {1,...,m-1} mapping i and i+1 to i. An element c in (C[m] x C[m]) satisfies the CPC condition if s_i(c) = 0 for all i.
## Conjecture
##### The only unhampered degree n CPC cycles in (C[m] x C[m]) for 0 < n < m-1 is 0.
The script construct counterexamples for any reasonably small n and m.