{"id":15914299,"url":"https://github.com/ammedmar/cpc_conjecture","last_synced_at":"2025-07-29T05:05:22.611Z","repository":{"id":141209508,"uuid":"141188386","full_name":"ammedmar/cpc_conjecture","owner":"ammedmar","description":"Searching for counterexamples to a conjecture about local cycles in the tensor product of a simplicial complex","archived":false,"fork":false,"pushed_at":"2019-08-16T20:47:35.000Z","size":21,"stargazers_count":0,"open_issues_count":0,"forks_count":0,"subscribers_count":0,"default_branch":"master","last_synced_at":"2025-04-03T03:28:45.772Z","etag":null,"topics":[],"latest_commit_sha":null,"homepage":"","language":"Python","has_issues":true,"has_wiki":null,"has_pages":null,"mirror_url":null,"source_name":null,"license":null,"status":null,"scm":"git","pull_requests_enabled":true,"icon_url":"https://github.com/ammedmar.png","metadata":{"files":{"readme":"README.md","changelog":null,"contributing":null,"funding":null,"license":null,"code_of_conduct":null,"threat_model":null,"audit":null,"citation":null,"codeowners":null,"security":null,"support":null,"governance":null,"roadmap":null,"authors":null,"dei":null,"publiccode":null,"codemeta":null}},"created_at":"2018-07-16T20:04:36.000Z","updated_at":"2020-01-22T16:21:41.000Z","dependencies_parsed_at":null,"dependency_job_id":"14f46d9b-5246-463f-bd42-57448c771e24","html_url":"https://github.com/ammedmar/cpc_conjecture","commit_stats":{"total_commits":27,"total_committers":2,"mean_commits":13.5,"dds":"0.37037037037037035","last_synced_commit":"cc0afe75a49dfdbebbbc2954c45bf618853639c9"},"previous_names":[],"tags_count":0,"template":false,"template_full_name":null,"purl":"pkg:github/ammedmar/cpc_conjecture","repository_url":"https://repos.ecosyste.ms/api/v1/hosts/GitHub/repositories/ammedmar%2Fcpc_conjecture","tags_url":"https://repos.ecosyste.ms/api/v1/hosts/GitHub/repositories/ammedmar%2Fcpc_conjecture/tags","releases_url":"https://repos.ecosyste.ms/api/v1/hosts/GitHub/repositories/ammedmar%2Fcpc_conjecture/releases","manifests_url":"https://repos.ecosyste.ms/api/v1/hosts/GitHub/repositories/ammedmar%2Fcpc_conjecture/manifests","owner_url":"https://repos.ecosyste.ms/api/v1/hosts/GitHub/owners/ammedmar","download_url":"https://codeload.github.com/ammedmar/cpc_conjecture/tar.gz/refs/heads/master","sbom_url":"https://repos.ecosyste.ms/api/v1/hosts/GitHub/repositories/ammedmar%2Fcpc_conjecture/sbom","host":{"name":"GitHub","url":"https://github.com","kind":"github","repositories_count":267632879,"owners_count":24118749,"icon_url":"https://github.com/github.png","version":null,"created_at":"2022-05-30T11:31:42.601Z","updated_at":"2022-07-04T15:15:14.044Z","status":"online","status_checked_at":"2025-07-29T02:00:12.549Z","response_time":2574,"last_error":null,"robots_txt_status":"success","robots_txt_updated_at":"2025-07-24T06:49:26.215Z","robots_txt_url":"https://github.com/robots.txt","online":true,"can_crawl_api":true,"host_url":"https://repos.ecosyste.ms/api/v1/hosts/GitHub","repositories_url":"https://repos.ecosyste.ms/api/v1/hosts/GitHub/repositories","repository_names_url":"https://repos.ecosyste.ms/api/v1/hosts/GitHub/repository_names","owners_url":"https://repos.ecosyste.ms/api/v1/hosts/GitHub/owners"}},"keywords":[],"created_at":"2024-10-06T17:01:41.038Z","updated_at":"2025-07-29T05:05:22.586Z","avatar_url":"https://github.com/ammedmar.png","language":"Python","funding_links":[],"categories":[],"sub_categories":[],"readme":"# Constructing a counterexample for the CPC-conjecture \n\nThe 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 \n\nThe following is a self-contained reformulation of the conjecture:\n\n## Definitions\n#### Chains\nFor any non-negative integer m let C_n[m] be the vector space generated by all sequences [v_0,...,v_n] with 0 \u003c= v_0 \u003c= ... \u003c= v_n \u003c= 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] --\u003e 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.\n\n#### Tensor product\nLet (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]) --\u003e (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. \n\n#### Unhamperedness\nLet T : (C[m] x C[m]) --\u003e (C[m] x C[m]) be the transposition map, i.e., T(a x b) = (b x a). An element \n(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.\n\n#### CPC condition\nLet s_i : C_n[m] --\u003e C_n[m-1] be the linear map induced by the only surjective order preserving function {1,...,m} --\u003e {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. \n\n## Conjecture\n##### The only unhampered degree n CPC cycles in (C[m] x C[m]) for 0 \u003c n \u003c m-1 is 0.\n\nThe script construct counterexamples for any reasonably small n and m.\n","project_url":"https://awesome.ecosyste.ms/api/v1/projects/github.com%2Fammedmar%2Fcpc_conjecture","html_url":"https://awesome.ecosyste.ms/projects/github.com%2Fammedmar%2Fcpc_conjecture","lists_url":"https://awesome.ecosyste.ms/api/v1/projects/github.com%2Fammedmar%2Fcpc_conjecture/lists"}