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Tropical moduli spaces can be identified with boundary complexes for $\\mathcal{M}_{g,n}$, as shown by Abramovich-Caporaso-Payne, so their reduced rational homology encodes top-weight rational cohomology of the complex moduli spaces $\\mathcal{M}_{g,n}$. We prove that $M^{\\mathrm{trop}}_{2,n}[1]$ is an $n$-connected topological space whose reduced integral homology is supported in the top two degrees only. We compute the reduced Euler characteristic of $M^{\\mathrm{trop}}_{g,n}[1]$ for"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1507.03878","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2015-07-14T15:04:31Z","cross_cats_sorted":["math.AG"],"title_canon_sha256":"e4d201ecb2136de2b84a1bffdf36c30799d25674528233a7cef37f49a867d9c4","abstract_canon_sha256":"c8ca02915f954dae0518175027e7b70c28f4d006c5b23dba54839fef8e2b1d34"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:36:52.345186Z","signature_b64":"UFG38PEAjM2sYO2N3oYk9KwyOsaBYeQQrCkvipKHVdOSudYvuMwBgrbocVEDjPY3Pq+rXQCNyv6ozCQLGlpqAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"3292d40027b87fab25d28b710f9f86de0ba0205f11aa44a4c27f5cba855269e2","last_reissued_at":"2026-05-18T01:36:52.344511Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:36:52.344511Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Topology of the tropical moduli spaces $M_{2,n}$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AG"],"primary_cat":"math.CO","authors_text":"Melody Chan","submitted_at":"2015-07-14T15:04:31Z","abstract_excerpt":"We study the topology of the link $M^{\\mathrm{trop}}_{g,n}[1]$ of the tropical moduli spaces of curves when g=2. Tropical moduli spaces can be identified with boundary complexes for $\\mathcal{M}_{g,n}$, as shown by Abramovich-Caporaso-Payne, so their reduced rational homology encodes top-weight rational cohomology of the complex moduli spaces $\\mathcal{M}_{g,n}$. We prove that $M^{\\mathrm{trop}}_{2,n}[1]$ is an $n$-connected topological space whose reduced integral homology is supported in the top two degrees only. 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