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This celebrated result was proven by Blind and Mani (Aequationes Math 34(2\u20133):287\u2013297, 1987, <jats:ext-link xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"10.1007\/BF01830678\" ext-link-type=\"doi\">https:\/\/doi.org\/10.1007\/BF01830678<\/jats:ext-link>), via a non-constructive proof using topological tools from homology theory. An elegant constructive proof was given by Kalai shortly after. In their original paper, Blind and Mani asked whether their result can be extended to simplicial spheres, and a positive answer to their question was conjectured by Kalai (2009, <jats:ext-link xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"https:\/\/gilkalai.wordpress.com\/2009\/01\/16\/telling-a-simple-polytope-from-its-graph\/\" ext-link-type=\"uri\">https:\/\/gilkalai.wordpress.com\/2009\/01\/16\/telling-a-simple-polytope-from-its-graph\/<\/jats:ext-link>). In this paper, we show that Kalai\u2019s conjecture holds in the particular case of Knutson and Miller\u2019s spherical subword complexes. This family of simplicial spheres arises in the context of Coxeter groups, and is conjectured to be polytopal. In contrast, not all manifolds are reconstructible. We show two explicit examples, namely the torus and the projective plane.<\/jats:p>","DOI":"10.1007\/s00454-025-00733-6","type":"journal-article","created":{"date-parts":[[2025,5,13]],"date-time":"2025-05-13T20:26:34Z","timestamp":1747167994000},"page":"23-48","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Subword Complexes and Kalai\u2019s Conjecture on Reconstruction of Spheres"],"prefix":"10.1007","volume":"74","author":[{"given":"Cesar","family":"Ceballos","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Joseph","family":"Doolittle","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2025,5,13]]},"reference":[{"key":"733_CR1","unstructured":"Adiprasito, K.: Personal communication (2017)"},{"issue":"1\u20134","key":"733_CR2","doi-asserted-by":"publisher","first-page":"46","DOI":"10.1016\/j.aam.2010.09.006","volume":"46","author":"D Armstrong","year":"2011","unstructured":"Armstrong, D., Hersh, P.: Sorting orders, subword complexes, Bruhat order and total positivity. 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