====References for Math 580A, Topological Combinatorics, Fall 2017==== A big chunk of the course will cover Matousek's book [[https://link.springer.com/book/10.1007%2F978-3-540-76649-0|Using the Borsuk-Ulam Theorem.]] Other course material is coming from: Welker, V., Ziegler, G., Zivaljevic, R., [[http://www.mi.fu-berlin.de/math/groups/discgeom/ziegler/Preprintfiles/041PREPRINT.pdf|Homotopy colimits -- comparison lemmas for combinatorial applications]] Barmak,J. [[http://www.maths.ed.ac.uk/~aar/papers/barmak2.pdf|Algebraic topology of finte spaces and applications]] ----- ==Some surveys of topological combinatorics:== Björner, A. [[http://ftp.cs.wisc.edu/pub/users/prem/for-prem/Comp.%20topology/bjorner-topological-methods-1995.pdf|Topological methods.]] Handbook of combinatorics, Vol. 1, 2, 1819–1872, Elsevier Sci. B. V., Amsterdam, 1995. Karasëv, R. N., [[http://www.ams.org/mathscinet/search/publdoc.html?arg3=&co4=AND&co5=AND&co6=AND&co7=AND&dr=all&pg4=AUCN&pg5=AUCN&pg6=TI&pg7=ALLF&pg8=ET&r=1&review_format=html&s4=&s5=&s6=topological%20methods%20in%20combinatorial%20geometry&s7=&s8=All&sort=Newest&vfpref=html&yearRangeFirst=&yearRangeSecond=&yrop=eq|Topological methods in combinatorial geometry]] Uspekhi Mat. Nauk 63 (2008), no. 6(384), 39--90; translation in Russian Math. Surveys 63 (2008), no. 6, 1031–1078 Zivaljevic, R., [[https://www.csun.edu/~ctoth/Handbook/chap21.pdf|Topological Methods In Discrete Geometry]], preliminary version ------ ==Applications of Borsuk-Ulam and equivariant topology== Anderson, Laura; Wenger, Rephael [[http://www.ams.org/mathscinet/search/publdoc.html?arg3=&co4=AND&co5=AND&co6=AND&co7=AND&dr=all&pg4=AUCN&pg5=AUCN&pg6=TI&pg7=ALLF&pg8=ET&r=1&review_format=html&s4=anderson&s5=wenger&s6=&s7=&s8=All&sort=Newest&vfpref=html&yearRangeFirst=&yearRangeSecond=&yrop=eq|Oriented matroids and hyperplane transversals.]] Adv. Math. 119 (1996), no. 1, 117–125. Bárány, I.; Lovász, L., [[http://www.renyi.hu/~barany/cikkek/10.pdf|Borsuk's theorem and the number of facets of centrally symmetric polytopes. ]] Acta Math. Acad. Sci. Hungar. 40 (1982), no. 3-4, 323–329. Blagojević, Pavle V. M.; Ziegler, Günter M. [[https://arxiv.org/abs/1202.5504|Convex equipartitions via equivariant obstruction theory.]] Israel J. Math. 200 (2014), no. 1, 49–77. Blagojević, Pavle V. M., Ziegler, Günter M. [[https://arxiv.org/abs/1605.07321|Beyond the Borsuk-Ulam theorem: The topological Tverberg story]], arxiv preprint Dobbins, Michael [[https://arxiv.org/abs/1312.4411| A point in a nd-polytope is the barycenter of n points in its d-faces.]] Invent. Math. 199 (2015), no. 1, 287–292. Lovász, L.[[http://www.ams.org/mathscinet/search/publdoc.html?arg3=&co4=AND&co5=AND&co6=AND&co7=AND&dr=all&pg4=AUCN&pg5=AUCN&pg6=TI&pg7=ALLF&pg8=ET&r=1&review_format=html&s4=&s5=&s6=Kneser%27s%20conjecture%2C%20chromatic%20number%2C%20and%20homotopy&s7=&s8=All&sort=Newest&vfpref=html&yearRangeFirst=&yearRangeSecond=&yrop=eq| Kneser's conjecture, chromatic number, and homotopy. ]] J. Combin. Theory Ser. A 25 (1978), no. 3, 319–324. Lovász, László; Schrijver, Alexander, [[http://www.ams.org/journals/proc/1998-126-05/S0002-9939-98-04244-0/S0002-9939-98-04244-0.pdf |A Borsuk theorem for antipodal links and a spectral characterization of linklessly embeddable graphs.]] ( Proc. Amer. Math. Soc. 126 (1998), no. 5, 1275–1285. Živaljević, Rade T.(YU-SAOS); Vrećica, Siniša T, [[http://www.ams.org/mathscinet/search/publdoc.html?r=1&pg1=MR&s1=1045292&loc=fromrevtext|An extension of the ham sandwich theorem]] Bull. London Math. Soc. 22 (1990), no. 2, 183–186. ------ ==Topology of posets== Anderson, Laura [[http://www.ams.org/mathscinet/search/publdoc.html?arg3=&co4=AND&co5=AND&co6=AND&co7=AND&dr=all&pg4=AUCN&pg5=AUCN&pg6=TI&pg7=ALLF&pg8=ET&r=1&review_format=html&s4=&s5=&s6=Homotopy%20Groups%20of%20the%20Combinatorial%20Grassmannian&s7=&s8=All&sort=Newest&vfpref=html&yearRangeFirst=&yearRangeSecond=&yrop=eq|Homotopy groups of the combinatorial Grassmannian. ]] Discrete Comput. Geom. 20 (1998), no. 4, 549–560. Anderson, Laura; Davis, James F., [[http://www.ams.org/mathscinet/search/publdoc.html?arg3=&co4=AND&co5=AND&co6=AND&co7=AND&dr=all&pg4=AUCN&pg5=AUCN&pg6=TI&pg7=ALLF&pg8=ET&review_format=html&s4=anderson%2C%20L%2A&s5=davis%2C%20j%2A&s6=&s7=&s8=All&sort=Newest&vfpref=html&yearRangeFirst=&yearRangeSecond=&yrop=eq&r=1&mx-pid=1913295| Mod 2 cohomology of combinatorial Grassmannians.]] Selecta Math. (N.S.) 8 (2002), no. 2, 161–200. Björner, Anders; Tancer, Martin [[http://www.ams.org/mathscinet/search/publdoc.html?arg3=&co4=AND&co5=AND&co6=AND&co7=AND&dr=all&pg4=AUCN&pg5=AUCN&pg6=TI&pg7=ALLF&pg8=ET&r=1&review_format=html&s4=&s5=&s6=Combinatorial%20Alexander%20Duality%E2%80%94A%20Short%20and%20Elementary%20Proof&s7=&s8=All&sort=Newest&vfpref=html&yearRangeFirst=&yearRangeSecond=&yrop=eq|Note: Combinatorial Alexander duality—a short and elementary proof.]] Discrete Comput. Geom. 42 (2009), no. 4, 586–593. Björner, Anders; Wachs, Michelle L.,[[http://www.ams.org/mathscinet/search/publdoc.html?arg3=&co4=AND&co5=AND&co6=AND&co7=AND&dr=all&pg4=AUCN&pg5=AUCN&pg6=TI&pg7=ALLF&pg8=ET&review_format=html&s4=&s5=wachs&s6=shellable%20nonpure&s7=&s8=All&sort=Newest&vfpref=html&yearRangeFirst=&yearRangeSecond=&yrop=eq&r=2&mx-pid=1333388| Shellable nonpure complexes and posets. I.]] Trans. Amer. Math. Soc. 348 (1996), no. 4, 1299–1327. Björner, Anders; Wachs, Michelle L.,[[http://www.ams.org/mathscinet/search/publdoc.html?arg3=&co4=AND&co5=AND&co6=AND&co7=AND&dr=all&pg4=AUCN&pg5=AUCN&pg6=TI&pg7=ALLF&pg8=ET&review_format=html&s4=&s5=wachs&s6=shellable%20nonpure&s7=&s8=All&sort=Newest&vfpref=html&yearRangeFirst=&yearRangeSecond=&yrop=eq&r=1&mx-pid=1401765| Shellable nonpure complexes and posets. II.]] Trans. Amer. Math. Soc. 349 (1997), no. 10, 3945–3975 McCammond, Jon; [[https://arxiv.org/abs/1707.06634|Noncrossing hypertrees]], arxiv preprint Shareshian, John; Woodroofe, Russ, [[http://www.ams.org/mathscinet/search/publdoc.html?arg3=&co4=AND&co5=AND&co6=AND&co7=AND&dr=all&pg4=AUCN&pg5=AUCN&pg6=TI&pg7=ALLF&pg8=ET&review_format=html&s4=shareshian&s5=&s6=&s7=poset&s8=All&sort=Newest&vfpref=html&yearRangeFirst=&yearRangeSecond=&yrop=eq&r=3&mx-pid=3459029| Order complexes of coset posets of finite groups are not contractible.]] Adv. Math. 291 (2016), 758–773.