{"id":1719,"date":"2025-09-19T16:11:09","date_gmt":"2025-09-19T23:11:09","guid":{"rendered":"https:\/\/dornsife.usc.edu\/mgsa\/?page_id=1719"},"modified":"2025-11-16T15:24:53","modified_gmt":"2025-11-16T23:24:53","slug":"topology","status":"publish","type":"page","link":"https:\/\/dornsife.usc.edu\/mgsa\/qualifying-exams\/topology\/","title":{"rendered":"Topology (540)"},"content":{"rendered":"\n\n \n \n\n\n\n\n\n\n\n
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To contribute solutions, corrections, notes, or other helpful resources, feel free to email mgsa@usc.edu<\/a> or submit a pull request to MGSA’s qualifying exams repository on GitHub<\/a>.<\/p>\n

Last update: 2025-11-16<\/em><\/p>\n

Back to Qualifying Exams Overview<\/em><\/a><\/p>\n<\/div>\n<\/div>\n<\/div>\n\n\n\n<\/div>\n\n\n <\/div><\/div>\n\n\n\n \n \n\n\n\n\n\n\n\n

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\n \n <\/h2>\n\n\n<\/div>\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
\n Exam\n <\/th>\n \n \n <\/th>\n \n Solutions\n <\/th>\n \n \n <\/th>\n \n Author\n <\/th>\n <\/tr>\n \n \n <\/thead>\n
\n \u2014\n <\/td>\n \n \n \n Spring 2025<\/a>\n <\/td>\n \n \n \n \u2014\n <\/td>\n \n \n \n Spring 2025<\/a>\n <\/td>\n \n \n \n Ivan Feng\n <\/td>\n \n <\/tr>\n
\n Fall 2025<\/a>
\nFall 2024<\/a>\n <\/td>\n \n \n 		\n Spring 2025<\/a>
\nSpring 2024<\/a>\n <\/td>\n \n \n 		\n \n <\/td>\n \n \n \n \n <\/td>\n \n \n \n \n <\/td>\n \n <\/tr>\n
\n Fall 2024<\/a>
\nFall 2023<\/a>\n <\/td>\n \n \n 		\n Spring 2024<\/a>
\nSpring 2023<\/a>\n <\/td>\n \n \n 		\n Fall 2024<\/a>
\nFall 2023<\/a>\n <\/td>\n \n \n 		\n Spring 2024
\nSpring 2023<\/a>\n <\/td>\n \n \n 		\n Hunter Hewitt\n <\/td>\n \n <\/tr>\n
\n Fall 2022<\/a>
\nFall 2021<\/a>\n <\/td>\n \n \n 		\n Spring 2022<\/a>
\n\u2014\n <\/td>\n \n \n 		\n \n <\/td>\n \n \n \n \n <\/td>\n \n \n \n \n <\/td>\n \n <\/tr>\n
\n \u2014
\nFall 2020<\/a>
\nFall 2019<\/a>
\nFall 2018<\/a>\n <\/td>\n \n \n 		\n Spring 2021<\/a>
\nSpring 2020<\/a>
\nSpring 2019<\/a>
\nSpring 2018<\/a>\n <\/td>\n \n \n 		\n \u2014
\nFall 2020<\/a>
\nFall 2019<\/a>
\nFall 2018<\/a>\n <\/td>\n \n \n 		\n Spring 2021<\/a>
\nSpring 2020<\/a>
\nSpring 2019<\/a>
\nSpring 2018<\/a>\n <\/td>\n \n \n 		\n Jonathan Michala\n <\/td>\n \n <\/tr>\n
\n Fall 2017<\/a>
\nFall 2016
\nFall 2015<\/a>
\nFall 2014<\/a>
\nFall 2013<\/a>
\nFall 2012<\/a>
\nFall 2011
\nFall 2010<\/a>\n <\/td>\n \n \n 		\n Spring 2017<\/a>
\nSpring 2016<\/a>
\nSpring 2015
\nSpring 2014<\/a>
\nSpring 2013<\/a>
\nSpring 2012<\/a>
\nSpring 2011<\/a>
\nSpring 2010\n <\/td>\n \n \n 		\n Fall 2017<\/a>
\nFall 2016
\nFall 2015<\/a>
\nFall 2014<\/a>
\nFall 2013<\/a>
\nFall 2012<\/a>
\nFall 2011
\nFall 2010<\/a>\n <\/td>\n \n \n 		\n Spring 2017<\/a>
\nSpring 2016<\/a>
\nSpring 2015
\nSpring 2014<\/a>
\nSpring 2013<\/a>
\nSpring 2012<\/a>
\nSpring 2011<\/a>
\nSpring 2010\n <\/td>\n \n \n 		\n Alec Sahakian<\/a>\n <\/td>\n \n <\/tr>\n
\n \n <\/td>\n \n \n \n \n <\/td>\n \n \n \n Fall 2017<\/a>
\nFall 2016
\nFall 2015<\/a>
\nFall 2014<\/a>
\nFall 2013<\/a>
\nFall 2012<\/a>
\nFall 2011
\nFall 2010<\/a>\n <\/td>\n \n \n 		\n Spring 2017<\/a>
\nSpring 2016<\/a>
\nSpring 2015
\nSpring 2014<\/a>
\nSpring 2013<\/a>
\nSpring 2012<\/a>
\nSpring 2011<\/a>
\nSpring 2010\n <\/td>\n \n \n 		\n Jonathan Michala<\/a>\n <\/td>\n \n <\/tr>\n
\n \n <\/td>\n \n \n \n \n <\/td>\n \n \n \n Fall 2016<\/a>
\nFall 2015<\/a>
\nFall 2014
\nFall 2011
\nFall 2010<\/a>\n <\/td>\n \n \n 		\n Spring 2016<\/a>
\nSpring 2015
\nSpring 2014<\/a>
\nSpring 2011<\/a>
\nSpring 2010\n <\/td>\n \n \n 		\n Dan Douglas\n <\/td>\n \n <\/tr>\n
\n \n <\/td>\n \n \n \n \n <\/td>\n \n \n \n Fall 2014<\/a>
\nFall 2013<\/a>
\nFall 2012<\/a>\n <\/td>\n \n \n 		\n Spring 2014<\/a>
\nSpring 2013<\/a>
\nSpring 2012<\/a>\n <\/td>\n \n \n 		\n Nicolle Gonz\u00e1lez\n <\/td>\n \n <\/tr>\n
\n Fall 2009<\/a>
\nFall 2008<\/a>
\nFall 2007<\/a>
\nFall 2006<\/a>
\nFall 2005<\/a>\n <\/td>\n \n \n 		\n Spring 2009<\/a>
\nSpring 2008<\/a>
\nSpring 2007<\/a>
\nSpring 2006<\/a>
\nSpring 2005<\/a>\n <\/td>\n \n \n 		\n Fall 2009<\/a>
\nFall 2008<\/a>
\nFall 2007<\/a>
\nFall 2006<\/a>
\nFall 2005<\/a>\n <\/td>\n \n \n 		\n Spring 2009<\/a>
\nSpring 2008<\/a>
\nSpring 2007
\nSpring 2006<\/a>
\nSpring 2005\n <\/td>\n \n \n 		\n Alec Sahakian<\/a>\n <\/td>\n \n <\/tr>\n
\n \n <\/td>\n \n \n \n \n <\/td>\n \n \n \n Fall 2009<\/a>
\nFall 2008<\/a>
\nFall 2007<\/a>\n <\/td>\n \n \n 		\n Spring 2009<\/a>
\nSpring 2008<\/a>
\nSpring 2007\n <\/td>\n \n \n 		\n Jonathan Michala<\/a>\n <\/td>\n \n <\/tr>\n
\n Fall 2012<\/a>
\nFall 2011
\nFall 2010<\/a>
\nFall 2009<\/a>
\nFall 2008<\/a>
\nFall 2007<\/a>
\nFall 2006<\/a>
\nFall 2005<\/a>
\nFall 2004<\/a>
\nFall 2003<\/a>
\nFall 2002
\nFall 2000<\/a>
\nFall 1999<\/a>\n <\/td>\n \n \n 		\n Spring 2012<\/a>
\nSpring 2011<\/a>
\nSpring 2010
\nSpring 2009<\/a>
\nSpring 2008<\/a>
\nSpring 2007<\/a>
\nSpring 2006<\/a>
\nSpring 2005<\/a>
\nSpring 2004<\/a>
\nSpring 2003<\/a>
\nSpring 2002<\/a>
\nSpring 2000
\nSpring 1999<\/a>\n <\/td>\n \n \n 		\n Fall 2012
\nFall 2011
\nFall 2010<\/a>
\nFall 2009<\/a>
\nFall 2008<\/a>
\nFall 2007<\/a>
\nFall 2006<\/a>
\nFall 2005<\/a>
\nFall 2004<\/a>
\nFall 2003<\/a>
\nFall 2002
\nFall 2000<\/a>
\nFall 1999<\/a>\n <\/td>\n \n \n 		\n Spring 2012<\/a>
\nSpring 2011<\/a>
\nSpring 2010
\nSpring 2009<\/a>
\nSpring 2008<\/a>
\nSpring 2007<\/a>
\nSpring 2006<\/a>
\nSpring 2005<\/a>
\nSpring 2004<\/a>
\nSpring 2003<\/a>
\nSpring 2002<\/a>
\nSpring 2000
\nSpring 1999<\/a>\n <\/td>\n \n \n 		\n Henry Tucker\n <\/td>\n \n <\/tr>\n
\n Fall 1998
\nFall 1997<\/a>
\nFall 1996
\nFall 1995
\nFall 1994<\/a>\n <\/td>\n \n \n 		\n Spring 1998<\/a>
\nSpring 1997<\/a>
\nSpring 1996<\/a>
\nSpring 1995<\/a>
\nSpring 1994<\/a>\n <\/td>\n \n \n 		\n \n <\/td>\n \n \n \n \n <\/td>\n \n \n \n \n <\/td>\n \n <\/tr>\n
\n \n <\/td>\n \n \n \n Spring 1993<\/a>\n <\/td>\n \n \n \n \n <\/td>\n \n \n \n Spring 1993<\/a>\n <\/td>\n \n \n \n William Chang\n <\/td>\n \n <\/tr>\n <\/tbody>\n <\/table>\n <\/div>\n\n <\/div><\/div>\n\n\n\n \n \n\n\n\n\n\n\n\n
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\n \n <\/h2>\n\n\n<\/div>\n\n \n \n \n \n \n \n \n \n \n \n \n
\n Extra material\n <\/th>\n \n Author\n <\/th>\n <\/tr>\n \n \n <\/thead>\n
\n Class exams and solutions<\/a>
\nNotes for MATH 535A (differential geometry)<\/a>
\nNotes for MATH 540<\/a>
\nNotes on algebraic topology<\/a>\n <\/td>\n \n \n 		\n Dan Douglas\n <\/td>\n \n <\/tr>\n
\n Notes and selected solutions<\/a>\n <\/td>\n \n \n \n Ivan Feng\n <\/td>\n \n <\/tr>\n
\n Cheat sheet<\/a>\n <\/td>\n \n \n \n Henry Tucker\n <\/td>\n \n <\/tr>\n <\/tbody>\n <\/table>\n <\/div>\n\n <\/div><\/div>\n","protected":false},"excerpt":{"rendered":"","protected":false},"author":990,"featured_media":0,"parent":192,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"_acf_changed":false,"footnotes":""},"class_list":["post-1719","page","type-page","status-publish","hentry"],"acf":[],"yoast_head":"\nTopology (540) - Mathematics Graduate Student Association<\/title>\n<meta name=\"description\" content=\"Student-contributed solutions to the Topology (540) qualifying exam for the Mathematics programs at USC.\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/dornsife.usc.edu\/mgsa\/qualifying-exams\/topology\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Topology (540) - 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