From 29557d1baa078eb3955551c78719765907e75d45 Mon Sep 17 00:00:00 2001 From: Joshua Moerman Date: Fri, 30 Jan 2015 12:10:52 +0100 Subject: [PATCH] Cleans up a bit --- thesis/chapters/Introduction.tex | 2 +- thesis/images/A_Preserves_WCof.png | Bin 86149 -> 0 bytes thesis/images/Apl_C_Quasi_Iso_LES.png | Bin 64508 -> 0 bytes thesis/images/Apl_C_Quasi_Iso_Limit.png | Bin 76342 -> 0 bytes thesis/images/Apl_C_Quasi_Iso_Pushout2.png | Bin 23019 -> 0 bytes thesis/images/Extend_Boundary_Form.png | Bin 53604 -> 0 bytes thesis/images/SnQ_Extension.png | Bin 93886 -> 0 bytes thesis/images/Sullivan_Lifting.png | Bin 34176 -> 0 bytes thesis/preamble.tex | 19 ------------------- thesis/thesis.tex | 2 -- 10 files changed, 1 insertion(+), 22 deletions(-) delete mode 100644 thesis/images/A_Preserves_WCof.png delete mode 100644 thesis/images/Apl_C_Quasi_Iso_LES.png delete mode 100644 thesis/images/Apl_C_Quasi_Iso_Limit.png delete mode 100644 thesis/images/Apl_C_Quasi_Iso_Pushout2.png delete mode 100644 thesis/images/Extend_Boundary_Form.png delete mode 100644 thesis/images/SnQ_Extension.png delete mode 100644 thesis/images/Sullivan_Lifting.png diff --git a/thesis/chapters/Introduction.tex b/thesis/chapters/Introduction.tex index 59a4ea8..79757f8 100644 --- a/thesis/chapters/Introduction.tex +++ b/thesis/chapters/Introduction.tex @@ -1,7 +1,7 @@ \chapter*{Introduction} -Homotopy theory is the study of topological spaces and homotopy equivalences. These equivalences are weaker than isomorphism. An isomorphism is given by two maps $f : X \leftadj Y : g$, such that the both compositions are equal to identities. A homotopy equivalence weakens this by requiring the compositions to be homotopic to identities. Some properties of spaces, such as some kinds of connectedness, only depend on the homotopy type. Such properties are homotopy invariants. \todo{not happy with this yet...} +Homotopy theory is the study of topological spaces and homotopy equivalences. These equivalences are weaker than isomorphism. An isomorphism is given by two maps $f : X \leftadj Y : g$, such that the both compositions are equal to identities. A homotopy equivalence weakens this by requiring the compositions to be homotopic to identities. Some properties of spaces, such as some kinds of connectedness, only depend on the homotopy type. Such properties are homotopy invariants. Examples of homotopy invariants are homology groups $H_n(X)$ and homotopy groups $\pi_n(X)$. The latter is defined as the set of continuous maps $S^n \to X$ up to homotopy. Despite the easy definition, the groups $\pi_n(S^k)$ are very hard to calculate and much of it is even unknown as of today. diff --git a/thesis/images/A_Preserves_WCof.png b/thesis/images/A_Preserves_WCof.png deleted file mode 100644 index c869dd0dc90308d162c4cfb307b95ef79daf9fee..0000000000000000000000000000000000000000 GIT binary patch literal 0 HcmV?d00001 literal 86149 zcmbTdg;yLwvp$Tw%LaG1;O;Jqy9QX?-7UDgEFJ;`cMq`vuNY zML`zk9G)x5;=P9IrfBF110xpspA&Yuw!!ONi0q}LDu=v{jDRUjtJnHv_g%zbuVdgP z{{^Qa1xTT@v*qF1hNF^ENEzY8alE#Wz1R_ zis(c*oCwlksgYg_A16%fbDUdv`XwHECoZSAw`R64k1lV6pb?v!7w4HBK|3eG@ivAi zN(NyJDN;BH@5kV0jugrNmZWg@z|@wm1pkq~mseQ<&AuN|!l~YmAsKgBsH3 z;aWqU+fxkGWowDTIZ@{3tkLD))gli#h;v^4ZO%{pgM!(E3W>=EJTUBS!#SBEOaE)` z5`~w*$)r5HKRI<@_+5` z-e_*VTbdSNj)y4p@@*M72p3N@Izcui0gY4#(_8@mYeL-QhzPx+vN&7+6iYzkrx~`e zI^Zarml`ZTcY0y%QSeNr55s@kLxBtqI@f~uDe(r)$un+_4vj4fEFkKkK;(^Fb?OYL zLvo)wPUIUs5QF?*(ZL=qO8BlNc5HP>(x*FIRfPF2Ff$9i`h(#J-2KM{RVT>Emqh)l 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