# Coverage Check for Hatcher 2.1

## ??
- Section: 2.1 Simplicial and Singular Homology
- PDF pages: 111--142
- ????: p.102--p.133
- ?? TeX: Hatcher_2.1_Chinese.tex
- ?? PDF: Hatcher_2.1_Chinese.pdf

## ???

| ???? | ???? | ???? | ???? TeX | TODO |
|---|---|---|---|---|
| p.102 | 2.1 ???singular/simplicial homology ???Delta-complexes ?? | ? | ? | ? |
| p.103 | simplex?standard simplex?face?boundary?open simplex?Delta-complex structure | ????? | ? | ? |
| p.104 | Delta-complex quotient construction?dunce cap?Simplicial Homology ?? | ? | ? | ? |
| p.105 | chain groups?boundary homomorphism?boundary orientation formulas | Lemma 2.1 | ? | ? |
| p.106 | chain complex?homology group?examples of S1, torus, RP2 | Examples 2.2--2.4 | ? | ? |
| p.107 | Sn example?simplicial complexes discussion?Singular Homology ?? | Example 2.5 | ? | ? |
| p.108 | singular simplex?singular chain groups?singular complex | ? | ? | ? |
| p.109 | geometric interpretation of singular cycles?path components | Proposition 2.6, Proposition 2.7 | ? | ? |
| p.110 | H0 proof?point space computation?reduced homology?Homotopy Invariance | Proposition 2.8 | ? | ? |
| p.111 | induced maps?chain maps?chain homotopy setup | Proposition 2.9, Theorem 2.10, Corollary 2.11 | ? | ? |
| p.112 | proof of Theorem 2.10 via prism operator | ? | ? | ? |
| p.113 | finish homotopy proof?Exact Sequences and Excision begins | Proposition 2.12 | ? | ? |
| p.114 | exactness; quotient long exact sequence for good pairs; sphere computation | Theorem 2.13, Corollary 2.14, Corollary 2.15 | ? | ? |
| p.115 | relative homology groups; relative cycles/boundaries | ? | ? | ? |
| p.116 | short exact sequence of chain complexes; connecting homomorphism | ? | ? | ? |
| p.117 | algebraic long exact sequence; relative long exact sequence; examples | Theorem 2.16, Examples 2.17--2.18, Proposition 2.19 | ? | ? |
| p.118 | triple sequence; Excision starts | Theorem 2.20 | ? | ? |
| p.119 | excision statement; barycentric subdivision proof begins | Proposition 2.21 | ? | ? |
| p.120 | barycentric subdivision of simplices; diameter estimate | ? | ? | ? |
| p.121 | linear chains; subdivision operator S | ? | ? | ? |
| p.122 | chain homotopy T; general chains | ? | ? | ? |
| p.123 | iterated subdivision; construction of rho | ? | ? | ? |
| p.124 | proof of excision; good pairs quotient result | Proposition 2.22 | ? | ? |
| p.125 | mapping cone relative homology; explicit generators | Example 2.23 | ? | ? |
| p.126 | CW subcomplex excision; wedge sums; invariance of dimension | Corollary 2.24, Corollary 2.25, Theorem 2.26 | ? | ? |
| p.127 | local homology; Naturality begins | ? | ? | ? |
| p.128 | naturality; equivalence of simplicial and singular homology begins | Theorem 2.27 | ? | ? |
| p.129 | proof of Theorem 2.27; Five-Lemma | Five-Lemma | ? | ? |
| p.130 | infinite-dimensional case; finite generation; historical note | ? | ? | ? |
| p.131 | historical note end; exercises 1--10 | Exercises 1--10 | ? | ? |
| p.132 | exercises 11--22 | Exercises 11--22 | ? | ? |
| p.133 | exercises 23--31 | Exercises 23--31 | ? | ?? 30 ?????????????????? |

## ??
- p.102--103 introductory/simplex figures: ????? `figures/hatcher_2_1_sample/surface_identifications.png`, `simplex_examples.png`?
- p.104 dunce-cap comparison: ????? `figures/hatcher_2_1_full/dunce_cap_comparison.png`?
- p.105 boundary orientation figure: ????? `figures/hatcher_2_1_full/boundary_orientations.png`?
- p.112 prism subdivision figure: ????? `figures/hatcher_2_1_full/prism_subdivision.png`?

## TODO
- ????????????? 30 ??????????????? TeX ?????????

## ??
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