It could be desirable to obtain explicit representatives of the persistent homology generators that "persist" through a significant sequence of filtration terms. There are two such generators in degree 1 and one such generator in degree 2. The explicit representatives in degree n could consist of an inclusion of pure cubical complexes Y_n ⊂ X_10 for which the incuced homology homomorphism H_n(Y_n, Z) → H_n(X_10, Z) is an isomorphism, and for which Y_n is minimal in the sense that its homotopy type changes if any one or more of its top dimensional cells are removed. Ideally the space Y_n should be as "close to the original dataset" X_0. The following commands first construct an explicit degree 2 homology generator representative Y_2⊂ X_10 where Y_2 is homotopy equivalent to X_10. They then construct an explicit degree 1 homology generators representative Y_1⊂ X_10 where Y_1 is homotopy equivalent to a wedge of two circles. The final command displays the homology generators representative Y_1.
+It could be desirable to obtain explicit representatives of the persistent homology generators that "persist" through a significant sequence of filtration terms. There are two such generators in degree 1 and one such generator in degree 2. The explicit representatives in degree n could consist of an inclusion of pure cubical complexes Y_n ⊂ X_10 for which the incuced homology homomorphism H_n(Y_n, Z) → H_n(X_10, Z) is an isomorphism, and for which Y_n is minimal in the sense that its homotopy type changes if any one or more of its top dimensional cells are removed. Ideally the space Y_n should be "close to the original dataset" X_0. The following commands first construct an explicit degree 2 homology generator representative Y_2⊂ X_10 where Y_2 is homotopy equivalent to X_10. They then construct an explicit degree 1 homology generators representative Y_1⊂ X_10 where Y_1 is homotopy equivalent to a wedge of two circles. The final command displays the homology generators representative Y_1.
diff --git a/tutorial/chap5.txt b/tutorial/chap5.txt index 81957758..0a890275 100644 --- a/tutorial/chap5.txt +++ b/tutorial/chap5.txt @@ -315,7 +315,7 @@ incuced homology homomorphism [22XH_n(Y_n, Z) → H_n(X_10, Z)[122X is an isomorphism, and for which [22XY_n[122X is minimal in the sense that its homotopy type changes if any one or more of its top dimensional cells are removed. Ideally the space - [22XY_n[122X should be as "close to the original dataset" [22XX_0[122X. The following commands + [22XY_n[122X should be "close to the original dataset" [22XX_0[122X. The following commands first construct an explicit degree [22X2[122X homology generator representative [22XY_2⊂ X_10[122X where [22XY_2[122X is homotopy equivalent to [22XX_10[122X. They then construct an explicit degree [22X1[122X homology generators representative [22XY_1⊂ X_10[122X where [22XY_1[122X is diff --git a/tutorial/chap5_mj.html b/tutorial/chap5_mj.html index e26ca4c6..113bc969 100644 --- a/tutorial/chap5_mj.html +++ b/tutorial/chap5_mj.html @@ -327,7 +327,7 @@5.5-1 Non-trivial cup product
5.5-2 Explicit homology generators
-It could be desirable to obtain explicit representatives of the persistent homology generators that "persist" through a significant sequence of filtration terms. There are two such generators in degree \(1\) and one such generator in degree \(2\). The explicit representatives in degree \(n\) could consist of an inclusion of pure cubical complexes \(Y_n \subset X_{10}\) for which the incuced homology homomorphism \(H_n(Y_n,\mathbb Z) \rightarrow H_n(X_{10},\mathbb Z)\) is an isomorphism, and for which \(Y_n\) is minimal in the sense that its homotopy type changes if any one or more of its top dimensional cells are removed. Ideally the space \(Y_n\) should be as "close to the original dataset" \(X_0\). The following commands first construct an explicit degree \(2\) homology generator representative \(Y_2\subset X_{10}\) where \(Y_2\) is homotopy equivalent to \(X_{10}\). They then construct an explicit degree \(1\) homology generators representative \(Y_1\subset X_{10}\) where \(Y_1\) is homotopy equivalent to a wedge of two circles. The final command displays the homology generators representative \(Y_1\).
+It could be desirable to obtain explicit representatives of the persistent homology generators that "persist" through a significant sequence of filtration terms. There are two such generators in degree \(1\) and one such generator in degree \(2\). The explicit representatives in degree \(n\) could consist of an inclusion of pure cubical complexes \(Y_n \subset X_{10}\) for which the incuced homology homomorphism \(H_n(Y_n,\mathbb Z) \rightarrow H_n(X_{10},\mathbb Z)\) is an isomorphism, and for which \(Y_n\) is minimal in the sense that its homotopy type changes if any one or more of its top dimensional cells are removed. Ideally the space \(Y_n\) should be "close to the original dataset" \(X_0\). The following commands first construct an explicit degree \(2\) homology generator representative \(Y_2\subset X_{10}\) where \(Y_2\) is homotopy equivalent to \(X_{10}\). They then construct an explicit degree \(1\) homology generators representative \(Y_1\subset X_{10}\) where \(Y_1\) is homotopy equivalent to a wedge of two circles. The final command displays the homology generators representative \(Y_1\).