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        "quote": "<span id=\"node-a-82e03117b53ccf7c18d065aa\" class=\"block-anchor\"></span>由于$Z_{r}(K)$与$B_{r}(K)$都是<a class=\"concept-reference\" href=\"01-section-3-1.md#bound-definition-13c6252e147d5e7e497b347d\"><u class=\"concept-term\" id=\"bound-mention-48dbbfbc0be7becca282fe8e\" data-concept-identity=\"urn:uuid:b405e339-ed38-485b-be7b-b30aab2271a7\" data-concept-role=\"discussion\" data-concept-first-mention=\"false\" data-concept-navigation-schema=\"1\" data-concept-source-node=\"a-82e03117b53ccf7c18d065aa\" data-concept-identity-status=\"formal\" data-concept-definition-status=\"mention\" data-concept-route=\"/concepts/by-id/b405e339-ed38-485b-be7b-b30aab2271a7\" title=\"查看本书定义\" data-concept-reference=\"bound-definition-13c6252e147d5e7e497b347d\" data-concept-reference-basis=\"definition\" data-concept-first-mention-target=\"/books/nakahara-geometry-topology-physics/second-edition-local/zh/chapter-03/01-section-3-1#bound-definition-13c6252e147d5e7e497b347d\">自由阿贝尔群</u></a>$C_r(K)$的子群，所以它们本身也是自由阿贝尔群。但这并不意味着$H_{r}(K) = Z_{r}(K) / B_{r}(K)$也是自由阿贝尔群。事实上，根据[定理 3.2](01-section-3-1.md#theorem-3-2)，$H_{r}(K)$最一般的形式为",
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        "quote": "> [!example] <a class=\"object-permalink\" href=\"04-section-4-4.md#example-4-4\">例 4.4</a>\n>\n> <span id=\"node-a-86b1dd328511a12699704cd1\" class=\"block-anchor\"></span><span id=\"example-4-4\" class=\"block-anchor\"></span>设 $\\mathbb{Z} \\oplus \\mathbb{Z} = \\{x^n y^m | n, m \\in \\mathbb{Z}\\}$ 是由 $X = \\{x, y\\}$ 生成的<a class=\"concept-reference\" href=\"../chapter-03/01-section-3-1.md#bound-definition-13c6252e147d5e7e497b347d\"><u class=\"concept-term\" id=\"bound-mention-0c7d5b0c6bdf19253b05e3a5\" data-concept-identity=\"urn:uuid:b405e339-ed38-485b-be7b-b30aab2271a7\" data-concept-role=\"discussion\" data-concept-first-mention=\"false\" data-concept-navigation-schema=\"1\" data-concept-source-node=\"a-86b1dd328511a12699704cd1\" data-concept-identity-status=\"formal\" data-concept-definition-status=\"mention\" data-concept-route=\"/concepts/by-id/b405e339-ed38-485b-be7b-b30aab2271a7\" title=\"查看本书定义\" data-concept-reference=\"bound-definition-13c6252e147d5e7e497b347d\" data-concept-reference-basis=\"definition\" data-concept-first-mention-target=\"/books/nakahara-geometry-topology-physics/second-edition-local/zh/chapter-03/01-section-3-1#bound-definition-13c6252e147d5e7e497b347d\">自由阿贝尔群</u></a>。于是有 $xy = yx$。由于 $xyx^{-1}y^{-1} = 1$，我们得到关系 $r = xyx^{-1}y^{-1}$。$\\mathbb{Z} \\oplus \\mathbb{Z}$ 的表现是 $(x, y : xyx^{-1}y^{-1})$。\n>",
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        "quote": "<span id=\"node-a-9f080f358e9c4a4e00bb1fce\" class=\"block-anchor\"></span>注意 $H_{1}(2$ 花束）是<a class=\"concept-reference\" href=\"../../../../../concepts/by-id/b405e339-ed38-485b-be7b-b30aab2271a7.md\"><u class=\"concept-term\" id=\"bound-definition-f2826a1478f054de706a6867\" data-concept-identity=\"urn:uuid:b405e339-ed38-485b-be7b-b30aab2271a7\" data-concept-role=\"definition\" data-concept-first-mention=\"false\" data-concept-navigation-schema=\"1\" data-concept-source-node=\"a-9f080f358e9c4a4e00bb1fce\" data-concept-identity-status=\"formal\" data-concept-definition-status=\"confirmed\" data-concept-route=\"/concepts/by-id/b405e339-ed38-485b-be7b-b30aab2271a7\" title=\"查看正式概念页\" data-concept-first-mention-target=\"/books/nakahara-geometry-topology-physics/second-edition-local/zh/chapter-03/01-section-3-1#bound-definition-13c6252e147d5e7e497b347d\">自由阿贝尔群(free Abelian group)</u></a>，而 $\\pi_{1}(2$ 花束）是自由群。下面的定理把 $\\pi_{1}(|K|)$ 与 $H_{1}(K)$ 联系起来。",
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