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I have a big problem and i don't know how to solve it i have no idea

So, let $i_2: X_2\rightarrow X$ an inclusion and $j_1: X\rightarrow (X,X_1)$ we have that $i_{2_*}: H_k(X_2)\rightarrow H_k(X)$ is injective and $j_{1_*}\circ i_{2_*}: H_k(X_2)\rightarrow H_k(X,X_1)$ is an isomorphism, so in the proof they say that because $i_{2_*}$ is injective then we have : $$H_k(X)=\ker j_{1_*}\oplus \text{Im}~ i_{2_*}$$

this is the proposition and the proof please help me:

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Vrouvrou
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1 Answers1

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What book is it you are quoting?

  • You need to consider the excision property (if you are to follow the given proof), which is given e.g. in Theorem 2.17 and Corollary 2.18 of the following links: http://www.math.toronto.edu/mgualt/MAT1300/Week%206-9%20Term%202.pdf

  • The idea is to use the 5-lemma to show the desired isomorphism.

  • Alternatively, one may show directly that $H_k(X)=H_k(X_1)\oplus H_k(X_2)$ simply because $X_1, X_2$ are a disjoint union and every singular simplex $\sigma:\Delta^k \to X$ is such that its image falls within $X_1$ or $X_2$ (use connectedness of the simplex). Likewise with $A$ and $A_i=A \cap X_i.$

Actually, it is all done here (if you use the 5-lemma): Homology of disjoint union is direct sum of homologies

You have $X=X_1 \sqcup X_2$ union of two disjoint subsets. In fact, by excision you have (here $S_*$ denotes the complex of singular chains of a space or of a pair) a canonical morphism

$$S_*(X_2)=S_*(X_2,\emptyset) \to S_*(X, X_1)$$ which is an isomorphism. If you write the exact sequence of relative homology (before taking homology, i.e. at the level of singular chains) you have

$$0\to S_*(X_1) \to S_*(X) \to S_*(X, X_1) \cong S_*(X_2) \to 0,$$ by the above. You can exchange both $X_1$ and $X_2$, and that means that the above exact sequence splits, i.e.

$$S_*(X)=S_*(X_1)\oplus S_*(X_2)$$ (there are other ways to do this, but let's leave it).

Likewise with $A$ and $A_i$. The direct sums you're looking for are there.

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Theon Alexander
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    D. Motreanu, V.V. Motreanu, and N. Papageorgiou, Topological and Variational Methods with Applications to Nonlinear Boundary Value Problems, Springer, 2014. – Vrouvrou Aug 31 '14 at 06:54
  • is methode given bye @MattN is correct in :http://math.stackexchange.com/questions/56286/homology-of-disjoint-union-is-direct-sum-of-homologies?rq=1 ? please – Vrouvrou Aug 31 '14 at 06:55
  • Sorry but i don't understand how we find $\ker j_{1_}$ and $\text{Im}~i_{2_}$ in direct somme using excision, please give me more details – Vrouvrou Aug 31 '14 at 08:34
  • The property of excesion just tel us that $H_k(X_2)\simeq H_k(X,X_1)$ what is the exact sequence witch give us that $H_k(X)=\ker j_{1_}\oplus \text{Im}~i_{2_}$ ? please please help me – Vrouvrou Aug 31 '14 at 09:38
  • This not the disered answer ....... – Vrouvrou Aug 31 '14 at 20:07
  • I have added some more details, but there is no replacement to a proper study of the tools. – Theon Alexander Sep 01 '14 at 03:45
  • The answer given in the link you provide is good, indeed (and simpler than the provided in the first reference). – Theon Alexander Sep 03 '14 at 16:46