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{$\Delta$}-complexes

Simplicial homology

ncountable, it is not at all clear that for a ∆ complex X with finitely many sim- plices,

Clear that

• {$H_n^{\Delta}(X)$} is finitely generated for all n
• {$H_n^{\Delta}(X)=0$} for all n larger than the dimension of X

Singular homology

Singular n-simplex in a space X is by definition just a map {$σ: ∆_n\rightarrow X$}

• {$C_n(X)$} is the free abelian group with basis the set of singular n-simplices in X
• singular n-chains are the elments of {$C_n(X)$}. They are finite formal sums {$\sum_i n_i σ_i$} for {$n_i \in\mathbb{Z}$} and {$σ_i : ∆_n\rightarrow X$}.
• boundary map {$\delta_n:C_n(X)\rightarrow C_{n-1}(X)$} is defined by {$\delta_n(\sigma)=\sum_i(-1)^i\sigma|[v_0,\cdots,\hat{v_i},\cdots v_n]$}
• singular homology group {$H_n(X) = \textrm{Ker}\; ∂_n / \textrm{Im}\; ∂_{n+1}$}

Homeomorphic spaces have isomorphic singular homology groups {$H_n$}

Want to show that

• {$H_n(X)$} is finitely generated for all n
• {$H_n(X)=0$} for all n larger than the dimension of X

Singular complex S(X) is the ∆-complex with one n simplex {$∆^n_σ$} for each singular n-complex {$\sigma :\Delta^n\rightarrow X$}, with {$∆^n_σ$} attached in the obvious way to the (n-1)-simplices of S(X) that are the restrictions of {$\sigma$} to the various (n-1)-simplices in {$\delta\Delta^n$}.

Corresponding to the decomposition of a space X into its path-L components {$X_α$} there is an isomorphism of {$H_n(X)$} with the direct sum {$H_n(X)=\bigoplus_α H_n(X_α)$}

If X is nonempty and path-connected, then {$H_0 (X) ≈ Z$} . Hence for any space X , {$H_0 (X)$} is a direct sum of {$\mathbb{Z}$} ’s, one for each path-component of X .

If X is a point, then {$H_n (X) = 0$} for n > 0 and {$H_0 (X) ≈ \mathbb{Z}$}

Reduced homology groups {$\tilde{H}_n(X)$} are the homology groups of the augmented chain complex. Here the point has trivial homology groups in all dimensions, including zero.

{$H_1 (X)$} is the abelianization of {$π_1 (X)$} whenever X is path-connected

For a map {$f : X →Y$} , an induced homomorphism {$f_♯ : C_n (X)→C_n (Y )$} is defined by composing each singular n-simplex {$σ : ∆_n →X$} with {$f$} to get a singular n-simplex {$f_♯ (σ ) = fσ:\Delta^n\rightarrow Y$}, then extending {$f_♯$} linearly via {$f_♯(\sum_i n_i σ_i) = \sum_i n_i f_♯ (σ_i ) = \sum_i n_if\sigma_i$}.

A chain map between chain complexes induces homomorphisms between the homology groups of the two complexes.

If two maps {$f , g : X\rightarrow Y$} are homotopic, then they induce the same homomorphism {$f_∗ = g_∗ : H_n (X)\rightarrow H_n (Y )$} .

The maps {$f_∗ : H_n (X)\rightarrow H_n (Y )$} induced by a homotopy equivalence {$f : X\rightarrow Y$} are isomorphisms for all n .

Chain-homotopic chain maps induce the same homomorphism on homology.

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