Epistemology
Introduction E9F5FC Questions FFFFC0 Software 
{$\Delta$}complexes Simplicial homology ncountable, it is not at all clear that for a ∆ complex X with finitely many sim plices, Clear that
Singular homology Singular nsimplex in a space X is by definition just a map {$σ: ∆_n\rightarrow X$}
Homeomorphic spaces have isomorphic singular homology groups {$H_n$} Want to show that
Singular complex S(X) is the ∆complex with one n simplex {$∆^n_σ$} for each singular ncomplex {$\sigma :\Delta^n\rightarrow X$}, with {$∆^n_σ$} attached in the obvious way to the (n1)simplices of S(X) that are the restrictions of {$\sigma$} to the various (n1)simplices in {$\delta\Delta^n$}. Corresponding to the decomposition of a space X into its pathL 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 pathconnected, then {$H_0 (X) ≈ Z$} . Hence for any space X , {$H_0 (X)$} is a direct sum of {$\mathbb{Z}$} ’s, one for each pathcomponent 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 pathconnected For a map {$f : X →Y$} , an induced homomorphism {$f_♯ : C_n (X)→C_n (Y )$} is defined by composing each singular nsimplex {$σ : ∆_n →X$} with {$f$} to get a singular nsimplex {$f_♯ (σ ) = fσ:\Delta^n\rightarrow Y$}, 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 . Chainhomotopic chain maps induce the same homomorphism on homology.
