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Vedi la Nota IV, apparsa alle pp. 402-409, voi. LIII di questi «Rendiconti».
Si continuano tre Note con lo stesso titolo apparse in questi «Rendiconti». Nella presente Nota IV ed in una successiva Nota V viene costruita una teoria omologica del tipo di Čech basata sullo schema cubico. Più precisamente, si definisce l'omologia cubica di Čech di uno spazio compatto X come il limite inverso delle omologie combinatorie di poliedri approssimativi QX (ved. [1]). Si mostra poi che la teoria omologica così costruita soddisfa agli assiomi di Eilenberg-Steenrod per le teorie omologiche...
Per ogni spazio topologico, X, viene assegnata una costruzione funtoriale di un complesso cubico QX. Quando X è compatto, QX risulta equivalente ad X (a meno di un'omotopia), ed è una dualizzazione del semisimpliciale S(X). Di tutto ciò verranno fatte numerose applicazioni in lavori successivi.
Si fa seguito ad una precedente Nota lincea [1], mostrando l'invarianza omotopica della nozione (qui introdotta) di contigua equivalenza fra morfismi cubici.
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