By James W. Vick (auth.)

ISBN-10: 1461269334

ISBN-13: 9781461269335

The twenty years because the book of this publication were an period of constant progress and improvement within the box of algebraic topology. New generations of younger mathematicians were educated, and classical difficulties were solved, really throughout the program of geometry and knot thought. varied new assets for introductory coursework have seemed, yet there's power curiosity in an intuitive therapy of the fundamental principles. This moment version has been elevated throughout the addition of a bankruptcy on masking areas. through research of the lifting challenge it introduces the funda psychological team and explores its homes, together with Van Kampen's Theorem and the connection with the 1st homology workforce. it's been inserted after the 3rd bankruptcy because it makes use of a few definitions and effects integrated sooner than that time. even if, a lot of the fabric is without delay available from a similar heritage as bankruptcy 1, so there will be a few flexibility in how those themes are built-in right into a path. The Bibliography has been supplemented by means of the addition of chosen books and old articles that experience seemed considering 1973.

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**Extra info for Homology Theory: An Introduction to Algebraic Topology**

**Example text**

Mittels einer Diagrammjagd zeigt man die Exaktheit der langen Homologiesequenz. Ihre Natiirlichkeit ist offen0 sichtlich. Sei {C*[iJ liE I} eine Familie von R-Kettenkomplexen. Man definiert die direkte Summe EBiEI C* [iJ und das direkte Produkt DiEI C* [iJ, indem man als n-ten Kettenmodul die direkte Summe bzw. das direkte Produkt der einzelnen Kettenmoduln Cn[iJ nimmt und das n-te Differential entsprechend definiert. 2 iEI (ffl ~ Hn ~ II Hn(C*[i]). 8) iEI Konstruktion der singuUiren Homologie Sei X ein topologischer Raum.

7fi/n) r-+ exp((l + t) ·7fi/n) exp((k + t) . 7fi/n) r-+ exp((l + 1 - t) ·7fi/n) {l, 2, ... , 2n}, k i- l. In diesem Sinne fUr t E [0,1] und feste Werte k, I E beschreiben die obigen Worte, wie man die Bogen paarweise identifiziert, urn die zugehorige FUiche zu erhalten. Wir bezeichnen im Folgenden mit Fo+ die Sphare 52, mit Fg+ die Flache zum Wort a1a2al1a;-1 ... a29-1a29a;-9~1 a;-gl und mit Fg- die Flache zum Wort a1a1b1b1 ... agag.

Man rechnet nun leicht nach, dass wiederum gilt n = O,d, sonst. 39 (Triviale Differentiale). Sei X ein endlicher CW-Komplex. Falls alle Inzidenzzahlen inzi,j fur nEZ, n 2: 0, i E In, j E I n- l verschwinden, sind aile Differentiale von C~· (X) trivial und es folgt 1i n (X) ~ EB1io({e}). ) Die Bedingung uber die Inzidenzzahlen ist trivialerweise erfUllt, falls X nur Zellen in geraden Dimensionen hat. 5). Daraus folgt o ~ n ~ 2d und n gerade, sonst. 40. (ZelluUire Kettenkornplex von lRJPd). Sei lRJPd der reelle projektive Raum fUr d E Z, d 2: 0 oder d = 00.

### Homology Theory: An Introduction to Algebraic Topology by James W. Vick (auth.)

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