Download A mathematical gift, 3, interplay between topology, by Kenji Ueno, Koji Shiga, Shigeyuki Morita, Toshikazu Sunada PDF

By Kenji Ueno, Koji Shiga, Shigeyuki Morita, Toshikazu Sunada

This e-book brings the sweetness and enjoyable of arithmetic to the school room. It bargains severe arithmetic in a full of life, reader-friendly kind. integrated are workouts and lots of figures illustrating the most ideas. the 1st bankruptcy talks in regards to the concept of manifolds. It contains dialogue of smoothness, differentiability, and analyticity, the belief of neighborhood coordinates and coordinate transformation, and an in depth clarification of the Whitney imbedding theorem (both in susceptible and in robust form). the second one bankruptcy discusses the idea of the world of a determine at the aircraft and the quantity of an outstanding physique in house. It comprises the facts of the Bolyai-Gerwien theorem approximately scissors-congruent polynomials and Dehn's resolution of the 3rd Hilbert challenge. this can be the 3rd quantity originating from a chain of lectures given at Kyoto collage (Japan). it truly is appropriate for school room use for prime institution arithmetic lecturers and for undergraduate arithmetic classes within the sciences and liberal arts.

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Extra resources for A mathematical gift, 3, interplay between topology, functions, geometry, and algebra

Example text

Wir versehen lRlP'oo mit der Direkte-Limesopologie bezuglich der Filtrierung lRlP'1 ~ lRlP'2 ~ .... Wir nennen lRlP'OO den unendlich dimensionalen reellen projektiven Raum. Die Filtrierung lRlP'1 ~ lRlP'2 ~ ... definiert die Struktur eines CW-Komplexes auf lRlP'OO. Sei ClP'd der komplexe d-dimensionale projektive Raum. Als Menge besteht er aus den 1-dimensionalen komplexen Untervektorraumen von ed + 1 . Als topologischen Raum definiert man ihn als den Quotientenraum von S2d+1 unter der S1-0peration, die durch komplexe Multiplikation gegeben ist, wobei wir S1 ~ e und S2d+1 ~ ed+1 auffassen.

J = falls q}«I,O)) = eJ und q}«-I,O)) =I- e6' falls q}«-I,O)) = e~ und q}«I,O)) =I- ej , sonst. 36. 35) ist. Beweis: Wir behandeln nur den Fall n 2': 2, der Fall n kommutative Diagramm = 1 geht analog. (Xn - 1 - er1)) 1"" H n - 1(sn-1) wobei [ jeweils Inklusionen und pr Projektionen von topologischen Raumen bezeichnet und prj die Projektion auf den j-ten Summanden ist. ,j gegeben ist. 19 (d). 28) eingefiihrt worden sind. 18) identifizieren. Daher stimmt die Komposition 50 3 CW-Komplexe mit der Komposition von 1-l n _dD n - l , sn-l) 1i n Pn-l : - 1-l o( {e}) ---+ 1-l n - 1 (D n - dp r\ 1-l n - 1 (D n 1i n - 1 / sn-2, l , sn-l) und der Abbildung {e}) d t ) - \ 1-l n - 1 (D n - l / sn-2) 1i n - d u-;=-j)) 1-l n _dS n - 1) iiberein.

X = Xn fur ein nEZ, dann ist der Beweis von (b) analog zu dem von (a). Urn den allgemeinen Fall zu behandeln, muss man noch beweisen, dass die Inklusion In: Xn --+ X fur k < n einen Isomorphismus 48 3 CW-Komplexe induziert. 7 (c)). B. 1 in IV auf Seite 149]). 0 Als nachstes berechnen wir den Kettenkomplex C,;t· (X, A) fur einen relativen CWKomplex (X, A). 26) ist. Fur n = 1, i E In und j E I n- 1 definieren wir Qr- inz~. ',J = falls q}«I,O)) = eJ und q}«-I,O)) =I- e6' falls q}«-I,O)) = e~ und q}«I,O)) =I- ej , sonst.

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