Документ взят из кэша поисковой машины. Адрес оригинального документа : http://www.mccme.ru/ium/postscript/f12/topology2-Sem7.pdf Дата изменения: Wed Nov 7 18:28:52 2012 Дата индексирования: Mon Feb 4 17:50:16 2013 Кодировка: Поисковые слова: m 5

INIDEPENDENT UNIVERSITY OF MOSCOW

7. SINGULAR HOMOLOGY

TOPOLOGY-2, FALL 2012

and B Rn are closed and homeomorphic then Hk (Rn \ A) Hk (Rn \ B ). = (b) Can one replace with S in the previous statement? (c) (Jordan's lemma in Rn ) If A Rn is homeomorphic to S n-1 , then Rn \ A consists of two connected components.
Rn
n

Problem 1. (a) (Alexander) If A
n

Rn

Problem 2. (a) Let

= {(x0 ; : : : ; xn ) | x0 ; : : : ; xn 0; x0 + · · · + xn = 1} be a standard simplex with the vertices ai = (0; : : : ; 1; : : : ; 0) (1 at the i-th position). Let Ks n в [0; 1] be a simplex with the vertices (ai ; 0), s i n и (ai ; 1), 0 i s. Prove that the simplices K0 ; : : : ; Kn have no common internal points and that their union is the whole n в [0; 1]. (b) Generalize the construction above and describe a splitting of the product n в m into simplices ([0; 1] is equivalent to the simplex 1 ).
Xn @ n;X Xn-1 @ n-1;X : : : and Y = · · · Yn @ n;Y Yn-1 @ n-1;Y : : : - - - - @ n;X @ n;Y @ n-1;X @ n-1;Y be complexes of Abelian groups, and X Y = · · · Xn Yn - Xn-1 Yn-1 - : : : be their tensor product. (a) Prove that if X and Y are complexes of vector spaces then H (X Y ) = H (X ) H (Y ) (i.e. Hn (X Y ) = n=0 Hk (X ) Hn-k (Y )). (b) Prove that in general H (X Y ) = H (X ) H (Y ) G for some k Abelian group G that depends on H (X ) and H (Y ) only. Give an example where G is nontrivial. Remark . Problem 3 is motivated by the fact that the CW-complex of the product P в Q of two CW-spaces P and Q is the tensor product of the CW-complexes of P and Q.

def

Problem 3 (Kunneth formula). Let X =

···

Problem 4. Let X be a topological space, and U1 ; : : : ; UN

X be its open subsets such that X = N Ui and i=1 any nite intersection Ui1 · · · Uik is either empty or homeomorphic to Rn . Let N (U1 ; : : : ; UN ) be a simplicial complex contatining vertices a1 ; : : : ; aN and a simplex ai1 : : : aik if and only if Ui1 · · · Uik = . (a) Prove that N (U1 ; : : : ; UN ) is homotopy equivalent to X . (b) Is the statement above still true if Ui1 · · · Uik is allowed to be homeomorphic to a disjoint union of several copies of Rn ? Consider, in particular, the case when X is a nite discrete space.