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INIDEPENDENT UNIVERSITY OF MOSCOW

8. APPLICATIONS OF HOMOLOGY. 1. Fundamental classes

TOPOLOGY-2, FALL 2012

One says that a compact oriented submanifold N M of dimension n realizes a homology class (N ) Hn (M ) where : N M is the tautological embedding and N Hn (N ) is the fundamental class of N chosen according to the orientation. Similarly, any compact submanifold N realizes a homology class mod2 (here N may be not oriented and even non-orientable). Problem 1. (a) Let N CP 2 be a smooth curve of degree n. Prove that N is orientable and realizes a class n Z = H2 (CP 2 ). (b) Prove a similar statement for RP 2 . (c) Prove a similar statement for H2n-2 (CP n ). (d) Realize all the classes in H (RP n ; Z=2Z) by smooth submanifolds. Which of them realize classes in H (RP n ; Z)? 2. Degree of a smooth map n n n Problem 2. (a) Let A : Rn Rn be linear and invertible. Prove that the map A = A|S1 1 : S1 -1 A(S1 -1 ), n where S1 -1 Rn is the unit sphere centered at the origin, is a di eomorphism, and that A : Hn-1 (S n-1 ) n-1 ) is the multiplication by ±1 = sign det A. (b) Let f : Rn Rn be a smooth map such that f (0) = 0 Hn-1 (S n n n and f (0) is nondegenerate. Prove that for " > 0 small enough the map f |S" 1 f (S" -1 ), where S" -1 Rn is the "-sphere centered in x, is a di eomorphism homotopic to f (0). Problem 3. Let M be a smooth n-manifold, and U M an open set di eomorphic to an n-ball. Prove that Hi (M ) = Hi (M \ U; @ U ) for all i. Let M , N be smooth manifolds of the same dimension n, f : M N be a smooth map, and y be its regular value: if f (x) = y then f (x) : Tx M Ty N is nondegenerate. Problem 4. (a) Prove that f -1 M is discrete; if M is compact then it is nite: f -1 (y) = {x1 ; : : : ; xN }. (b) Let U" N be an open "-ball (in some Riemannian metric) centered in y. Prove that for " small enough the preimage f -1 (U" ) is a nite disjoint union N Vi where for every i one has xi Vi and the restriction f |@ Vi : @ Vi @ U" i=1 is a di eomorphism homotopic to f (xi ). Problem 5. In the notation of Problem 4 consider the diagram Hn (M ) = Hn (M \ i Vi ; i @ Vi ) - Hn-1 ( i @ Vi ) f f Hn (N ) = Hn (N \ U" ; @ U" ) - Hn-1 (@ U" ) where the horizontal arrows are part of the exact sequence of the pairs (M \ i Vi ; i @ Vi ) and (N \ U" ; @ U" ), respectively, and prove that deg f = N sign det f (xi ). i=1
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3. Euler characteristic Problem 6. Using the Mayer{Vietoris sequence, prove that (X ) = (X1 ) + (X2 ) - (X1 X2 ), where X1 and X2 are simplicial subspaces of a simplicial space X . Prove the same equality \by counting simplices". Problem 7. Compute the Euler characteristics of (a) RP n ; (b) CP n ; (c) all compact 2-manifolds. Problem 8. Prove that the Euler characteristic of a compact smooth manifold of odd dimension is 0. Problem 9. Prove that if p : E B is a ber bundle with a ber F , and B , E and F are simplicial spaces then (E ) = (B )(F ). In particular, (B в F ) = (B )(F ) and (E ) = n(B ) if p is an n-sheeted covering. Problem 10. Let f : CP 1 CP 1 be a meromorphic function of degree n, and a1 ; : : : ; ak be its critical points. Call di Z0 the multiplicity of f at ai if f (z ) - f (ai ) = (z - ai )di + o((z - ai )di ); here z is any local holomorphic coordinate on CP 1 near ai . Prove that k=1 di = 2n - 2 + k. What does this formula give if f is a polynomial? i n be a compact oriented hypersurface (smooth submanifold of dimension n - 1). For Problem 11. Let M R x M denote by v(x) the unit vector normal to M at x; the choice between two such vectors is dictated (how?) by the orientation of M . So, v is the map M S n-1 . Prove that if n is even then deg v = 0, otherwise deg v = 2(M ). 4. Fixed points Problem 12. Construct a continuous map f : X X without xed points or prove that it does not exist. Can f be homotopic to the identity map? (a) X = S n ; (b) X = RP n ; (c) X = CP n ; (d) X is a sphere with g handles; (e) X is a 2-disk with n holes, n > 0.