MATH 636, HW5. DUE MAY 29.
1. (a) Let X be a topological space such that Hq (X, Z) is finitely generated for
every q and zero for q large enough. Prove that for every field F , one has
X
(−1)q dim H q (X, F ) = χ(X).
q
(b) Prove that the Euler characteristic of any odd-dimensional compact manifold is
equal to 0. (You may use the following fact: for every compact manifold M the groups
Hq (M, Z) are finitely generated.)
2. (a) Read pages 252-254 of Hatcher’s book on manifolds with boundary.
(b) Let N be a compact connected oriented 4n-dimensional manifold. The signature
of N is defined to be the signature of the symmetric bilinear form1
∨
H 2n (N, R) × H 2n (N, R) −→ H 4n (N, R) = R.
Prove that if N is the boundary of a compact orientable manifold M then the signature
of N is zero. (Hint: show that the signature of a non-degenerate symmetric bilinear
on a real vector space V is 0 if and only if there is a subspace W ⊂ V , with 2 dim W =
dim V , such that the restriction of the form to W is 0. Then use the Poincaré duality
for manifolds with boundary.)
(c) Show that CP 2n1 × CP 2n2 × · · · × CP 2nk is not homotopy equivalent to the
boundary of a compact orientable manifold.
3. Let M be a manifold. A d-dimensional submanifold N ⊂ M is a closed subspace
such that for every point x ∈ N there exists an open neighborhood x ∈ V ⊂ M and
∼
and a homeomorphism V −→ Rd × Rn−d which maps N ∩ V homeomorphically to
d
d
d
n−d
R =R ×0⊂R ×R
.
(a) Let N ⊂ M be as above. Prove that for an open cover M = U ∪ W , one has the
Mayor-Vietoris long exact sequence
· · · → H q (M, M − N, A) → H q (U, U − U ∩ N, A) ⊕ H q (W, W − W ∩ N, A)
→ H q (U ∩ W, U ∩ W − U ∩ W ∩ N, A) → H q+1 (M, M − N, A) → · · ·
(b) Define the homomorphism
φ : H q (N, A) ⊗ H p (M, M − N, A) → H p+q (M, M − N, A)
as follows: given α ∈ H q (N, A) choose an open neighborhood N ⊂ U ⊂ M such
∼
that α is a restriction of some class α̃ ∈ H q (U, A). Given β ∈ H p (M, M − N, A) −→
p
H (U, U − N, A), set
∼
φ(α ⊗ β) = α̃ ∪ β ∈ H p+q (U, U − N, A) −→ H p+q (M, M − N, A).
Show that φ is well defined (i.e. does not depend on the choice of U and α̃ and defines
a structure of a H ∗ (N, A)-module on H ∗ (M, M − N, A).
(c) Assume that M and N are A-oriented. Prove that there exists a unique element
]N [∈ H n−d (M, M − N, A) with the following property: for every point x ∈ N and a
1The signature of a non-degenerate symmetric bilinear form on a real vector space is computed
as follows: choose a basis in which the form is represented by a diagonal matrix. Then the signature
is the number of positive diagonal entries minus the number of negative diagonal entries.
1
2
MATH 636, HW5. DUE MAY 29.
coordinate neighborhood x ∈ V ⊂ M , the restriction ]N ∩ V [ of ]N [ to H n−d (V, V −
∼
N ∩ V, A) −→ H n−d (Rd × Rn−d , Rd × Rn−d − Rd , A) = A is the generator determined
by the A-orientations of M and N , that is
N
]N ∩ V [ ∩µM
x = µx .
(The image of ]N [ in H n−d (M, A) is called the fundamental class of N . In class, we
have discussed another construction of the fundamental class in the special case when
N and M are compact.)
(d) Moreover, show that H ∗ (M, M − N, A) is a free H ∗ (N, A)-module generated by
]N [. That is H q (M, M − N, A) = 0, for q < n − d and the map
H i (N, A) → H n−d+i (M, M − N, A),
which takes α ∈ H i (N, A) to φ(α, ]N [) is an isomorphism. Remark: this is a difficult
problem; you may first try to prove this assuming the existence of an open neighborhood N ⊂ U ⊂ M , such that N is a deformation retract of U , and that A is a field.
As a corollary, we obtain the Gysin long exact sequence
· · · → H q−n+d (N, A) → H q (M, A) → H q (M − N, A) → H q−n+d+1 (N, A) → · · ·