Math 527/427
Homework 6—Final
14 December 2015
1. Calculate the following cohomology groups:
(a) H∗ (RP 2 × RP 2 ; Z);
(b) H∗ (RP 2 × RP 4 ; Z).
Describe, with proof, the ring structure on H∗ (RP 2 ×RP 2 ; Z). What is the ring structure on H∗ (RP 2 ×
RP 4 ; Z)? (Warning: I think this last part is very hard using only the techniques from this course. Do
not treat this part of the question as mandatory.)
2. Throughout this question, we use projective coordinates [z 0 ; z 1 ; . . . ; z n ] for points in CP n . Write
x n = [1; 0; . . . ; 0], and use this as a basepoint for CP n .
(a) Suppose π is a permutation of {0, . . . , n}. Show that the map f π : CP n → CP n given by sending
[z 0 ; z 1 ; . . . ; z n ] = [z π(0) ; z π(1) ; . . . ; z π(n) ] is homotopic to the identity map. (It may be helpful to
consider π as represented by an element of GLn+1 (C)).
(b) Given CP n and CP m , one may define a map s n,m : CP n × CP m → CP nm+n+m (the Segre embedding) by the formula
([z 0 ; z 1 ; . . . ; z n ], [w 0 ; w 1 ; . . . ; w m ]) 7→ ([z 0 w 0 ; z 0 w 1 ; . . . ; z 0 ; w m ; z 1 w 0 ; . . . ; z 1 w m ; z 2 w 0 ; . . . ; z n w m ]).
Calculate the map in cohomology induced by the map f n : CP n → CP nm+n+m given by composing the inclusion CP n × {x m } ,→ CP n × CP m and the Segre embedding s n,m .
∗
(c) Describe the map of cohomology rings s n,m
: H∗ (CP nm+n+m ; Z) → H∗ (CP n × CP m ; Z).
3. Let F be a field.
(a) Suppose
0
0
/B
/A
f
/A
/B
/C
g
/C
/0
h
/0
is a commutative diagram of finite dimensional vector spaces, in which the rows are exact.
Show that the traces of these maps satisfy Tr(g ) = Tr( f ) + Tr(h).
(b) Suppose f : C • → C • is a map of bounded (i.e. only finitely many nonzero terms) chain complexes of finite-dimensional F vector spaces. Define the Lefschetz number, Λ f , of f to be
∞
X
(−1)i Tr( f ).
i =−∞
If f ∗ is the induced map on homology, f ∗ : H∗ (C • ) → H∗ (C • )— here interpreted as a chain
complex with 0 differentials—show that Λ f = Λ f ∗ .
1
(c) Let X be a finite simplicial complex (i.e. a ∆-complex where the vertices of each simplex determine the simplex uniquely) such that X ' RP 2 × RP 4 . Recall that a map f : X → X is said
to be simplicial if f takes simplices to simplices (and therefore induces a map on simplicial
chain complexes). Suppose f : X → X is a simplicial map. Show there is some point in x such
that f (x) = x.1
(d) Repeat the previous question but with X ' RP 2 × S 4 , and with the additional hypothesis that
f 3 = id.
1 The ’simplicial’ hypotheses may all be lifted & replaced by something like ‘X is a compact, locally contractible subset of Rn ’—
see Section 2.C and Appendix A in Hatcher.
2