MAT 215B: PROBLEM SET 1
Abstract. This problem set corresponds to the first week of the course MAT-215B Spring 2024. It is due Friday Apr 12 at 9:00pm submitted via Gradescope.
Task: Solve two of the problems below and submit it through Gradescope by Friday Apr 12 at 9pm. Be rigorous and precise in writing your solutions.
Problem 1. First, check that the following chain complexes (C•, ∂) of Z-modules are indeed chain complexes (i.e. the differential square to zero). Second, compute the homology of each of them. Notation: A map between Z-modules will be given in its matrix expression, so that the image of an elements is given by left multiplication by that matrix. Elements in a Z-modules are represented by vertical vectors. The 0th graded piece is indicated by [0].
(1) 0 Z Z[0] 0 (0)
(2) 0 Z Z[0] 0 (2)
That is, the map Z −→ Z is x 7−→ (2) · x = 2x.
(3) 0 Z Z8[0] 0 (5) (mod 8)
That is, the map Z −→ Z8 is reduction modulo 8 and multiplication by 5.
(4) 0 Z Z8[0] 0 (2) (mod 8)
(5) 0 Z Z Z Z7 Z7[0] 0 (0) (7) (1) (mod 7) (0)
(6) 0 Z Z 2 Z 2 Z[0] 0 ( 3 1 ) −1 3 −2 6 ( 2 −1 ) 1 2 DUE TO FRIDAY APR 12 2024
Problem 2. Compute the homology of the following chain complexes of Z-modules:
(1) 0 Z 3 Z 4 [0] 0 1 3 −1 −1 2 1 0 −1 1 2 −1 0
(2) 0 Z 3 Z 3 [0] 0 5 6 −3 −2 4 3 0 −1 0 !
(3) 0 Z 4 Z 2 [0] 0 4 5 −11 2 1 −2 4 4
Problem 3. Find a ∆-complex structure on each of the following topological spaces X:
(1) X = Σg, the orientable genus g surface.
(2) X = S n, the n-sphere, n ∈ N.
(3) X = RPn, the real projective n-space, n ∈ N.
Problem 4. Compute the simplicial homology1 of each of the following topological spaces X:
(1) X = S 1 .
(2) X = Wn S 1 , the wedge of n circles.
(3) X = Σg, the orientable genus g surface.
(4) X = RP2 , the real projective plane.
(5) X = RP2#RP2 , the connected sum of two real projective planes.
Problem 5. Find a ∆-complex structure on each of the following topological spaces X:
(1) The submanifold X = {(z, w) ∈ C 2 : z 2 = w 3} ∩ {(z, w) ∈ C 2 : |z| 2 + |w| 2 = 1} ⊆ C 2 .
(2) The submanifold X = {[z1 : z2 : z3] ∈ CP2 : z1 + z2 − 4z3 = 0} ⊆ CP2 .
(3) X = SO(3), the special group of orthogonal (3 × 3)-matrices, i.e. X = A ∈ M3×3(R) : A · A t = Id3, det(A) = 1 ⊆ M3×3(R) ∼= R 9 .
Problem 6. Compute the simplicial homology of each of the following topological spaces X:
(1) The submanifold X = {(z1, z2, z3) ∈ C 3 : z 2 1 + z 3 2 + z 3 = 0} ∩ {(z1, z2, z3) ∈ C 3 : |z1| 2 + |z2| 2 + |z3| 2 = 1} ⊆ C 3 .
(2) The submanifold X = {[z1 : z2 : z3] ∈ CP2 : z 2 1 + z 2 2 + z 2 3 = 0} ⊆ CP2 .
(3) (Optional) The smooth manifold
Gr2,4(R) = {V ⊆ R 4 : V ⊆ R 4 oriented vector subspace of dimension dimR(V ) = 2}, i.e. the real Grassmannian of oriented 2-planes through the origin inside R4 .