Homology and Degree Theory

1 (Spring ’09) #topology/qual/work

Compute the homology of the one-point union of S1×S1 and S1.

2 (Fall ’06) #topology/qual/work

  • State the Mayer-Vietoris theorem.

  • Use it to compute the homology of the space X obtained by gluing two solid tori along their boundary as follows. Let D2 be the unit disk and let S1 be the unit circle in the complex plane C. Let A=S1×D2 and B=D2×S1.

Then X is the quotient space of the disjoint union AB obtained by identifying (z,w)A with (zw3,w)B for all (z,w)S1×S1.

3 (Fall ’12) #topology/qual/work

Let A and B be circles bounding disjoint disks in the plane z=0 in R3. Let X be the subset of the upper half-space of R3 that is the union of the plane z=0 and a (topological) cylinder that intersects the plane in C=AB.

Compute H(X) using the Mayer–Vietoris sequence.

4 (Fall ’14) #topology/qual/work

Compute the integral homology groups of the space X=YZ which is the union of the sphere Y={x2+y2+z2=1} and the ellipsoid Z={x2+y2+z24=1}.

5 (Spring ’08) #topology/qual/work

Let X consist of two copies of the solid torus D2×S1, glued together by the identity map along the boundary torus S1×S1. Compute the homology groups of X.

6 (Spring ’17) #topology/qual/work

Use the circle along which the connected sum is performed and the Mayer-Vietoris long exact sequence to compute the homology of {\mathbf{RP}}^2 # {\mathbf{RP}}^2.

7 (Fall ’15) #topology/qual/work

Express a Klein bottle as the union of two annuli.

Use the Mayer Vietoris sequence and this decomposition to compute its homology.

8 (Spring ’09) #topology/qual/work

Let X be the topological space obtained by identifying three distinct points on S2. Calculate H(X;Z).

9 (Fall ’05) #topology/qual/work

Compute H0 and H1 of the complete graph K5 formed by taking five points and joining each pair with an edge.

10 (Fall ’18) #topology/qual/work

Compute the homology of the subset XR3 formed as the union of the unit sphere, the z-axis, and the xy-plane.

11 (Spring ’05/Fall ’13) #topology/qual/work

Let X be the topological space formed by filling in two circles S1×{p1} and S1×{p2} in the torus S1×S1 with disks.

Calculate the fundamental group and the homology groups of X.

12 (Spring ’19) #topology/qual/work

  • Consider the quotient space T2=R2/where(x,y)(x+m,y+n) for m,nZ, and let A be any 2×2 matrix whose entries are integers such that detA=1.

Prove that the action of A on R2 descends via the quotient R2T2 to induce a homeomorphism T2T2.

  • Using this homeomorphism of T2, we define a new quotient space T3A:=T2×Rwhere((x,y),t)(A(x,y),t+1)

Compute H1(T3A) if A=(1101).

13 (Spring ’12) #topology/qual/work

Give a self-contained proof that the zeroth homology H0(X) is isomorphic to Z for every path-connected space X.

14 (Fall ’18) #topology/qual/work

It is a fact that if X is a single point then H1(X)={0}.

One of the following is the correct justification of this fact in terms of the singular chain complex.

Which one is correct and why is it correct?

  • C1(X)={0}.

  • C1(X){0} but ker1=0 with 1:C1(X)C0(X).

  • ker10 but ker1=im2 with 2:C2(X)C1(X).

15 (Fall ’10) #topology/qual/work

Compute the homology groups of S2×S2.

16 (Fall ’16) #topology/qual/work

Let Σ be a closed orientable surface of genus g. Compute Hi(S1×Σ;Z) for i=0,1,2,3.

17 (Spring ’07) #topology/qual/work

Prove that if A is a retract of the topological space X, then for all nonnegative integers n there is a group Gn such that Hn(X)Hn(A)Gn.

Here Hn denotes the nth singular homology group with integer coefficients.

18 (Spring ’13) #topology/qual/work

Does there exist a map of degree 2013 from S2S2.

19 (Fall ’18) #topology/qual/work

For each nZ give an example of a map fn:S2S2.

For which n must any such map have a fixed point?

20 (Spring ’09) #topology/qual/work

  • What is the degree of the antipodal map on the n-sphere?

(No justification required)

  • Define a CW complex homeomorphic to the real projective n-space RPn.
  • Let π:RPnX be a covering map. Show that if n is even, π is a homeomorphism.

21 (Fall ’17) #topology/qual/work

Let AX. Prove that the relative homology group H0(X,A) is trivial if and only if A intersects every path component of X.

22 (Fall ’18) #topology/qual/work

Let D be a closed disk embedded in the torus T=S1×S1 and let X be the result of removing the interior of D from T . Let B be the boundary of X, i.e. the circle boundary of the original closed disk D.

Compute Hi(T,B) for all i.

23 (Fall ’11) #topology/qual/work

For any n1 let Sn={(x0,,xn)|x2i=1} denote the n dimensional unit sphere and let E={(x0,...,xn)|xn=0} denote the “equator”.

Find, for all k, the relative homology Hk(Sn,E).

24 (Spring ’12/Spring ’15) #topology/qual/work

Suppose that U and V are open subsets of a space X, with X=UV. Find, with proof, a general formula relating the Euler characteristics of X,U,V, and UV.

You may assume that the homologies of U,V,UV,X are finite-dimensional so that their Euler characteristics are well defined.

Spring 2021 #6

problem (Spring 2021, 6):

For each of the following spaces, compute the fundamental group and the homology groups.

  • The graph Θ consisting of two edges and three vertices connecting them.

  • The 2-dimensional cell complex Θ2 consisting of a closed circle and three 2-dimensional disks each having boundary running once around that circle.

Spring 2021 #7

problem (Spring 2021, 7):

Prove directly from the definition that the 0th singular homology of a nonempty path-connected space is isomorphic to Z.

Spring 2021 #9

problem (Spring 2021, 9):

Prove that for every continuous map f:S2nS2n there is a point xS2n such that either f(x)=x or f(x)=x.

You may use standard facts about degrees of maps of spheres, including that the antipodal map on S2n has degree d=1.

#topology/qual/work #6 #7 #9