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
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State the Mayer-Vietoris theorem.
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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 A∐B 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=A∪B.
Compute H∗(X) using the Mayer–Vietoris sequence.
4 (Fall ’14) #topology/qual/work
Compute the integral homology groups of the space X=Y∪Z 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 X⊂R3 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,n∈Z, 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 R2→T2 to induce a homeomorphism T2→T2.
- Using this homeomorphism of T2, we define a new quotient space T3A:=T2×R∼where((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?
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C1(X)={0}.
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C1(X)≠{0} but ker∂1=0 with ∂1:C1(X)→C0(X).
- ker∂1≠0 but ker∂1=im∂2 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 S2→S2.
19 (Fall ’18) #topology/qual/work
For each n∈Z give an example of a map fn:S2→S2.
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:RPn→X be a covering map. Show that if n is even, π is a homeomorphism.
21 (Fall ’17) #topology/qual/work
Let A⊂X. 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 n≥1 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=U∪V. Find, with proof, a general formula relating the Euler characteristics of X,U,V, and U∩V.
You may assume that the homologies of U,V,U∩V,X are finite-dimensional so that their Euler characteristics are well defined.
Spring 2021 #6
For each of the following spaces, compute the fundamental group and the homology groups.
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The graph Θ consisting of two edges and three vertices connecting them.
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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
Prove directly from the definition that the 0th singular homology of a nonempty path-connected space is isomorphic to Z.
Spring 2021 #9
Prove that for every continuous map f:S2n→S2n there is a point x∈S2n 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.