### 1 construct an explicit deformation retraction of the torus

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1. Construct an explicit deformation retraction of the torus with one point deleted onto a graph consisting of two circles intersecting in a point, namely, longitude and meridian circles of the torus.

2. Construct an explicit deformation retraction of Rn- {0} onto Sn-1

3. (a) Show that the composition of homotopy equivalences XY and YZ is ahomotopy equivalence XZ . Deduce that homotopy equivalence is an equivalencerelation.

(b) Show that the relation of homotopy among maps XY is an equivalence relation.

(c) Show that a map homotopic to a homotopy equivalence is a homotopyequivalence.

4. A deformation retraction in the weak sense of a space X to a subspace A is ahomotopyft:XX such that f0= 11, f1(X) A, and ft(A) A for all t . Show that if X deformation retracts to A in this weak sense, then the inclusion mapA?Xis a homotopy equivalence.

5. Show that if a space X deformation retracts to a point x X, then for each neighborhood U of x in X there exists a neighborhood V U of x such that the inclusion map V? U is nullhomotopic

6. Show that a retract of a contractible space is contractible.

7. 10. Show that a space X is contractible iff every map f :XY , for arbitrary Y , is nullhomotopic. Similarly, show X is contractible iff every map f: YX is nullhomotopic.

Group 2 :

1. Show that composition of paths satisfies the following cancellation property: If

i.      f0.g0?f1.g1 and g0?g1 then f0?f1.

2. Show that the change-of-basepoint homomorphism βhdepends only on the homotopyclass of h.

3. For a path-connected space X, show that π1(X) is abelian iff all basepoint-changehomomorphismsβhdepend only on the endpoints of the path h.

4. Show that for a space X, the following three conditions are equivalent:

(a) Every map S1X is homotopic to a constant map, with image a point.

(b) Every map S1X extends to a map D2X.

(c) π1(X,x0) = 0 for all x0X.

5. Does the Borsuk-Ulam theorem hold for the torus? In other words, for every mapf :S1×S1→R2 must there exist (x,y) S1×S1 such that f (x,y) = f (-x,-y)?

6. From the isomorphism π1(X×Y, (x0,y0)) π1(X,x0)×π1(Y,y0) it follows that

7. loops in X×{y0} and {x0Y represent commuting elements of π1(X×Y, (x0,y0).Construct an explicit homotopy demonstrating this.

8. If X0 is the path-component of a space X containing the basepointx0 , show thatthe inclusion X0?X induces an isomorphism π1(X0,x0)π1(X,x0).

9. Show that every homomorphism π1(S1)π1(S1) can be realized as the inducedhomomorphism ?∗of a map ?: S1S1 .

10. Show that there are no retractions r :XA in the following cases:

(a)  X = R3 with Aany subspace homeomorphic to S1 .

(b)  X = S1×D2 with A its boundary torus S1×S1 .

(c)   X = S1×D2 and A the circle shown in the figure.

(d)  X = D2D2 with A its boundary S1S1 .

(e) X the Möbius band and Aits boundary circle.

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