##### Reference no: EM13371631

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 R^{n}- {0} onto *S*^{n}^{-}^{1}

3. (a) Show that the composition of homotopy equivalences *X*→*Y *and *Y*→*Z *is ahomotopy equivalence *X*→*Z *. Deduce that homotopy equivalence is an equivalencerelation.

(b) Show that the relation of homotopy among maps *X*→*Y *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 ahomotopy*f*_{t}:*X*→*X *such that *f*_{0}= 11, *f*_{1}*(X) *⊂*A*, and *f*_{t}(A) ⊂*A *for all *t *. Show that if *X *deformation retracts to *A *in this weak sense, then the inclusion map*A*?*X*is 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 *:*X*→*Y *, for arbitrary *Y *, is nullhomotopic. Similarly, show *X *is contractible **iff** every map *f*: *Y*→*X *is nullhomotopic.

Group 2 :

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

i. *f*_{0}.*g*_{0}?*f*_{1}.*g*_{1} and *g*_{0}?*g*_{1} then *f*_{0}?*f*_{1}.

2. Show that the change-of-basepoint homomorphism *β*_{h}depends only on the homotopyclass of *h*.

3. For a path-connected space *X*, show that *π*_{1}*(X) *is abelian **iff** all basepoint-changehomomorphisms*β*_{h}depend only on the endpoints of the path *h*.

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

(a) Every map *S*^{1}→*X *is homotopic to a constant map, with image a point.

(b) Every map *S*^{1}→*X *extends to a map *D*^{2}→*X*.

(c) *π*_{1}*(X,x*_{0}*) *= 0 for all *x*_{0}∈*X*.

5. Does the Borsuk-Ulam theorem hold for the torus? In other words, for every map*f *:*S*^{1}×*S*^{1}→R^{2} must there exist *(x,y) *∈*S*^{1}×*S*^{1} such that *f (x,y) *= *f (*-*x,*-*y)*?

6. From the isomorphism *π*_{1}*(X*×*Y, (x*_{0}*,y*_{0}*)) *≈ *π*_{1}*(X,x*_{0}*)*×*π*_{1}*(Y,y*_{0}*) *it follows that

7. loops in *X*×{*y*_{0}} and {*x*_{0}}×*Y *represent commuting elements of *π*_{1}*(X*×*Y, (x*0*,y*0*)*.Construct an explicit homotopy demonstrating this.

8. If *X*_{0} is the path-component of a space *X *containing the basepoint*x*_{0} , show thatthe inclusion *X*_{0}?*X *induces an isomorphism *π*_{1}*(X*_{0}*,x*_{0}*)*→*π*_{1}*(X,x*_{0}*)*.

9. Show that every homomorphism *π*_{1}*(S*^{1}*)*→*π*_{1}*(S*^{1}*) *can be realized as the inducedhomomorphism *?*∗of a map *?*: *S*^{1}→*S*^{1} .

10. Show that there are no retractions *r *:*X*→*A *in the following cases:

(a) *X *= R^{3} with *A*any subspace homeomorphic to *S*^{1} .

(b) *X *= *S*^{1}×*D*^{2} with *A *its boundary torus *S*^{1}×*S*^{1} .

(c) *X *= *S*^{1}×*D*^{2} and *A *the circle shown in the figure.

(d) *X *= *D*^{2}∨*D*^{2} with *A *its boundary *S*^{1}∨*S*^{1} .

(e) *X *the Möbius band and *A*its boundary circle.