Is taking adjunction space compatible with topological product?

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I am reading John W. Milnor, Morse Theory, though it might be a bit difficult, and I was showing in Lemma 3.6 that $kcirc l$ is homotopic to identity map. (Until here is for those who have the specified book.) There, I found I have to construct a homotopy $(D^lambdacup_varphiX)times [0, 1]to D^lambdacup_varphiX$.



In order to define this continuous map, it seems to me that if homeomorphism like $(Xcup_varphiY)times Zapprox (Xtimes Z)cup_varphitimes id_Z(Ytimes Z)
$ holds, then it is easier to define the map because I can make a better use of the universal mapping property. And, this is apparently true according to my intuition. Therefore, I would like to ask if that holds. Any comments appreciated.




general-topology algebraic-topology cw-complexes




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  • I'm not sure what your notation means, but in general quotient maps are not respected by products. If $Ato B$ is a quotient map, the induced map $Atimes Cto Btimes C$ need not be, but if $C$ is locally compact Hausdorff it will be.
    – Lord Shark the Unknown
    Jul 26 at 10:11










  • Thank you very much!
    – neander
    Jul 26 at 10:16














up vote
1
down vote

favorite












I am reading John W. Milnor, Morse Theory, though it might be a bit difficult, and I was showing in Lemma 3.6 that $kcirc l$ is homotopic to identity map. (Until here is for those who have the specified book.) There, I found I have to construct a homotopy $(D^lambdacup_varphiX)times [0, 1]to D^lambdacup_varphiX$.



In order to define this continuous map, it seems to me that if homeomorphism like $(Xcup_varphiY)times Zapprox (Xtimes Z)cup_varphitimes id_Z(Ytimes Z)
$ holds, then it is easier to define the map because I can make a better use of the universal mapping property. And, this is apparently true according to my intuition. Therefore, I would like to ask if that holds. Any comments appreciated.




general-topology algebraic-topology cw-complexes




share|cite|improve this question



















  • I'm not sure what your notation means, but in general quotient maps are not respected by products. If $Ato B$ is a quotient map, the induced map $Atimes Cto Btimes C$ need not be, but if $C$ is locally compact Hausdorff it will be.
    – Lord Shark the Unknown
    Jul 26 at 10:11










  • Thank you very much!
    – neander
    Jul 26 at 10:16












up vote
1
down vote

favorite









up vote
1
down vote

favorite











I am reading John W. Milnor, Morse Theory, though it might be a bit difficult, and I was showing in Lemma 3.6 that $kcirc l$ is homotopic to identity map. (Until here is for those who have the specified book.) There, I found I have to construct a homotopy $(D^lambdacup_varphiX)times [0, 1]to D^lambdacup_varphiX$.



In order to define this continuous map, it seems to me that if homeomorphism like $(Xcup_varphiY)times Zapprox (Xtimes Z)cup_varphitimes id_Z(Ytimes Z)
$ holds, then it is easier to define the map because I can make a better use of the universal mapping property. And, this is apparently true according to my intuition. Therefore, I would like to ask if that holds. Any comments appreciated.




general-topology algebraic-topology cw-complexes




share|cite|improve this question











I am reading John W. Milnor, Morse Theory, though it might be a bit difficult, and I was showing in Lemma 3.6 that $kcirc l$ is homotopic to identity map. (Until here is for those who have the specified book.) There, I found I have to construct a homotopy $(D^lambdacup_varphiX)times [0, 1]to D^lambdacup_varphiX$.



In order to define this continuous map, it seems to me that if homeomorphism like $(Xcup_varphiY)times Zapprox (Xtimes Z)cup_varphitimes id_Z(Ytimes Z)
$ holds, then it is easier to define the map because I can make a better use of the universal mapping property. And, this is apparently true according to my intuition. Therefore, I would like to ask if that holds. Any comments appreciated.





general-topology algebraic-topology cw-complexes





share|cite|improve this question










share|cite|improve this question




share|cite|improve this question









asked Jul 26 at 10:01









neander

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  • I'm not sure what your notation means, but in general quotient maps are not respected by products. If $Ato B$ is a quotient map, the induced map $Atimes Cto Btimes C$ need not be, but if $C$ is locally compact Hausdorff it will be.
    – Lord Shark the Unknown
    Jul 26 at 10:11










  • Thank you very much!
    – neander
    Jul 26 at 10:16
















  • I'm not sure what your notation means, but in general quotient maps are not respected by products. If $Ato B$ is a quotient map, the induced map $Atimes Cto Btimes C$ need not be, but if $C$ is locally compact Hausdorff it will be.
    – Lord Shark the Unknown
    Jul 26 at 10:11










  • Thank you very much!
    – neander
    Jul 26 at 10:16















I'm not sure what your notation means, but in general quotient maps are not respected by products. If $Ato B$ is a quotient map, the induced map $Atimes Cto Btimes C$ need not be, but if $C$ is locally compact Hausdorff it will be.
– Lord Shark the Unknown
Jul 26 at 10:11




I'm not sure what your notation means, but in general quotient maps are not respected by products. If $Ato B$ is a quotient map, the induced map $Atimes Cto Btimes C$ need not be, but if $C$ is locally compact Hausdorff it will be.
– Lord Shark the Unknown
Jul 26 at 10:11












Thank you very much!
– neander
Jul 26 at 10:16




Thank you very much!
– neander
Jul 26 at 10:16















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