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user43326
user43326
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Consider $H_*(X\wedge Y;Z)$, where $X=Y=BZ/2$ for concreteness' sake. If we write $e_i$ the generator of $H_i(BZ/2;Z/2)$., we see that the $E_2=E_{\infty}$ term of the Bockstein spectral sequence for $X\wedge Y$ is trivial, thus the permanent cycles are the image of $\beta$. As we have $$\beta e_{2j}=e_{2j-1}$$ one can choose a basis of the set of permanent cycles as $$\{e_{2i-1}\otimes e_{2j-1}\}\cup \{e_{2i}\otimes e_{2j-1}+e_{2i-1}\otimes e_{2j}\}.$$ If we still denote by $e_{2i-1}$ the lift of $e_{2i-1}\in H_{2i-1}(BZ/2;Z/2)$ to $H_{2i-1}(BZ/2;Z)$, then we see that the elements of the first set lifts simply to the "products $e_{2i-1}\otimes e_{2j-1}$" of integral homology classes, whereas those in the second lift to "some sort of Massey products $\langle e_{2i-1} , 2, e_{2j-1}\rangle $".

Now, my questions are

  1. Is there any reference for this kind of facts, that is the description of the homology of the product of spaces using "Massey products"?

  2. Is there a setting in which one can "really" consider the obvious lifts of the elements $e_{2i}\otimes e_{2j-1}+e_{2i-1}\otimes e_{2j}$ as Massey product?

Consider $H_*(X\wedge Y;Z)$, where $X=Y=BZ/2$ for concreteness' sake. If we write $e_i$ the generator of $H_i(BZ/2;Z/2)$., we see that the $E_2=E_{\infty}$ term of the Bockstein spectral sequence for $X\wedge Y$ is trivial, thus the permanent cycles are the image of $\beta$. As we have $$\beta e_{2j}=e_{2j-1}$$ one can choose a basis of the set of permanent cycles as $$\{e_{2i-1}\otimes e_{2j-1}\}\cup \{e_{2i}\otimes e_{2j-1}+e_{2i-1}\otimes e_{2j}\}.$$ If we still denote by $e_{2i-1}$ the lift of $e_{2i-1}\in H_{2i-1}(BZ/2;Z/2)$ to $H_{2i-1}(BZ/2;Z)$, then we see that the elements of the first set lifts simply to the "products $e_{2i-1}\otimes e_{2j-1}$" of integral homology classes, whereas those in the second lift to "some sort of Massey products $\langle e_{2i-1} , 2, e_{2j-1}\rangle $".

Now, my questions are

  1. Is there any reference for this kind of facts?

  2. Is there a setting in which one can "really" consider the obvious lifts of the elements $e_{2i}\otimes e_{2j-1}+e_{2i-1}\otimes e_{2j}$ as Massey product?

Consider $H_*(X\wedge Y;Z)$, where $X=Y=BZ/2$ for concreteness' sake. If we write $e_i$ the generator of $H_i(BZ/2;Z/2)$., we see that the $E_2=E_{\infty}$ term of the Bockstein spectral sequence for $X\wedge Y$ is trivial, thus the permanent cycles are the image of $\beta$. As we have $$\beta e_{2j}=e_{2j-1}$$ one can choose a basis of the set of permanent cycles as $$\{e_{2i-1}\otimes e_{2j-1}\}\cup \{e_{2i}\otimes e_{2j-1}+e_{2i-1}\otimes e_{2j}\}.$$ If we still denote by $e_{2i-1}$ the lift of $e_{2i-1}\in H_{2i-1}(BZ/2;Z/2)$ to $H_{2i-1}(BZ/2;Z)$, then we see that the elements of the first set lifts simply to the "products $e_{2i-1}\otimes e_{2j-1}$" of integral homology classes, whereas those in the second lift to "some sort of Massey products $\langle e_{2i-1} , 2, e_{2j-1}\rangle $".

Now, my questions are

  1. Is there any reference for this kind of facts, that is the description of the homology of the product of spaces using "Massey products"?

  2. Is there a setting in which one can "really" consider the obvious lifts of the elements $e_{2i}\otimes e_{2j-1}+e_{2i-1}\otimes e_{2j}$ as Massey product?

Source Link
user43326
user43326
  • 3.1k
  • 17
  • 25

Homology of the product of spaces with integer coefficients and the Massey products

Consider $H_*(X\wedge Y;Z)$, where $X=Y=BZ/2$ for concreteness' sake. If we write $e_i$ the generator of $H_i(BZ/2;Z/2)$., we see that the $E_2=E_{\infty}$ term of the Bockstein spectral sequence for $X\wedge Y$ is trivial, thus the permanent cycles are the image of $\beta$. As we have $$\beta e_{2j}=e_{2j-1}$$ one can choose a basis of the set of permanent cycles as $$\{e_{2i-1}\otimes e_{2j-1}\}\cup \{e_{2i}\otimes e_{2j-1}+e_{2i-1}\otimes e_{2j}\}.$$ If we still denote by $e_{2i-1}$ the lift of $e_{2i-1}\in H_{2i-1}(BZ/2;Z/2)$ to $H_{2i-1}(BZ/2;Z)$, then we see that the elements of the first set lifts simply to the "products $e_{2i-1}\otimes e_{2j-1}$" of integral homology classes, whereas those in the second lift to "some sort of Massey products $\langle e_{2i-1} , 2, e_{2j-1}\rangle $".

Now, my questions are

  1. Is there any reference for this kind of facts?

  2. Is there a setting in which one can "really" consider the obvious lifts of the elements $e_{2i}\otimes e_{2j-1}+e_{2i-1}\otimes e_{2j}$ as Massey product?