Skip to main content
MathOverflow

Return to Question

added 8 characters in body
Source Link
Will Sawin
Will Sawin
  • 148.5k
  • 9
  • 324
  • 563

From $0\rightarrow \mathbb{Z}\rightarrow \mathbb{Z}\rightarrow \mathbb{Z}_{2}\rightarrow 0$, we have the long exact sequence $H^{1}(X,\mathbb{Z})\rightarrow H^{1}(X,\mathbb{Z}_{2})\rightarrow H^{2}(X,\mathbb{Z})$. Meanwhile $H^{1}(X,\mathbb{Z}_{2})\cong H^{1}(X,\mathbb{Z})\otimes\mathbb{Z}_{2} \oplus Tor(H^{2}(X,\mathbb{Z}),\mathbb{Z}_{2})$.

My question is: if $L$ is a real line bundle on $X$, does $w_{1}(L)$ have component in    $H^{1}(X,\mathbb{Z})\otimes\mathbb{Z}_{2}$ or $Tor(H^{2}(X,\mathbb{Z}),\mathbb{Z}_{2})$ under the above isomorphism.

From $0\rightarrow \mathbb{Z}\rightarrow \mathbb{Z}\rightarrow \mathbb{Z}_{2}\rightarrow 0$, we have the long exact sequence $H^{1}(X,\mathbb{Z})\rightarrow H^{1}(X,\mathbb{Z}_{2})\rightarrow H^{2}(X,\mathbb{Z})$. Meanwhile $H^{1}(X,\mathbb{Z}_{2})\cong H^{1}(X,\mathbb{Z})\otimes\mathbb{Z}_{2} \oplus Tor(H^{2}(X,\mathbb{Z}),\mathbb{Z}_{2})$

My question is: if $L$ is a real line bundle on $X$, does $w_{1}(L)$ have component in  $H^{1}(X,\mathbb{Z})\otimes\mathbb{Z}_{2}$ or $Tor(H^{2}(X,\mathbb{Z}),\mathbb{Z}_{2})$ under the above isomorphism

From $0\rightarrow \mathbb{Z}\rightarrow \mathbb{Z}\rightarrow \mathbb{Z}_{2}\rightarrow 0$, we have the long exact sequence $H^{1}(X,\mathbb{Z})\rightarrow H^{1}(X,\mathbb{Z}_{2})\rightarrow H^{2}(X,\mathbb{Z})$. Meanwhile $H^{1}(X,\mathbb{Z}_{2})\cong H^{1}(X,\mathbb{Z})\otimes\mathbb{Z}_{2} \oplus Tor(H^{2}(X,\mathbb{Z}),\mathbb{Z}_{2})$.

My question is: if $L$ is a real line bundle on $X$, does $w_{1}(L)$ have component in  $H^{1}(X,\mathbb{Z})\otimes\mathbb{Z}_{2}$ or $Tor(H^{2}(X,\mathbb{Z}),\mathbb{Z}_{2})$ under the above isomorphism.

added 8 characters in body
Source Link
Allen
Allen
  • 399
  • 1
  • 10

From $0\rightarrow \mathbb{Z}\rightarrow \mathbb{Z}\rightarrow \mathbb{Z}_{2}\rightarrow 0$, we have the long exact sequence $H^{1}(X,\mathbb{Z})\rightarrow H^{1}(X,\mathbb{Z}_{2})\rightarrow H^{2}(X,\mathbb{Z})$. Meanwhile $H^{1}(X,\mathbb{Z}_{2})\cong H^{1}(X,\mathbb{Z})\otimes\mathbb{Z}_{2} \oplus Tor(H^{2}(X,\mathbb{Z}),\mathbb{Z}_{2})$.

My question is: if $L$ is a real line bundle on $X$, does $w_{1}(L)$ have component in    $H^{1}(X,\mathbb{Z})\otimes\mathbb{Z}_{2}$ or $Tor(H^{2}(X,\mathbb{Z}),\mathbb{Z}_{2})$ under the above isomorphism.

From $0\rightarrow \mathbb{Z}\rightarrow \mathbb{Z}\rightarrow \mathbb{Z}_{2}\rightarrow 0$, we have the long exact sequence $H^{1}(X,\mathbb{Z})\rightarrow H^{1}(X,\mathbb{Z}_{2})\rightarrow H^{2}(X,\mathbb{Z})$. Meanwhile $H^{1}(X,\mathbb{Z}_{2})\cong H^{1}(X,\mathbb{Z})\otimes\mathbb{Z}_{2} \oplus Tor(H^{2}(X,\mathbb{Z}),\mathbb{Z}_{2})$.

My question is: if $L$ is a real line bundle on $X$, does $w_{1}(L)$ have component in  $H^{1}(X,\mathbb{Z})\otimes\mathbb{Z}_{2}$ or $Tor(H^{2}(X,\mathbb{Z}),\mathbb{Z}_{2})$ under the above isomorphism.

From $0\rightarrow \mathbb{Z}\rightarrow \mathbb{Z}\rightarrow \mathbb{Z}_{2}\rightarrow 0$, we have the long exact sequence $H^{1}(X,\mathbb{Z})\rightarrow H^{1}(X,\mathbb{Z}_{2})\rightarrow H^{2}(X,\mathbb{Z})$. Meanwhile $H^{1}(X,\mathbb{Z}_{2})\cong H^{1}(X,\mathbb{Z})\otimes\mathbb{Z}_{2} \oplus Tor(H^{2}(X,\mathbb{Z}),\mathbb{Z}_{2})$

My question is: if $L$ is a real line bundle on $X$, does $w_{1}(L)$ have component in  $H^{1}(X,\mathbb{Z})\otimes\mathbb{Z}_{2}$ or $Tor(H^{2}(X,\mathbb{Z}),\mathbb{Z}_{2})$ under the above isomorphism

Source Link
Allen
Allen
  • 399
  • 1
  • 10

One question on first Stiefel-Whitney class

From $0\rightarrow \mathbb{Z}\rightarrow \mathbb{Z}\rightarrow \mathbb{Z}_{2}\rightarrow 0$, we have the long exact sequence $H^{1}(X,\mathbb{Z})\rightarrow H^{1}(X,\mathbb{Z}_{2})\rightarrow H^{2}(X,\mathbb{Z})$. Meanwhile $H^{1}(X,\mathbb{Z}_{2})\cong H^{1}(X,\mathbb{Z})\otimes\mathbb{Z}_{2} \oplus Tor(H^{2}(X,\mathbb{Z}),\mathbb{Z}_{2})$.

My question is: if $L$ is a real line bundle on $X$, does $w_{1}(L)$ have component in $H^{1}(X,\mathbb{Z})\otimes\mathbb{Z}_{2}$ or $Tor(H^{2}(X,\mathbb{Z}),\mathbb{Z}_{2})$ under the above isomorphism.