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Will Sawin
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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.

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Allen
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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

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Allen
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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.