Wu formula for manifolds with boundary
$begingroup$
The classical Wu formula claims that if $M$ is a smooth closed $n$-manifold with fundamental class $zin H_n(M;mathbb{Z}_2)$, then the total Stiefel-Whitney class $w(M)$ is equal to $Sq(v)$, where $v=sum v_iin H^*(M;mathbb{Z}_2)$ is the unique cohomology class such that
$$langle vcup x,zrangle=langle Sq(x),zrangle$$
for all $xin H^*(M;mathbb{Z}_2)$. Thus, for $kge0$, $v_kcup x=Sq^k(x)$ for all $xin H^{n-k}(M;mathbb{Z}_2)$, and
$$w_k(M)=sum_{i+j=k}Sq^i(v_j).$$
Here the Poincare duality guarantees the existence and uniqueness of $v$.
My question: if $M$ is a smooth compact $n$-manifold with boundary, is there a similar Wu formula?
In this case, there is a fundamental class $zin H_n(M,partial M;mathbb{Z}_2)$ and the relative Poincare duality claims that capping with $z$ yields duality isomorphisms
$$D:H^p(M,partial M;mathbb{Z}_2)to H_{n-p}(M;mathbb{Z}_2)$$
and
$$D:H^p(M;mathbb{Z}_2)to H_{n-p}(M,partial M;mathbb{Z}_2).$$
Thank you!
at.algebraic-topology gt.geometric-topology cohomology smooth-manifolds characteristic-classes
asked yesterday
BorromeanBorromean
600312
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$begingroup$
The classical Wu formula claims that if $M$ is a smooth closed $n$-manifold with fundamental class $zin H_n(M;mathbb{Z}_2)$, then the total Stiefel-Whitney class $w(M)$ is equal to $Sq(v)$, where $v=sum v_iin H^*(M;mathbb{Z}_2)$ is the unique cohomology class such that
$$langle vcup x,zrangle=langle Sq(x),zrangle$$
for all $xin H^*(M;mathbb{Z}_2)$. Thus, for $kge0$, $v_kcup x=Sq^k(x)$ for all $xin H^{n-k}(M;mathbb{Z}_2)$, and
$$w_k(M)=sum_{i+j=k}Sq^i(v_j).$$
Here the Poincare duality guarantees the existence and uniqueness of $v$.
My question: if $M$ is a smooth compact $n$-manifold with boundary, is there a similar Wu formula?
In this case, there is a fundamental class $zin H_n(M,partial M;mathbb{Z}_2)$ and the relative Poincare duality claims that capping with $z$ yields duality isomorphisms
$$D:H^p(M,partial M;mathbb{Z}_2)to H_{n-p}(M;mathbb{Z}_2)$$
and
$$D:H^p(M;mathbb{Z}_2)to H_{n-p}(M,partial M;mathbb{Z}_2).$$
Thank you!
at.algebraic-topology gt.geometric-topology cohomology smooth-manifolds characteristic-classes
asked yesterday
BorromeanBorromean
600312
$endgroup$
add a comment |
$begingroup$
The classical Wu formula claims that if $M$ is a smooth closed $n$-manifold with fundamental class $zin H_n(M;mathbb{Z}_2)$, then the total Stiefel-Whitney class $w(M)$ is equal to $Sq(v)$, where $v=sum v_iin H^*(M;mathbb{Z}_2)$ is the unique cohomology class such that
$$langle vcup x,zrangle=langle Sq(x),zrangle$$
for all $xin H^*(M;mathbb{Z}_2)$. Thus, for $kge0$, $v_kcup x=Sq^k(x)$ for all $xin H^{n-k}(M;mathbb{Z}_2)$, and
$$w_k(M)=sum_{i+j=k}Sq^i(v_j).$$
Here the Poincare duality guarantees the existence and uniqueness of $v$.
My question: if $M$ is a smooth compact $n$-manifold with boundary, is there a similar Wu formula?
In this case, there is a fundamental class $zin H_n(M,partial M;mathbb{Z}_2)$ and the relative Poincare duality claims that capping with $z$ yields duality isomorphisms
$$D:H^p(M,partial M;mathbb{Z}_2)to H_{n-p}(M;mathbb{Z}_2)$$
and
$$D:H^p(M;mathbb{Z}_2)to H_{n-p}(M,partial M;mathbb{Z}_2).$$
Thank you!
at.algebraic-topology gt.geometric-topology cohomology smooth-manifolds characteristic-classes
asked yesterday
BorromeanBorromean
600312
$endgroup$
The classical Wu formula claims that if $M$ is a smooth closed $n$-manifold with fundamental class $zin H_n(M;mathbb{Z}_2)$, then the total Stiefel-Whitney class $w(M)$ is equal to $Sq(v)$, where $v=sum v_iin H^*(M;mathbb{Z}_2)$ is the unique cohomology class such that
$$langle vcup x,zrangle=langle Sq(x),zrangle$$
for all $xin H^*(M;mathbb{Z}_2)$. Thus, for $kge0$, $v_kcup x=Sq^k(x)$ for all $xin H^{n-k}(M;mathbb{Z}_2)$, and
$$w_k(M)=sum_{i+j=k}Sq^i(v_j).$$
Here the Poincare duality guarantees the existence and uniqueness of $v$.
My question: if $M$ is a smooth compact $n$-manifold with boundary, is there a similar Wu formula?
In this case, there is a fundamental class $zin H_n(M,partial M;mathbb{Z}_2)$ and the relative Poincare duality claims that capping with $z$ yields duality isomorphisms
$$D:H^p(M,partial M;mathbb{Z}_2)to H_{n-p}(M;mathbb{Z}_2)$$
and
$$D:H^p(M;mathbb{Z}_2)to H_{n-p}(M,partial M;mathbb{Z}_2).$$
Thank you!
at.algebraic-topology gt.geometric-topology cohomology smooth-manifolds characteristic-classes
at.algebraic-topology gt.geometric-topology cohomology smooth-manifolds characteristic-classes
asked yesterday
BorromeanBorromean
600312
asked yesterday
BorromeanBorromean
600312
asked yesterday
BorromeanBorromean
600312
asked yesterday
BorromeanBorromean
600312
asked yesterday
BorromeanBorromean
600312
600312
add a comment |
add a comment |
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$begingroup$
A relative Wu formula for manifolds with boundary is discussed in Section 7 of
Kervaire, Michel A., Relative characteristic classes, Am. J. Math. 79, 517-558 (1957). ZBL0173.51201.
In particular, there are relative Wu classes $U^qin H^q(M;mathbb{Z}/2)$ for $q=0,1,ldots , n$ defined by the property that
$$
Sq^q(x)=U^qcup x in H^n(M,partial M;mathbb{Z}/2)
$$
for all $xin H^{n-q}(M,partial M;mathbb{Z}/2)$. Kervaire deduces the relative Wu formula $w(M)=Sq(U)$ from the absolute Wu formula for the double $N=Mcup_{partial M} M$ (the closed manifold obtained by gluing two copies of $M$ along the identity map of $partial M$), using naturality with respect to the inclusion $i:Mhookrightarrow N$.
answered yesterday
Mark GrantMark Grant
22.6k658136
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$begingroup$
A relative Wu formula for manifolds with boundary is discussed in Section 7 of
Kervaire, Michel A., Relative characteristic classes, Am. J. Math. 79, 517-558 (1957). ZBL0173.51201.
In particular, there are relative Wu classes $U^qin H^q(M;mathbb{Z}/2)$ for $q=0,1,ldots , n$ defined by the property that
$$
Sq^q(x)=U^qcup x in H^n(M,partial M;mathbb{Z}/2)
$$
for all $xin H^{n-q}(M,partial M;mathbb{Z}/2)$. Kervaire deduces the relative Wu formula $w(M)=Sq(U)$ from the absolute Wu formula for the double $N=Mcup_{partial M} M$ (the closed manifold obtained by gluing two copies of $M$ along the identity map of $partial M$), using naturality with respect to the inclusion $i:Mhookrightarrow N$.
answered yesterday
Mark GrantMark Grant
22.6k658136
$endgroup$
add a comment |
$begingroup$
A relative Wu formula for manifolds with boundary is discussed in Section 7 of
Kervaire, Michel A., Relative characteristic classes, Am. J. Math. 79, 517-558 (1957). ZBL0173.51201.
In particular, there are relative Wu classes $U^qin H^q(M;mathbb{Z}/2)$ for $q=0,1,ldots , n$ defined by the property that
$$
Sq^q(x)=U^qcup x in H^n(M,partial M;mathbb{Z}/2)
$$
for all $xin H^{n-q}(M,partial M;mathbb{Z}/2)$. Kervaire deduces the relative Wu formula $w(M)=Sq(U)$ from the absolute Wu formula for the double $N=Mcup_{partial M} M$ (the closed manifold obtained by gluing two copies of $M$ along the identity map of $partial M$), using naturality with respect to the inclusion $i:Mhookrightarrow N$.
answered yesterday
Mark GrantMark Grant
22.6k658136
$endgroup$
add a comment |
$begingroup$
A relative Wu formula for manifolds with boundary is discussed in Section 7 of
Kervaire, Michel A., Relative characteristic classes, Am. J. Math. 79, 517-558 (1957). ZBL0173.51201.
In particular, there are relative Wu classes $U^qin H^q(M;mathbb{Z}/2)$ for $q=0,1,ldots , n$ defined by the property that
$$
Sq^q(x)=U^qcup x in H^n(M,partial M;mathbb{Z}/2)
$$
for all $xin H^{n-q}(M,partial M;mathbb{Z}/2)$. Kervaire deduces the relative Wu formula $w(M)=Sq(U)$ from the absolute Wu formula for the double $N=Mcup_{partial M} M$ (the closed manifold obtained by gluing two copies of $M$ along the identity map of $partial M$), using naturality with respect to the inclusion $i:Mhookrightarrow N$.
answered yesterday
Mark GrantMark Grant
22.6k658136
$endgroup$
A relative Wu formula for manifolds with boundary is discussed in Section 7 of
Kervaire, Michel A., Relative characteristic classes, Am. J. Math. 79, 517-558 (1957). ZBL0173.51201.
In particular, there are relative Wu classes $U^qin H^q(M;mathbb{Z}/2)$ for $q=0,1,ldots , n$ defined by the property that
$$
Sq^q(x)=U^qcup x in H^n(M,partial M;mathbb{Z}/2)
$$
for all $xin H^{n-q}(M,partial M;mathbb{Z}/2)$. Kervaire deduces the relative Wu formula $w(M)=Sq(U)$ from the absolute Wu formula for the double $N=Mcup_{partial M} M$ (the closed manifold obtained by gluing two copies of $M$ along the identity map of $partial M$), using naturality with respect to the inclusion $i:Mhookrightarrow N$.
answered yesterday
Mark GrantMark Grant
22.6k658136
answered yesterday
Mark GrantMark Grant
22.6k658136
answered yesterday
Mark GrantMark Grant
22.6k658136
answered yesterday
Mark GrantMark Grant
22.6k658136
22.6k658136
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