Projection formula for proper maps of manifolds
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Let $X,Y$ be (not necessarily compact) orientable manifolds without boundaries of dimension $m$ and $n$. Let $f : X rightarrow Y$ be a proper map. Assume that all the cohomologies have coefficients in $mathbb{Q}$. We could define the following pullback maps
$$f^* : H^*(Y) rightarrow H^*(X), f^*_c : H^*_c(Y) rightarrow H^*_c(X)$$
We also have the following maps defined by their poincare duals
$$ f_{!,c} : H^*_c(X) rightarrow H^{*+(n-m)}_c(Y), f_! : H^*(X) rightarrow H^{*+(n-m)}(Y) $$
I am interested in knowing the projection formulas that are available in this setup. A little bit of googling up gives the following link
https://mathoverflow.net/questions/67228/where-do-all-these-projection-formulas-come-from
and
https://mathoverflow.net/questions/18799/ubiquity-of-the-push-pull-formula
which seems useful but they are not the kind of equality I am interested in. I am wondering if the following is true for $alpha in H^*_c(Y), beta in H^*(X)$.
$$f_!(f^*_c(alpha) cup beta) = alpha cup f_!(beta)$$
Q1. Are there other kinds of projection formulas available?
Q2. Which of them are poincare dual or related to the others?
It would be extremely helpful if the answers contain references if not the proofs themselves. I am not sure if the notations I have used are the correct ones. Please feel free to edit accordingly.
Thanks!
general-topology algebraic-topology duality-theorems poincare-duality
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2
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Let $X,Y$ be (not necessarily compact) orientable manifolds without boundaries of dimension $m$ and $n$. Let $f : X rightarrow Y$ be a proper map. Assume that all the cohomologies have coefficients in $mathbb{Q}$. We could define the following pullback maps
$$f^* : H^*(Y) rightarrow H^*(X), f^*_c : H^*_c(Y) rightarrow H^*_c(X)$$
We also have the following maps defined by their poincare duals
$$ f_{!,c} : H^*_c(X) rightarrow H^{*+(n-m)}_c(Y), f_! : H^*(X) rightarrow H^{*+(n-m)}(Y) $$
I am interested in knowing the projection formulas that are available in this setup. A little bit of googling up gives the following link
https://mathoverflow.net/questions/67228/where-do-all-these-projection-formulas-come-from
and
https://mathoverflow.net/questions/18799/ubiquity-of-the-push-pull-formula
which seems useful but they are not the kind of equality I am interested in. I am wondering if the following is true for $alpha in H^*_c(Y), beta in H^*(X)$.
$$f_!(f^*_c(alpha) cup beta) = alpha cup f_!(beta)$$
Q1. Are there other kinds of projection formulas available?
Q2. Which of them are poincare dual or related to the others?
It would be extremely helpful if the answers contain references if not the proofs themselves. I am not sure if the notations I have used are the correct ones. Please feel free to edit accordingly.
Thanks!
general-topology algebraic-topology duality-theorems poincare-duality
add a comment |
up vote
2
down vote
favorite
up vote
2
down vote
favorite
Let $X,Y$ be (not necessarily compact) orientable manifolds without boundaries of dimension $m$ and $n$. Let $f : X rightarrow Y$ be a proper map. Assume that all the cohomologies have coefficients in $mathbb{Q}$. We could define the following pullback maps
$$f^* : H^*(Y) rightarrow H^*(X), f^*_c : H^*_c(Y) rightarrow H^*_c(X)$$
We also have the following maps defined by their poincare duals
$$ f_{!,c} : H^*_c(X) rightarrow H^{*+(n-m)}_c(Y), f_! : H^*(X) rightarrow H^{*+(n-m)}(Y) $$
I am interested in knowing the projection formulas that are available in this setup. A little bit of googling up gives the following link
https://mathoverflow.net/questions/67228/where-do-all-these-projection-formulas-come-from
and
https://mathoverflow.net/questions/18799/ubiquity-of-the-push-pull-formula
which seems useful but they are not the kind of equality I am interested in. I am wondering if the following is true for $alpha in H^*_c(Y), beta in H^*(X)$.
$$f_!(f^*_c(alpha) cup beta) = alpha cup f_!(beta)$$
Q1. Are there other kinds of projection formulas available?
Q2. Which of them are poincare dual or related to the others?
It would be extremely helpful if the answers contain references if not the proofs themselves. I am not sure if the notations I have used are the correct ones. Please feel free to edit accordingly.
Thanks!
general-topology algebraic-topology duality-theorems poincare-duality
Let $X,Y$ be (not necessarily compact) orientable manifolds without boundaries of dimension $m$ and $n$. Let $f : X rightarrow Y$ be a proper map. Assume that all the cohomologies have coefficients in $mathbb{Q}$. We could define the following pullback maps
$$f^* : H^*(Y) rightarrow H^*(X), f^*_c : H^*_c(Y) rightarrow H^*_c(X)$$
We also have the following maps defined by their poincare duals
$$ f_{!,c} : H^*_c(X) rightarrow H^{*+(n-m)}_c(Y), f_! : H^*(X) rightarrow H^{*+(n-m)}(Y) $$
I am interested in knowing the projection formulas that are available in this setup. A little bit of googling up gives the following link
https://mathoverflow.net/questions/67228/where-do-all-these-projection-formulas-come-from
and
https://mathoverflow.net/questions/18799/ubiquity-of-the-push-pull-formula
which seems useful but they are not the kind of equality I am interested in. I am wondering if the following is true for $alpha in H^*_c(Y), beta in H^*(X)$.
$$f_!(f^*_c(alpha) cup beta) = alpha cup f_!(beta)$$
Q1. Are there other kinds of projection formulas available?
Q2. Which of them are poincare dual or related to the others?
It would be extremely helpful if the answers contain references if not the proofs themselves. I am not sure if the notations I have used are the correct ones. Please feel free to edit accordingly.
Thanks!
general-topology algebraic-topology duality-theorems poincare-duality
general-topology algebraic-topology duality-theorems poincare-duality
edited Nov 16 at 7:48
asked Nov 12 at 17:30
random123
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993719
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