Talk:S-duality II (Ex)
The map we use is closely related to the previous exercise so we will copy and paste some terminology used there - note that we will use where is used there, and have changed the diagonal map appropriately:
Let be a finite CW complex. Embed with regular neighbourhood , so that
with a homotopy-equivalence and an -dimensional manifold-with-boundary embedded in .
Now let
be the collapse map,
be the map induced by
and let be the composite
where is a chosen homotopy-inverse for .
Exercise 0.1 (first part).
First, take where is the dimension of . Fix a Thom class and a class determining the Poincaré duality for .
Take reduced chain complexes (denoted ) for the spaces in the composite . We then make a choice of Eilenberg-Zilber map (unique up to chain homotopy) so that
The chain-level slant map now involves a choice - in general if we have two chains and then the slant involves a choice of ordering to form or . When considering -duality, the choice represents the symmetry that iff . We will need to choose the slant map so that
this is followed with the map induced by so that the overall slant map isomorphism induced is (see the previous exercise for a justification of this, albeit for the opposite choice of slant).
Considering the composite again, note that before we use the smash map , it is still valid to use the identity for slant product that for any , we have
as this is still a (relative) definition of cap product on the chain level.
We now want to know how the slant map interacts with the smash map . In general for any and , we have a commutative diagram:
Hence
Exercise 0.2 (second part).
This now follows easily from breaking the diagram down in a careful fashion to
and noting that following the top side of the diagram round gives
Let be a finite CW complex. Embed with regular neighbourhood , so that
with a homotopy-equivalence and an -dimensional manifold-with-boundary embedded in .
Now let
be the collapse map,
be the map induced by
and let be the composite
where is a chosen homotopy-inverse for .
Exercise 0.1 (first part).
First, take where is the dimension of . Fix a Thom class and a class determining the Poincaré duality for .
Take reduced chain complexes (denoted ) for the spaces in the composite . We then make a choice of Eilenberg-Zilber map (unique up to chain homotopy) so that
The chain-level slant map now involves a choice - in general if we have two chains and then the slant involves a choice of ordering to form or . When considering -duality, the choice represents the symmetry that iff . We will need to choose the slant map so that
this is followed with the map induced by so that the overall slant map isomorphism induced is (see the previous exercise for a justification of this, albeit for the opposite choice of slant).
Considering the composite again, note that before we use the smash map , it is still valid to use the identity for slant product that for any , we have
as this is still a (relative) definition of cap product on the chain level.
We now want to know how the slant map interacts with the smash map . In general for any and , we have a commutative diagram:
Hence
Exercise 0.2 (second part).
This now follows easily from breaking the diagram down in a careful fashion to
and noting that following the top side of the diagram round gives
Let be a finite CW complex. Embed with regular neighbourhood , so that
with a homotopy-equivalence and an -dimensional manifold-with-boundary embedded in .
Now let
be the collapse map,
be the map induced by
and let be the composite
where is a chosen homotopy-inverse for .
Exercise 0.1 (first part).
First, take where is the dimension of . Fix a Thom class and a class determining the Poincaré duality for .
Take reduced chain complexes (denoted ) for the spaces in the composite . We then make a choice of Eilenberg-Zilber map (unique up to chain homotopy) so that
The chain-level slant map now involves a choice - in general if we have two chains and then the slant involves a choice of ordering to form or . When considering -duality, the choice represents the symmetry that iff . We will need to choose the slant map so that
this is followed with the map induced by so that the overall slant map isomorphism induced is (see the previous exercise for a justification of this, albeit for the opposite choice of slant).
Considering the composite again, note that before we use the smash map , it is still valid to use the identity for slant product that for any , we have
as this is still a (relative) definition of cap product on the chain level.
We now want to know how the slant map interacts with the smash map . In general for any and , we have a commutative diagram:
Hence
Exercise 0.2 (second part).
This now follows easily from breaking the diagram down in a careful fashion to
and noting that following the top side of the diagram round gives
Let be a finite CW complex. Embed with regular neighbourhood , so that
with a homotopy-equivalence and an -dimensional manifold-with-boundary embedded in .
Now let
be the collapse map,
be the map induced by
and let be the composite
where is a chosen homotopy-inverse for .
Exercise 0.1 (first part).
First, take where is the dimension of . Fix a Thom class and a class determining the Poincaré duality for .
Take reduced chain complexes (denoted ) for the spaces in the composite . We then make a choice of Eilenberg-Zilber map (unique up to chain homotopy) so that
The chain-level slant map now involves a choice - in general if we have two chains and then the slant involves a choice of ordering to form or . When considering -duality, the choice represents the symmetry that iff . We will need to choose the slant map so that
this is followed with the map induced by so that the overall slant map isomorphism induced is (see the previous exercise for a justification of this, albeit for the opposite choice of slant).
Considering the composite again, note that before we use the smash map , it is still valid to use the identity for slant product that for any , we have
as this is still a (relative) definition of cap product on the chain level.
We now want to know how the slant map interacts with the smash map . In general for any and , we have a commutative diagram:
Hence
Exercise 0.2 (second part).
This now follows easily from breaking the diagram down in a careful fashion to
and noting that following the top side of the diagram round gives