Plumbing (Ex)
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− | Choose $S^1\subseteq \partial X$ representing a generator of $\pi_1(\partial X)$ and let $V$ be the trace of a surgery on this $S^1$. Define $X':=X\cup_{\partial X}V$. Show | + | Choose $S^1\subseteq \partial X$ representing a generator of $\pi_1(\partial X)$ and let $V$ be the trace of a surgery on this $S^1$. Define $X':=X\cup_{\partial X}V$. Show the following: |
− | #$H_i(X')\cong H_i(X)$ for $i\neq 1$ | + | #$H_i(X')\cong H_i(X)$ for $i\neq 1$. |
− | #$H_i(\partial X')\cong H_i(\partial X)$ for $1<i<2q-2$ | + | #$H_i(\partial X')\cong H_i(\partial X)$ for $1<i<2q-2$. |
{{beginrem|Hint}} | {{beginrem|Hint}} | ||
For (1) use the long exact sequence of the pair $(X',X)$ and $X'\simeq X\cup D^2$. For (2) use the long exact sequence of the pair $(V,\partial X')$ as well as Poincaré Duality and the Universal Coefficient Theorem. | For (1) use the long exact sequence of the pair $(X',X)$ and $X'\simeq X\cup D^2$. For (2) use the long exact sequence of the pair $(V,\partial X')$ as well as Poincaré Duality and the Universal Coefficient Theorem. |
Revision as of 21:45, 23 March 2012
The exercises below are about plumbing manifolds. For the details of the construction, see the page Plumbing.
In this page we use slightly different notation. For let be a closed connected oriented manifold of dimension and let
be an oriented -bundle over where is fixed and each . Let be a graph with verticies where the edge set between and is non-empty only if and .
Starting from the dijoint union of the total spaces we form a manifolds as follows: given an edge in connecting and , for and , let and let be a neighbourhood of , such that and such that if the fiber of . Let and be orientation preserving (resp. reversing) diffeomorphisms. We define the plumbing of and at and by taking and identifying and via
Proceeding in this way for each edge of we obtain the plumbing manifold
Exercise 0.1. Let be connected. Show the following:
- is free.
- for ??
Hint 0.2. The statement is trivial for , since is homotopy equivalent to a wedge of 1-spheres and -spheres. Now use van Kampen's theorem for and for use the Mayer-Vietories Sequence with and show that all components involved are connected.
Exercise 0.3. Choose representing a generator of and let be the trace of a surgery on this . Define . Show the following:
- for .
- for .
Hint 0.4. For (1) use the long exact sequence of the pair and . For (2) use the long exact sequence of the pair as well as Poincaré Duality and the Universal Coefficient Theorem.
Exercise 0.5. Assume now that each and that each bundle is some multiple of , the unit disc bundle of the tangent bundle of the -sphere. Show that there is a degree 1 normal map .
Hint 0.6. Use that is homotopy equivalent to a wedge of 1-spheres and -spheres and that the tangent bundle of is stabily trivial (since it is so on every component of the wedge), so that the normal bundle of is trivial for large. Then show that the map can be extend over .
References
let be a closed connected oriented manifold of dimension and letbe an oriented -bundle over where is fixed and each . Let be a graph with verticies where the edge set between and is non-empty only if and .
Starting from the dijoint union of the total spaces we form a manifolds as follows: given an edge in connecting and , for and , let and let be a neighbourhood of , such that and such that if the fiber of . Let and be orientation preserving (resp. reversing) diffeomorphisms. We define the plumbing of and at and by taking and identifying and via
Proceeding in this way for each edge of we obtain the plumbing manifold
Exercise 0.1. Let be connected. Show the following:
- is free.
- for ??
Hint 0.2. The statement is trivial for , since is homotopy equivalent to a wedge of 1-spheres and -spheres. Now use van Kampen's theorem for and for use the Mayer-Vietories Sequence with and show that all components involved are connected.
Exercise 0.3. Choose representing a generator of and let be the trace of a surgery on this . Define . Show the following:
- for .
- for .
Hint 0.4. For (1) use the long exact sequence of the pair and . For (2) use the long exact sequence of the pair as well as Poincaré Duality and the Universal Coefficient Theorem.
Exercise 0.5. Assume now that each and that each bundle is some multiple of , the unit disc bundle of the tangent bundle of the -sphere. Show that there is a degree 1 normal map .
Hint 0.6. Use that is homotopy equivalent to a wedge of 1-spheres and -spheres and that the tangent bundle of is stabily trivial (since it is so on every component of the wedge), so that the normal bundle of is trivial for large. Then show that the map can be extend over .
References
== References == {{#RefList:}} [[Category:Exercises]]i = 1, \dots, n let be a closed connected oriented manifold of dimension and letbe an oriented -bundle over where is fixed and each . Let be a graph with verticies where the edge set between and is non-empty only if and .
Starting from the dijoint union of the total spaces we form a manifolds as follows: given an edge in connecting and , for and , let and let be a neighbourhood of , such that and such that if the fiber of . Let and be orientation preserving (resp. reversing) diffeomorphisms. We define the plumbing of and at and by taking and identifying and via
Proceeding in this way for each edge of we obtain the plumbing manifold
Exercise 0.1. Let be connected. Show the following:
- is free.
- for ??
Hint 0.2. The statement is trivial for , since is homotopy equivalent to a wedge of 1-spheres and -spheres. Now use van Kampen's theorem for and for use the Mayer-Vietories Sequence with and show that all components involved are connected.
Exercise 0.3. Choose representing a generator of and let be the trace of a surgery on this . Define . Show the following:
- for .
- for .
Hint 0.4. For (1) use the long exact sequence of the pair and . For (2) use the long exact sequence of the pair as well as Poincaré Duality and the Universal Coefficient Theorem.
Exercise 0.5. Assume now that each and that each bundle is some multiple of , the unit disc bundle of the tangent bundle of the -sphere. Show that there is a degree 1 normal map .
Hint 0.6. Use that is homotopy equivalent to a wedge of 1-spheres and -spheres and that the tangent bundle of is stabily trivial (since it is so on every component of the wedge), so that the normal bundle of is trivial for large. Then show that the map can be extend over .