Mixing
Gluing cells along generators of $pi_n(X)$
Clash Royale CLAN TAG#URR8PPP
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Given a representative $alpha$ for $[alpha] in pi_q(X)$ we may attach a cell along $alpha$ to form a new space $C(alpha)$ (This really is the mapping cone). One would hope that this has the effect of killing off $alpha$, that is,
$$pi_q(C(alpha)) cong pi_q(X)/langle alpha rangle.$$
In the case $q=1$ this follows quickly from the Seifert–van Kampen Theorem, since then one has
beginalign*
pi_1(C(alpha)) &cong pi_1(X) *_pi_1S^1 pi_1(D^2)\
&cong pi_1(X) *_pi_1S^1 *\
&cong pi_1(X) / langle alpha rangle,\
endalign*
as the amalgamation relates $alpha$ to the trivial map. One of the comments points out this is not true in general for $q ge 2$.
What is the analogous result for $q ge 2$ (with references please)?
algebraic-topology homotopy-theory
asked Jul 25 at 18:18
Chris
9316
add a comment |Â
up vote
2
down vote
favorite
Given a representative $alpha$ for $[alpha] in pi_q(X)$ we may attach a cell along $alpha$ to form a new space $C(alpha)$ (This really is the mapping cone). One would hope that this has the effect of killing off $alpha$, that is,
$$pi_q(C(alpha)) cong pi_q(X)/langle alpha rangle.$$
In the case $q=1$ this follows quickly from the Seifert–van Kampen Theorem, since then one has
beginalign*
pi_1(C(alpha)) &cong pi_1(X) *_pi_1S^1 pi_1(D^2)\
&cong pi_1(X) *_pi_1S^1 *\
&cong pi_1(X) / langle alpha rangle,\
endalign*
as the amalgamation relates $alpha$ to the trivial map. One of the comments points out this is not true in general for $q ge 2$.
What is the analogous result for $q ge 2$ (with references please)?
algebraic-topology homotopy-theory
asked Jul 25 at 18:18
Chris
9316
I'm not sure what you're claiming the cofibration assumption improves about the situation. Why is it easy to see the effect of squishing the image of $alpha$ on the homotopy groups? In any case, your claim is not actually true. Your mapping cone kills the subgroup of $pi_q$ generated by the orbit of $alpha$ under the $pi_1$-action.
– Kevin Carlson
Jul 26 at 18:34
I see what you mean about the the quotient not really giving you what you want. If you post your remark about the action of $pi_q$ as an answer with a little more details and a reference, then I would consider this question answered.
– Chris
Jul 26 at 21:00
add a comment |Â
up vote
2
down vote
favorite
up vote
2
down vote
favorite
Given a representative $alpha$ for $[alpha] in pi_q(X)$ we may attach a cell along $alpha$ to form a new space $C(alpha)$ (This really is the mapping cone). One would hope that this has the effect of killing off $alpha$, that is,
$$pi_q(C(alpha)) cong pi_q(X)/langle alpha rangle.$$
In the case $q=1$ this follows quickly from the Seifert–van Kampen Theorem, since then one has
beginalign*
pi_1(C(alpha)) &cong pi_1(X) *_pi_1S^1 pi_1(D^2)\
&cong pi_1(X) *_pi_1S^1 *\
&cong pi_1(X) / langle alpha rangle,\
endalign*
as the amalgamation relates $alpha$ to the trivial map. One of the comments points out this is not true in general for $q ge 2$.
What is the analogous result for $q ge 2$ (with references please)?
algebraic-topology homotopy-theory
asked Jul 25 at 18:18
Chris
9316
Given a representative $alpha$ for $[alpha] in pi_q(X)$ we may attach a cell along $alpha$ to form a new space $C(alpha)$ (This really is the mapping cone). One would hope that this has the effect of killing off $alpha$, that is,
$$pi_q(C(alpha)) cong pi_q(X)/langle alpha rangle.$$
In the case $q=1$ this follows quickly from the Seifert–van Kampen Theorem, since then one has
beginalign*
pi_1(C(alpha)) &cong pi_1(X) *_pi_1S^1 pi_1(D^2)\
&cong pi_1(X) *_pi_1S^1 *\
&cong pi_1(X) / langle alpha rangle,\
endalign*
as the amalgamation relates $alpha$ to the trivial map. One of the comments points out this is not true in general for $q ge 2$.
What is the analogous result for $q ge 2$ (with references please)?
algebraic-topology homotopy-theory
asked Jul 25 at 18:18
Chris
9316
edited Jul 26 at 20:57
asked Jul 25 at 18:18
Chris
9316
asked Jul 25 at 18:18
Chris
9316
asked Jul 25 at 18:18
Chris
9316
9316
I'm not sure what you're claiming the cofibration assumption improves about the situation. Why is it easy to see the effect of squishing the image of $alpha$ on the homotopy groups? In any case, your claim is not actually true. Your mapping cone kills the subgroup of $pi_q$ generated by the orbit of $alpha$ under the $pi_1$-action.
– Kevin Carlson
Jul 26 at 18:34
I see what you mean about the the quotient not really giving you what you want. If you post your remark about the action of $pi_q$ as an answer with a little more details and a reference, then I would consider this question answered.
– Chris
Jul 26 at 21:00
add a comment |Â
I'm not sure what you're claiming the cofibration assumption improves about the situation. Why is it easy to see the effect of squishing the image of $alpha$ on the homotopy groups? In any case, your claim is not actually true. Your mapping cone kills the subgroup of $pi_q$ generated by the orbit of $alpha$ under the $pi_1$-action.
– Kevin Carlson
Jul 26 at 18:34
I see what you mean about the the quotient not really giving you what you want. If you post your remark about the action of $pi_q$ as an answer with a little more details and a reference, then I would consider this question answered.
– Chris
Jul 26 at 21:00
I'm not sure what you're claiming the cofibration assumption improves about the situation. Why is it easy to see the effect of squishing the image of $alpha$ on the homotopy groups? In any case, your claim is not actually true. Your mapping cone kills the subgroup of $pi_q$ generated by the orbit of $alpha$ under the $pi_1$-action.
– Kevin Carlson
Jul 26 at 18:34
I'm not sure what you're claiming the cofibration assumption improves about the situation. Why is it easy to see the effect of squishing the image of $alpha$ on the homotopy groups? In any case, your claim is not actually true. Your mapping cone kills the subgroup of $pi_q$ generated by the orbit of $alpha$ under the $pi_1$-action.
– Kevin Carlson
Jul 26 at 18:34
I see what you mean about the the quotient not really giving you what you want. If you post your remark about the action of $pi_q$ as an answer with a little more details and a reference, then I would consider this question answered.
– Chris
Jul 26 at 21:00
I see what you mean about the the quotient not really giving you what you want. If you post your remark about the action of $pi_q$ as an answer with a little more details and a reference, then I would consider this question answered.
– Chris
Jul 26 at 21:00
add a comment |Â
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I'm not sure what you're claiming the cofibration assumption improves about the situation. Why is it easy to see the effect of squishing the image of $alpha$ on the homotopy groups? In any case, your claim is not actually true. Your mapping cone kills the subgroup of $pi_q$ generated by the orbit of $alpha$ under the $pi_1$-action.
– Kevin Carlson
Jul 26 at 18:34
I see what you mean about the the quotient not really giving you what you want. If you post your remark about the action of $pi_q$ as an answer with a little more details and a reference, then I would consider this question answered.
– Chris
Jul 26 at 21:00