Confused by Hatchers proof of Corollary 2.24Where is the inclusion map being used in the proof of Corollary 2.24 from Hatcher's AT?Expressing $mathbbR$ as the quotient of a disjoint union of unit intervalsSimultaneous CW ApproximationHelp Understanding/Completing Proof of Prop 3.18/3.19 in Hatcher's Algebraic TopologyCW complex is contractible if union of contractible subcomplexes with contractible intersectionGiven a group $G$, the existence of a space such that $pi_1(X)simeq G$.If p is a covering map of a connected space, does p evenly cover the whole space?Mayer-Vietoris sequence from Eilenberg–Steenrod axiomsIdea behind the proof of Whitehead's Theorem and Compression Lemmaunderstanding proof of excision theoremproof of excision theorem: commutativity of a diagram
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Confused by Hatchers proof of Corollary 2.24
Where is the inclusion map being used in the proof of Corollary 2.24 from Hatcher's AT?Expressing $mathbbR$ as the quotient of a disjoint union of unit intervalsSimultaneous CW ApproximationHelp Understanding/Completing Proof of Prop 3.18/3.19 in Hatcher's Algebraic TopologyCW complex is contractible if union of contractible subcomplexes with contractible intersectionGiven a group $G$, the existence of a space such that $pi_1(X)simeq G$.If p is a covering map of a connected space, does p evenly cover the whole space?Mayer-Vietoris sequence from Eilenberg–Steenrod axiomsIdea behind the proof of Whitehead's Theorem and Compression Lemmaunderstanding proof of excision theoremproof of excision theorem: commutativity of a diagram
$begingroup$
Corollary 2.24 says: If the CW complex $X$ is the union of subcomplexes $A$ and $B$, then the inclusion $(B,A cap B) rightarrow (X,A)$ induces isomorphisms $H_n(B,A cap B) rightarrow H_n(X,A)$ for all $n$.
Hatcher then gives a very brief proof that I don't comprehend:
Since the $CW$ pairs are good, Proposition 2.22 allows us to pass to the quotient spaces $B/A cap B$ and $X/A$ which are homeomorphic, assuming we are not in the trivial case $A cap B = emptyset$.
Can someone please elaborate on this proof? In particular, why are those two spaces homeomorphic and how exactly do we apply proposition 2.22 here?
algebraic-topology
asked May 7 '17 at 18:52
TuoTuoTuoTuo
1,781516
$endgroup$
add a comment |
$begingroup$
Corollary 2.24 says: If the CW complex $X$ is the union of subcomplexes $A$ and $B$, then the inclusion $(B,A cap B) rightarrow (X,A)$ induces isomorphisms $H_n(B,A cap B) rightarrow H_n(X,A)$ for all $n$.
Hatcher then gives a very brief proof that I don't comprehend:
Since the $CW$ pairs are good, Proposition 2.22 allows us to pass to the quotient spaces $B/A cap B$ and $X/A$ which are homeomorphic, assuming we are not in the trivial case $A cap B = emptyset$.
Can someone please elaborate on this proof? In particular, why are those two spaces homeomorphic and how exactly do we apply proposition 2.22 here?
algebraic-topology
asked May 7 '17 at 18:52
TuoTuoTuoTuo
1,781516
$endgroup$
add a comment |
$begingroup$
Corollary 2.24 says: If the CW complex $X$ is the union of subcomplexes $A$ and $B$, then the inclusion $(B,A cap B) rightarrow (X,A)$ induces isomorphisms $H_n(B,A cap B) rightarrow H_n(X,A)$ for all $n$.
Hatcher then gives a very brief proof that I don't comprehend:
Since the $CW$ pairs are good, Proposition 2.22 allows us to pass to the quotient spaces $B/A cap B$ and $X/A$ which are homeomorphic, assuming we are not in the trivial case $A cap B = emptyset$.
Can someone please elaborate on this proof? In particular, why are those two spaces homeomorphic and how exactly do we apply proposition 2.22 here?
algebraic-topology
asked May 7 '17 at 18:52
TuoTuoTuoTuo
1,781516
$endgroup$
Corollary 2.24 says: If the CW complex $X$ is the union of subcomplexes $A$ and $B$, then the inclusion $(B,A cap B) rightarrow (X,A)$ induces isomorphisms $H_n(B,A cap B) rightarrow H_n(X,A)$ for all $n$.
Hatcher then gives a very brief proof that I don't comprehend:
Since the $CW$ pairs are good, Proposition 2.22 allows us to pass to the quotient spaces $B/A cap B$ and $X/A$ which are homeomorphic, assuming we are not in the trivial case $A cap B = emptyset$.
Can someone please elaborate on this proof? In particular, why are those two spaces homeomorphic and how exactly do we apply proposition 2.22 here?
algebraic-topology
algebraic-topology
asked May 7 '17 at 18:52
TuoTuoTuoTuo
1,781516
asked May 7 '17 at 18:52
TuoTuoTuoTuo
1,781516
asked May 7 '17 at 18:52
TuoTuoTuoTuo
1,781516
asked May 7 '17 at 18:52
TuoTuoTuoTuo
1,781516
asked May 7 '17 at 18:52
TuoTuoTuoTuo
1,781516
1,781516
add a comment |
add a comment |
1 Answer
1
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$begingroup$
The inclusion $B to X$ induces a map $B/(A cap B) to X/A$, and its inverse is induced by the map $X to B/(A cap B)$ that is the identity on $B - A$ and that sends $A$ to $A cap B$.
You may then apply the proposition which states that for a good pair $(X,A)$, $H_n(X,A) = tilde H_n(X/A)$.
edited 2 days ago
Wolfgang
4,31943377
answered May 7 '17 at 19:00
Alex ProvostAlex Provost
15.5k22350
$endgroup$
add a comment |
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1 Answer
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$begingroup$
The inclusion $B to X$ induces a map $B/(A cap B) to X/A$, and its inverse is induced by the map $X to B/(A cap B)$ that is the identity on $B - A$ and that sends $A$ to $A cap B$.
You may then apply the proposition which states that for a good pair $(X,A)$, $H_n(X,A) = tilde H_n(X/A)$.
edited 2 days ago
Wolfgang
4,31943377
answered May 7 '17 at 19:00
Alex ProvostAlex Provost
15.5k22350
$endgroup$
add a comment |
$begingroup$
The inclusion $B to X$ induces a map $B/(A cap B) to X/A$, and its inverse is induced by the map $X to B/(A cap B)$ that is the identity on $B - A$ and that sends $A$ to $A cap B$.
You may then apply the proposition which states that for a good pair $(X,A)$, $H_n(X,A) = tilde H_n(X/A)$.
edited 2 days ago
Wolfgang
4,31943377
answered May 7 '17 at 19:00
Alex ProvostAlex Provost
15.5k22350
$endgroup$
add a comment |
$begingroup$
The inclusion $B to X$ induces a map $B/(A cap B) to X/A$, and its inverse is induced by the map $X to B/(A cap B)$ that is the identity on $B - A$ and that sends $A$ to $A cap B$.
You may then apply the proposition which states that for a good pair $(X,A)$, $H_n(X,A) = tilde H_n(X/A)$.
edited 2 days ago
Wolfgang
4,31943377
answered May 7 '17 at 19:00
Alex ProvostAlex Provost
15.5k22350
$endgroup$
The inclusion $B to X$ induces a map $B/(A cap B) to X/A$, and its inverse is induced by the map $X to B/(A cap B)$ that is the identity on $B - A$ and that sends $A$ to $A cap B$.
You may then apply the proposition which states that for a good pair $(X,A)$, $H_n(X,A) = tilde H_n(X/A)$.
edited 2 days ago
Wolfgang
4,31943377
answered May 7 '17 at 19:00
Alex ProvostAlex Provost
15.5k22350
edited 2 days ago
Wolfgang
4,31943377
edited 2 days ago
Wolfgang
4,31943377
edited 2 days ago
Wolfgang
4,31943377
4,31943377
answered May 7 '17 at 19:00
Alex ProvostAlex Provost
15.5k22350
answered May 7 '17 at 19:00
Alex ProvostAlex Provost
15.5k22350
answered May 7 '17 at 19:00
Alex ProvostAlex Provost
15.5k22350
15.5k22350
add a comment |
add a comment |
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