Sphere immersions in 4-manifolds The Next CEO of Stack OverflowFramed manifolds and framed knotsCellular Homology of the 3-TorusIf a smooth map between manifolds is injective, is the induced map on the tangent spaces injective too?Some questions about homology with local coefficients.smooth embedding between manifoldsFundamental group of the sphere with n-points identifiedConfusion with immersions, embeddings, local homeomorphisms, and local diffeomorphisms.Fundamental group of common Klein bottle immersionDoes any isomorphism between $pi_1(X,x_0)$ and $pi_1(Y,y_0)$ always induce a homeomorphism between $(X,x_0)$ and $(Y,y_0)$?I'm having trouble understanding manifolds.
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Sphere immersions in 4-manifolds
The Next CEO of Stack OverflowFramed manifolds and framed knotsCellular Homology of the 3-TorusIf a smooth map between manifolds is injective, is the induced map on the tangent spaces injective too?Some questions about homology with local coefficients.smooth embedding between manifoldsFundamental group of the sphere with n-points identifiedConfusion with immersions, embeddings, local homeomorphisms, and local diffeomorphisms.Fundamental group of common Klein bottle immersionDoes any isomorphism between $pi_1(X,x_0)$ and $pi_1(Y,y_0)$ always induce a homeomorphism between $(X,x_0)$ and $(Y,y_0)$?I'm having trouble understanding manifolds.
$begingroup$
I'm having trouble understanding the homomorphisms $pi_1g(S^2)longrightarrow pi_1 M^4$ where $g: S^2longrightarrow M^4$ can be a framed or unframed immersion.
Any reference on this?
algebraic-topology manifolds spheres
asked Mar 18 at 20:39
sarahsarah
1
$endgroup$
add a comment |
$begingroup$
I'm having trouble understanding the homomorphisms $pi_1g(S^2)longrightarrow pi_1 M^4$ where $g: S^2longrightarrow M^4$ can be a framed or unframed immersion.
Any reference on this?
algebraic-topology manifolds spheres
asked Mar 18 at 20:39
sarahsarah
1
$endgroup$
$begingroup$
You typically learn this in an algebraic topology class/textbook: A continuous map of pointed topological spaces $(X,x)to (Y,y)$ induces a homomorphism $pi_1(X,x)to pi_1(Y,y)$. The extra assumptions ($g$ is an immersion etc) are irrelevant here. In your case, $X=S^2$ is simply-connected, hence, your homomorphism is trivial. My suggestion is to read Chapter 1.1 of Hatcher's "Algebraic Topology".
$endgroup$
– Moishe Kohan
Mar 20 at 17:51
$begingroup$
No, my homomorphism is $pi_1g(S^2)rightlongarrowpi_1M. The immersed sphere would have finitely many self intersections.@MoisheKohan
$endgroup$
– sarah
Mar 21 at 3:07
add a comment |
$begingroup$
I'm having trouble understanding the homomorphisms $pi_1g(S^2)longrightarrow pi_1 M^4$ where $g: S^2longrightarrow M^4$ can be a framed or unframed immersion.
Any reference on this?
algebraic-topology manifolds spheres
asked Mar 18 at 20:39
sarahsarah
1
$endgroup$
I'm having trouble understanding the homomorphisms $pi_1g(S^2)longrightarrow pi_1 M^4$ where $g: S^2longrightarrow M^4$ can be a framed or unframed immersion.
Any reference on this?
algebraic-topology manifolds spheres
algebraic-topology manifolds spheres
asked Mar 18 at 20:39
sarahsarah
1
asked Mar 18 at 20:39
sarahsarah
1
asked Mar 18 at 20:39
sarahsarah
1
asked Mar 18 at 20:39
sarahsarah
1
asked Mar 18 at 20:39
sarahsarah
1
1
$begingroup$
You typically learn this in an algebraic topology class/textbook: A continuous map of pointed topological spaces $(X,x)to (Y,y)$ induces a homomorphism $pi_1(X,x)to pi_1(Y,y)$. The extra assumptions ($g$ is an immersion etc) are irrelevant here. In your case, $X=S^2$ is simply-connected, hence, your homomorphism is trivial. My suggestion is to read Chapter 1.1 of Hatcher's "Algebraic Topology".
$endgroup$
– Moishe Kohan
Mar 20 at 17:51
$begingroup$
No, my homomorphism is $pi_1g(S^2)rightlongarrowpi_1M. The immersed sphere would have finitely many self intersections.@MoisheKohan
$endgroup$
– sarah
Mar 21 at 3:07
add a comment |
$begingroup$
You typically learn this in an algebraic topology class/textbook: A continuous map of pointed topological spaces $(X,x)to (Y,y)$ induces a homomorphism $pi_1(X,x)to pi_1(Y,y)$. The extra assumptions ($g$ is an immersion etc) are irrelevant here. In your case, $X=S^2$ is simply-connected, hence, your homomorphism is trivial. My suggestion is to read Chapter 1.1 of Hatcher's "Algebraic Topology".
$endgroup$
– Moishe Kohan
Mar 20 at 17:51
$begingroup$
No, my homomorphism is $pi_1g(S^2)rightlongarrowpi_1M. The immersed sphere would have finitely many self intersections.@MoisheKohan
$endgroup$
– sarah
Mar 21 at 3:07
$begingroup$
You typically learn this in an algebraic topology class/textbook: A continuous map of pointed topological spaces $(X,x)to (Y,y)$ induces a homomorphism $pi_1(X,x)to pi_1(Y,y)$. The extra assumptions ($g$ is an immersion etc) are irrelevant here. In your case, $X=S^2$ is simply-connected, hence, your homomorphism is trivial. My suggestion is to read Chapter 1.1 of Hatcher's "Algebraic Topology".
$endgroup$
– Moishe Kohan
Mar 20 at 17:51
$begingroup$
You typically learn this in an algebraic topology class/textbook: A continuous map of pointed topological spaces $(X,x)to (Y,y)$ induces a homomorphism $pi_1(X,x)to pi_1(Y,y)$. The extra assumptions ($g$ is an immersion etc) are irrelevant here. In your case, $X=S^2$ is simply-connected, hence, your homomorphism is trivial. My suggestion is to read Chapter 1.1 of Hatcher's "Algebraic Topology".
$endgroup$
– Moishe Kohan
Mar 20 at 17:51
$begingroup$
No, my homomorphism is $pi_1g(S^2)rightlongarrowpi_1M. The immersed sphere would have finitely many self intersections.@MoisheKohan
$endgroup$
– sarah
Mar 21 at 3:07
$begingroup$
No, my homomorphism is $pi_1g(S^2)rightlongarrowpi_1M. The immersed sphere would have finitely many self intersections.@MoisheKohan
$endgroup$
– sarah
Mar 21 at 3:07
add a comment |
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$begingroup$
You typically learn this in an algebraic topology class/textbook: A continuous map of pointed topological spaces $(X,x)to (Y,y)$ induces a homomorphism $pi_1(X,x)to pi_1(Y,y)$. The extra assumptions ($g$ is an immersion etc) are irrelevant here. In your case, $X=S^2$ is simply-connected, hence, your homomorphism is trivial. My suggestion is to read Chapter 1.1 of Hatcher's "Algebraic Topology".
$endgroup$
– Moishe Kohan
Mar 20 at 17:51
$begingroup$
No, my homomorphism is $pi_1g(S^2)rightlongarrowpi_1M. The immersed sphere would have finitely many self intersections.@MoisheKohan
$endgroup$
– sarah
Mar 21 at 3:07