What is the induced isomorphism of a handle decomposition $X_k=X_k-1cup H^lambda_k$, $X_0=D^m$ given by the long exact Mayer-Vietoris sequence?

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Let $X_k=X_k-1cup_theta H^lambda_k$, for $kge 1$, be the $lambda_k$-handle attached to a $dim X_k-1$-manifold along the embedding map $theta:S^lambda_k-1times D^dim X_k-1-lambda_ktopartial X_k-1$ with $X_0=D^m$. Then the long exact Mayer-Vietoris sequence is $$dotslongrightarrow H_lambda_k+1(X_k)oversetpartial_*longrightarrow H_lambda_k(X_k-1cap H^lambda_k)overset(i^k_*,j^k_*)longrightarrow H_lambda_k(X_k-1)oplus H_lambda_k(H^lambda_k)oversetk^k_*-ell^k_*longrightarrow H_lambda_k(X_k)oversetpartial_*longrightarrow H_lambda_k-1(X_k-1cap H^lambda_k)overset(i_*^k,j_*^k)longrightarrow H_lambda_k-1(X_k-1)oplus H_lambda_k-1(H^lambda_k)oversetk^k_*-ell^k_*longrightarrow H_lambda_k-1(X_k)oversetpartial_*longrightarrowH_lambda_k-2(S^lambda_k-1)overset(i_*^k,j_*^k)longrightarrow dots overset(i_*^k,j_*^k)longrightarrow H_0(X_k-1)oplus H_0(H^lambda_k)oversetk^k_*-ell^k_*longrightarrow H_0(X_k)longrightarrow 0$$ or equivalently, $$dotslongrightarrow H_lambda_k+1(X_k)oversetpartial_*longrightarrow H_lambda_k(S^lambda_k-1)overset(i^k_*,j^k_*)longrightarrow H_lambda_k(X_k-1)oplus H_lambda_k(H^lambda_k)oversetk^k_*-ell^k_*longrightarrow H_lambda_k(X_k)oversetpartial_*longrightarrow H_lambda_k-1(S^lambda_k-1)overset(i_*^k,j_*^k)longrightarrow H_lambda_k-1(X_k-1)oplus H_lambda_k-1(H^lambda_k)oversetk^k_*-ell^k_*longrightarrow H_lambda_k-1(X_k)oversetpartial_*longrightarrowH_lambda_k-2(S^lambda_k-1)overset(i_*^k,j_*^k)longrightarrow dots overset(i_*^k,j_*^k)longrightarrow H_0(X_k-1)oplus H_0(H^lambda_k)oversetk^k_*-ell^k_*longrightarrow H_0(X_k)longrightarrow 0.$$ Observing that $H^k=D^ktimes D^n−kcong*$ has trivial homology, the exact sequence reduces to $$0overset(i^k_*,j^k_*)longrightarrow H_lambda_k(X_k-1)oversetk^k_*-ell^k_*longrightarrow H_lambda_k(X_k)oversetpartial_*longrightarrow mathbbZoverset(i_*^k,j_*^k)longrightarrow H_lambda_k-1(X_k-1)oversetk^k_*-ell^k_*longrightarrow H_lambda_k-1(X_k)oversetpartial_*longrightarrow 0.$$




What is the induced isomorphism, from the sequence of short exact sequences obtained from this long exact sequence, which involves $H_lambda_k(X_k)$? Is it correct to write $0overset(i^k_*,j^k_*)longrightarrow H_lambda_k(X_k-1)oversetk^k_*-ell^k_*longrightarrow H_lambda_k(X_k)oversetpartial_*longrightarrow mathbbZoverset(i_*^k,j_*^k)longrightarrow0$?




I know that $textim(H_lambda_k(X_k-1))=ker(H_lambda_k(X_k)tomathbbZ)$; however, I am not sure how to determine this kernel. I believe this long exact sequence may be decomposed into a sequence of short exact sequences, one of which being $0overset(i^k_*,j^k_*)longrightarrow H_lambda_k(X_k-1)oversetk^k_*-ell^k_*longrightarrow H_lambda_k(X_k)oversetpartial_*longrightarrow mathbbZoverset(i_*^k,j_*^k)longrightarrow0$.



Any help would be much appreciated. Thanks in advance!






algebraic-topology homology-cohomology







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  • 1




    You know that $H^k=D^ktimes D^n-ksimeqast$ has trivial cohomology. Also $H^kcap Mcong S^k-1times D^nsimeq S^k-1$ has (reduced) cohomology in only a single dimension. These observations should simplify your sequences somewhat.
    – Tyrone
    Sep 2 at 13:34










  • Thanks @Tyrone! The sequence reduces to $0overset(i^k_*,j^k_*)longrightarrow H_lambda_k(X_k-1)oversetk^n_*-ell^n_*longrightarrow H_lambda_k(X_k)oversetpartial_*longrightarrow mathbbZoverset(i_*^k,j_*^k)longrightarrow H_lambda_k-1(X_k-1)oversetk^n_*-ell^n_*longrightarrow H_lambda_k-1(X_k)oversetpartial_*longrightarrow 0$.
    – Multivariablecalculus
    Sep 2 at 15:50










  • Is there a way to proceed from here? @Tyrone
    – Multivariablecalculus
    Sep 2 at 20:19






  • 1




    You can simplitfy your maps as well. The point is that there is that there is a deformation retraction of $Mcup H^k$ onto the core $Mcup_S^k-1 (D^ktimes 0)$ so homotopically attaching a $k$-handle is just the same as attaching a $k$-cell. If you want to study CW homology then maybe Hatcher's book would be a good place to start for a fairly modern treatment.
    – Tyrone
    Sep 3 at 8:55














up vote
3
down vote

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Let $X_k=X_k-1cup_theta H^lambda_k$, for $kge 1$, be the $lambda_k$-handle attached to a $dim X_k-1$-manifold along the embedding map $theta:S^lambda_k-1times D^dim X_k-1-lambda_ktopartial X_k-1$ with $X_0=D^m$. Then the long exact Mayer-Vietoris sequence is $$dotslongrightarrow H_lambda_k+1(X_k)oversetpartial_*longrightarrow H_lambda_k(X_k-1cap H^lambda_k)overset(i^k_*,j^k_*)longrightarrow H_lambda_k(X_k-1)oplus H_lambda_k(H^lambda_k)oversetk^k_*-ell^k_*longrightarrow H_lambda_k(X_k)oversetpartial_*longrightarrow H_lambda_k-1(X_k-1cap H^lambda_k)overset(i_*^k,j_*^k)longrightarrow H_lambda_k-1(X_k-1)oplus H_lambda_k-1(H^lambda_k)oversetk^k_*-ell^k_*longrightarrow H_lambda_k-1(X_k)oversetpartial_*longrightarrowH_lambda_k-2(S^lambda_k-1)overset(i_*^k,j_*^k)longrightarrow dots overset(i_*^k,j_*^k)longrightarrow H_0(X_k-1)oplus H_0(H^lambda_k)oversetk^k_*-ell^k_*longrightarrow H_0(X_k)longrightarrow 0$$ or equivalently, $$dotslongrightarrow H_lambda_k+1(X_k)oversetpartial_*longrightarrow H_lambda_k(S^lambda_k-1)overset(i^k_*,j^k_*)longrightarrow H_lambda_k(X_k-1)oplus H_lambda_k(H^lambda_k)oversetk^k_*-ell^k_*longrightarrow H_lambda_k(X_k)oversetpartial_*longrightarrow H_lambda_k-1(S^lambda_k-1)overset(i_*^k,j_*^k)longrightarrow H_lambda_k-1(X_k-1)oplus H_lambda_k-1(H^lambda_k)oversetk^k_*-ell^k_*longrightarrow H_lambda_k-1(X_k)oversetpartial_*longrightarrowH_lambda_k-2(S^lambda_k-1)overset(i_*^k,j_*^k)longrightarrow dots overset(i_*^k,j_*^k)longrightarrow H_0(X_k-1)oplus H_0(H^lambda_k)oversetk^k_*-ell^k_*longrightarrow H_0(X_k)longrightarrow 0.$$ Observing that $H^k=D^ktimes D^n−kcong*$ has trivial homology, the exact sequence reduces to $$0overset(i^k_*,j^k_*)longrightarrow H_lambda_k(X_k-1)oversetk^k_*-ell^k_*longrightarrow H_lambda_k(X_k)oversetpartial_*longrightarrow mathbbZoverset(i_*^k,j_*^k)longrightarrow H_lambda_k-1(X_k-1)oversetk^k_*-ell^k_*longrightarrow H_lambda_k-1(X_k)oversetpartial_*longrightarrow 0.$$




What is the induced isomorphism, from the sequence of short exact sequences obtained from this long exact sequence, which involves $H_lambda_k(X_k)$? Is it correct to write $0overset(i^k_*,j^k_*)longrightarrow H_lambda_k(X_k-1)oversetk^k_*-ell^k_*longrightarrow H_lambda_k(X_k)oversetpartial_*longrightarrow mathbbZoverset(i_*^k,j_*^k)longrightarrow0$?




I know that $textim(H_lambda_k(X_k-1))=ker(H_lambda_k(X_k)tomathbbZ)$; however, I am not sure how to determine this kernel. I believe this long exact sequence may be decomposed into a sequence of short exact sequences, one of which being $0overset(i^k_*,j^k_*)longrightarrow H_lambda_k(X_k-1)oversetk^k_*-ell^k_*longrightarrow H_lambda_k(X_k)oversetpartial_*longrightarrow mathbbZoverset(i_*^k,j_*^k)longrightarrow0$.



Any help would be much appreciated. Thanks in advance!






algebraic-topology homology-cohomology







share|cite|improve this question



















  • 1




    You know that $H^k=D^ktimes D^n-ksimeqast$ has trivial cohomology. Also $H^kcap Mcong S^k-1times D^nsimeq S^k-1$ has (reduced) cohomology in only a single dimension. These observations should simplify your sequences somewhat.
    – Tyrone
    Sep 2 at 13:34










  • Thanks @Tyrone! The sequence reduces to $0overset(i^k_*,j^k_*)longrightarrow H_lambda_k(X_k-1)oversetk^n_*-ell^n_*longrightarrow H_lambda_k(X_k)oversetpartial_*longrightarrow mathbbZoverset(i_*^k,j_*^k)longrightarrow H_lambda_k-1(X_k-1)oversetk^n_*-ell^n_*longrightarrow H_lambda_k-1(X_k)oversetpartial_*longrightarrow 0$.
    – Multivariablecalculus
    Sep 2 at 15:50










  • Is there a way to proceed from here? @Tyrone
    – Multivariablecalculus
    Sep 2 at 20:19






  • 1




    You can simplitfy your maps as well. The point is that there is that there is a deformation retraction of $Mcup H^k$ onto the core $Mcup_S^k-1 (D^ktimes 0)$ so homotopically attaching a $k$-handle is just the same as attaching a $k$-cell. If you want to study CW homology then maybe Hatcher's book would be a good place to start for a fairly modern treatment.
    – Tyrone
    Sep 3 at 8:55












up vote
3
down vote

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Let $X_k=X_k-1cup_theta H^lambda_k$, for $kge 1$, be the $lambda_k$-handle attached to a $dim X_k-1$-manifold along the embedding map $theta:S^lambda_k-1times D^dim X_k-1-lambda_ktopartial X_k-1$ with $X_0=D^m$. Then the long exact Mayer-Vietoris sequence is $$dotslongrightarrow H_lambda_k+1(X_k)oversetpartial_*longrightarrow H_lambda_k(X_k-1cap H^lambda_k)overset(i^k_*,j^k_*)longrightarrow H_lambda_k(X_k-1)oplus H_lambda_k(H^lambda_k)oversetk^k_*-ell^k_*longrightarrow H_lambda_k(X_k)oversetpartial_*longrightarrow H_lambda_k-1(X_k-1cap H^lambda_k)overset(i_*^k,j_*^k)longrightarrow H_lambda_k-1(X_k-1)oplus H_lambda_k-1(H^lambda_k)oversetk^k_*-ell^k_*longrightarrow H_lambda_k-1(X_k)oversetpartial_*longrightarrowH_lambda_k-2(S^lambda_k-1)overset(i_*^k,j_*^k)longrightarrow dots overset(i_*^k,j_*^k)longrightarrow H_0(X_k-1)oplus H_0(H^lambda_k)oversetk^k_*-ell^k_*longrightarrow H_0(X_k)longrightarrow 0$$ or equivalently, $$dotslongrightarrow H_lambda_k+1(X_k)oversetpartial_*longrightarrow H_lambda_k(S^lambda_k-1)overset(i^k_*,j^k_*)longrightarrow H_lambda_k(X_k-1)oplus H_lambda_k(H^lambda_k)oversetk^k_*-ell^k_*longrightarrow H_lambda_k(X_k)oversetpartial_*longrightarrow H_lambda_k-1(S^lambda_k-1)overset(i_*^k,j_*^k)longrightarrow H_lambda_k-1(X_k-1)oplus H_lambda_k-1(H^lambda_k)oversetk^k_*-ell^k_*longrightarrow H_lambda_k-1(X_k)oversetpartial_*longrightarrowH_lambda_k-2(S^lambda_k-1)overset(i_*^k,j_*^k)longrightarrow dots overset(i_*^k,j_*^k)longrightarrow H_0(X_k-1)oplus H_0(H^lambda_k)oversetk^k_*-ell^k_*longrightarrow H_0(X_k)longrightarrow 0.$$ Observing that $H^k=D^ktimes D^n−kcong*$ has trivial homology, the exact sequence reduces to $$0overset(i^k_*,j^k_*)longrightarrow H_lambda_k(X_k-1)oversetk^k_*-ell^k_*longrightarrow H_lambda_k(X_k)oversetpartial_*longrightarrow mathbbZoverset(i_*^k,j_*^k)longrightarrow H_lambda_k-1(X_k-1)oversetk^k_*-ell^k_*longrightarrow H_lambda_k-1(X_k)oversetpartial_*longrightarrow 0.$$




What is the induced isomorphism, from the sequence of short exact sequences obtained from this long exact sequence, which involves $H_lambda_k(X_k)$? Is it correct to write $0overset(i^k_*,j^k_*)longrightarrow H_lambda_k(X_k-1)oversetk^k_*-ell^k_*longrightarrow H_lambda_k(X_k)oversetpartial_*longrightarrow mathbbZoverset(i_*^k,j_*^k)longrightarrow0$?




I know that $textim(H_lambda_k(X_k-1))=ker(H_lambda_k(X_k)tomathbbZ)$; however, I am not sure how to determine this kernel. I believe this long exact sequence may be decomposed into a sequence of short exact sequences, one of which being $0overset(i^k_*,j^k_*)longrightarrow H_lambda_k(X_k-1)oversetk^k_*-ell^k_*longrightarrow H_lambda_k(X_k)oversetpartial_*longrightarrow mathbbZoverset(i_*^k,j_*^k)longrightarrow0$.



Any help would be much appreciated. Thanks in advance!






algebraic-topology homology-cohomology







share|cite|improve this question















Let $X_k=X_k-1cup_theta H^lambda_k$, for $kge 1$, be the $lambda_k$-handle attached to a $dim X_k-1$-manifold along the embedding map $theta:S^lambda_k-1times D^dim X_k-1-lambda_ktopartial X_k-1$ with $X_0=D^m$. Then the long exact Mayer-Vietoris sequence is $$dotslongrightarrow H_lambda_k+1(X_k)oversetpartial_*longrightarrow H_lambda_k(X_k-1cap H^lambda_k)overset(i^k_*,j^k_*)longrightarrow H_lambda_k(X_k-1)oplus H_lambda_k(H^lambda_k)oversetk^k_*-ell^k_*longrightarrow H_lambda_k(X_k)oversetpartial_*longrightarrow H_lambda_k-1(X_k-1cap H^lambda_k)overset(i_*^k,j_*^k)longrightarrow H_lambda_k-1(X_k-1)oplus H_lambda_k-1(H^lambda_k)oversetk^k_*-ell^k_*longrightarrow H_lambda_k-1(X_k)oversetpartial_*longrightarrowH_lambda_k-2(S^lambda_k-1)overset(i_*^k,j_*^k)longrightarrow dots overset(i_*^k,j_*^k)longrightarrow H_0(X_k-1)oplus H_0(H^lambda_k)oversetk^k_*-ell^k_*longrightarrow H_0(X_k)longrightarrow 0$$ or equivalently, $$dotslongrightarrow H_lambda_k+1(X_k)oversetpartial_*longrightarrow H_lambda_k(S^lambda_k-1)overset(i^k_*,j^k_*)longrightarrow H_lambda_k(X_k-1)oplus H_lambda_k(H^lambda_k)oversetk^k_*-ell^k_*longrightarrow H_lambda_k(X_k)oversetpartial_*longrightarrow H_lambda_k-1(S^lambda_k-1)overset(i_*^k,j_*^k)longrightarrow H_lambda_k-1(X_k-1)oplus H_lambda_k-1(H^lambda_k)oversetk^k_*-ell^k_*longrightarrow H_lambda_k-1(X_k)oversetpartial_*longrightarrowH_lambda_k-2(S^lambda_k-1)overset(i_*^k,j_*^k)longrightarrow dots overset(i_*^k,j_*^k)longrightarrow H_0(X_k-1)oplus H_0(H^lambda_k)oversetk^k_*-ell^k_*longrightarrow H_0(X_k)longrightarrow 0.$$ Observing that $H^k=D^ktimes D^n−kcong*$ has trivial homology, the exact sequence reduces to $$0overset(i^k_*,j^k_*)longrightarrow H_lambda_k(X_k-1)oversetk^k_*-ell^k_*longrightarrow H_lambda_k(X_k)oversetpartial_*longrightarrow mathbbZoverset(i_*^k,j_*^k)longrightarrow H_lambda_k-1(X_k-1)oversetk^k_*-ell^k_*longrightarrow H_lambda_k-1(X_k)oversetpartial_*longrightarrow 0.$$




What is the induced isomorphism, from the sequence of short exact sequences obtained from this long exact sequence, which involves $H_lambda_k(X_k)$? Is it correct to write $0overset(i^k_*,j^k_*)longrightarrow H_lambda_k(X_k-1)oversetk^k_*-ell^k_*longrightarrow H_lambda_k(X_k)oversetpartial_*longrightarrow mathbbZoverset(i_*^k,j_*^k)longrightarrow0$?




I know that $textim(H_lambda_k(X_k-1))=ker(H_lambda_k(X_k)tomathbbZ)$; however, I am not sure how to determine this kernel. I believe this long exact sequence may be decomposed into a sequence of short exact sequences, one of which being $0overset(i^k_*,j^k_*)longrightarrow H_lambda_k(X_k-1)oversetk^k_*-ell^k_*longrightarrow H_lambda_k(X_k)oversetpartial_*longrightarrow mathbbZoverset(i_*^k,j_*^k)longrightarrow0$.



Any help would be much appreciated. Thanks in advance!






algebraic-topology homology-cohomology




algebraic-topology homology-cohomology






share|cite|improve this question















share|cite|improve this question













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edited Sep 6 at 0:45

























asked Sep 2 at 5:26









Multivariablecalculus

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  • 1




    You know that $H^k=D^ktimes D^n-ksimeqast$ has trivial cohomology. Also $H^kcap Mcong S^k-1times D^nsimeq S^k-1$ has (reduced) cohomology in only a single dimension. These observations should simplify your sequences somewhat.
    – Tyrone
    Sep 2 at 13:34










  • Thanks @Tyrone! The sequence reduces to $0overset(i^k_*,j^k_*)longrightarrow H_lambda_k(X_k-1)oversetk^n_*-ell^n_*longrightarrow H_lambda_k(X_k)oversetpartial_*longrightarrow mathbbZoverset(i_*^k,j_*^k)longrightarrow H_lambda_k-1(X_k-1)oversetk^n_*-ell^n_*longrightarrow H_lambda_k-1(X_k)oversetpartial_*longrightarrow 0$.
    – Multivariablecalculus
    Sep 2 at 15:50










  • Is there a way to proceed from here? @Tyrone
    – Multivariablecalculus
    Sep 2 at 20:19






  • 1




    You can simplitfy your maps as well. The point is that there is that there is a deformation retraction of $Mcup H^k$ onto the core $Mcup_S^k-1 (D^ktimes 0)$ so homotopically attaching a $k$-handle is just the same as attaching a $k$-cell. If you want to study CW homology then maybe Hatcher's book would be a good place to start for a fairly modern treatment.
    – Tyrone
    Sep 3 at 8:55












  • 1




    You know that $H^k=D^ktimes D^n-ksimeqast$ has trivial cohomology. Also $H^kcap Mcong S^k-1times D^nsimeq S^k-1$ has (reduced) cohomology in only a single dimension. These observations should simplify your sequences somewhat.
    – Tyrone
    Sep 2 at 13:34










  • Thanks @Tyrone! The sequence reduces to $0overset(i^k_*,j^k_*)longrightarrow H_lambda_k(X_k-1)oversetk^n_*-ell^n_*longrightarrow H_lambda_k(X_k)oversetpartial_*longrightarrow mathbbZoverset(i_*^k,j_*^k)longrightarrow H_lambda_k-1(X_k-1)oversetk^n_*-ell^n_*longrightarrow H_lambda_k-1(X_k)oversetpartial_*longrightarrow 0$.
    – Multivariablecalculus
    Sep 2 at 15:50










  • Is there a way to proceed from here? @Tyrone
    – Multivariablecalculus
    Sep 2 at 20:19






  • 1




    You can simplitfy your maps as well. The point is that there is that there is a deformation retraction of $Mcup H^k$ onto the core $Mcup_S^k-1 (D^ktimes 0)$ so homotopically attaching a $k$-handle is just the same as attaching a $k$-cell. If you want to study CW homology then maybe Hatcher's book would be a good place to start for a fairly modern treatment.
    – Tyrone
    Sep 3 at 8:55







1




1




You know that $H^k=D^ktimes D^n-ksimeqast$ has trivial cohomology. Also $H^kcap Mcong S^k-1times D^nsimeq S^k-1$ has (reduced) cohomology in only a single dimension. These observations should simplify your sequences somewhat.
– Tyrone
Sep 2 at 13:34




You know that $H^k=D^ktimes D^n-ksimeqast$ has trivial cohomology. Also $H^kcap Mcong S^k-1times D^nsimeq S^k-1$ has (reduced) cohomology in only a single dimension. These observations should simplify your sequences somewhat.
– Tyrone
Sep 2 at 13:34












Thanks @Tyrone! The sequence reduces to $0overset(i^k_*,j^k_*)longrightarrow H_lambda_k(X_k-1)oversetk^n_*-ell^n_*longrightarrow H_lambda_k(X_k)oversetpartial_*longrightarrow mathbbZoverset(i_*^k,j_*^k)longrightarrow H_lambda_k-1(X_k-1)oversetk^n_*-ell^n_*longrightarrow H_lambda_k-1(X_k)oversetpartial_*longrightarrow 0$.
– Multivariablecalculus
Sep 2 at 15:50




Thanks @Tyrone! The sequence reduces to $0overset(i^k_*,j^k_*)longrightarrow H_lambda_k(X_k-1)oversetk^n_*-ell^n_*longrightarrow H_lambda_k(X_k)oversetpartial_*longrightarrow mathbbZoverset(i_*^k,j_*^k)longrightarrow H_lambda_k-1(X_k-1)oversetk^n_*-ell^n_*longrightarrow H_lambda_k-1(X_k)oversetpartial_*longrightarrow 0$.
– Multivariablecalculus
Sep 2 at 15:50












Is there a way to proceed from here? @Tyrone
– Multivariablecalculus
Sep 2 at 20:19




Is there a way to proceed from here? @Tyrone
– Multivariablecalculus
Sep 2 at 20:19




1




1




You can simplitfy your maps as well. The point is that there is that there is a deformation retraction of $Mcup H^k$ onto the core $Mcup_S^k-1 (D^ktimes 0)$ so homotopically attaching a $k$-handle is just the same as attaching a $k$-cell. If you want to study CW homology then maybe Hatcher's book would be a good place to start for a fairly modern treatment.
– Tyrone
Sep 3 at 8:55




You can simplitfy your maps as well. The point is that there is that there is a deformation retraction of $Mcup H^k$ onto the core $Mcup_S^k-1 (D^ktimes 0)$ so homotopically attaching a $k$-handle is just the same as attaching a $k$-cell. If you want to study CW homology then maybe Hatcher's book would be a good place to start for a fairly modern treatment.
– Tyrone
Sep 3 at 8:55















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How to combine Bézier curves to a surface?

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Clash Royale CLAN TAG #URR8PPP up vote 2 down vote favorite My aim is to smooth the terrain in a video game. Therefore I contrived an algorithm that makes use of Bézier curves of different orders. But this algorithm is defined in a two dimensional space for now. To shift it into the third dimension I need to somehow combine the Bézier curves. Given two curves in each two dimensions, how can I combine them to build a surface? I thought about something like a curve of a curve. Or maybe I could multiply them somehow. How can I combine them to cause the wanted behavior? Is there a common approach? In the image you can see the input, on the left, and the output, on the right, of the algorithm that works in a two dimensional space. For the real application there is a three dimensional input. The algorithm relays only on the adjacent tiles. Given these it will define the control points of the mentioned Bézier cur...
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1st Magritte Awards

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1st Magritte Awards Official poster Date 5 February 2011  ( 2011-02-05 ) Site Square Mont des Arts, Brussels, Belgium Hosted by Helena Noguerra Produced by José Bouquiaux Directed by Vincent J. Gustin Highlights Best Film Mr. Nobody Most awards Mr. Nobody (6) Most nominations Illegal (8) Television coverage Network BeTV Magritte Awards 2nd → The 1st Magritte Awards ceremony, presented by the Académie André Delvaux, honored the best films of 2010 in Belgium and took place on 5 February 2011 at the Square in the historic site of Mont des Arts, Brussels, beginning at 7:30 p.m. CET. During the ceremony, the Académie André Delvaux presented Magritte Awards in twenty categories. The ceremony, televised in Belgium by BeTV, was produced by José Bouquiaux and directed by Vincent J. Gustin. [1] Film director Jaco Van Dormael presided the ceremony, while actress Helena Noguerra hosted the evening. [2] The pre-show ceremony was hoste...
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