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dg.differential geometry – Vector bailiwick tangent to a submanifold and transverse to the zero part Answer

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dg.differential geometry – Vector bailiwick tangent to a submanifold and transverse to the zero part

In Hirsch’s Differential Topology there’s the next :

Suppose a compact $n$-manifold can breathe expressed as $Acup B$ the place $A,B$ are compact $n$-dimensional submanifolds and $Acap B$ is an $(n-1)$-dimensional submanifold. Then $chi(Acup B)=chi(A)+chi(B)-chi(Acap B)$.

Trying to decipher this a query the next got here to my intellect :

Suppose we’ve got a submanifold $N$ of $M$.Is it workable to have a bit $f:Mrightarrow TM$ such that $f|_Nin TN$, the place we make the canonical identifications, and such that $f$ is transverse to the zero part ?

Now I belive we will assemble a vector bailiwick $f$ in $M$ that’s tangent to $N$ utilizing native charts however for the added requirement of it being transverse to the zero part I’m not positive the way it might breathe finished, positive we will employ the transversality theorem to get maps $h_krightarrow f$ which might be transverse to the zero part , however no signify how immediate I can approximate $f$ I can at all times raze the truth that is tangent to $N$, not positive if there may be anymore circumstances I can put to cease this from hapenning.

Does anybody have any ideas on this ? Thanks in forward.

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