of the inclusion $\partial M \to  M$ together with the unit section $M\to  E\bigstar M$. The manifold $\hat{W}$ together with the map $F\cup_{\partial M} f$, and the class $\hat{\Sigma}= \Sigma \cup_{\partial M}  \sigma$ give a geometric cycle in ${K^{\rm geo}_ {G, *}({X},{P})}$ which maps to the given cycle for $(X,A)$.

    Exactness at other points of the sequence is verified analogously.

4.  We will construct an inverse to the map induced by the inclusion $l$. Recall the extension of Morse Theory to proper actions [@Hingstonequivariantmorsetheory], Theorems A and B in page 94. It can be concluded that given a geometric cycle $(M, f, \sigma) $ for $(X,A)$, there exists a smooth $G$-invariant function $\epsilon :M\to  {{\mathbb R}}$ with a number of critical orbits, which are all generic.

    We will assume that the function $\epsilon$ separates $f^{-1}(  \overline{U})$ and $M - f^{-1}(A)$. Then, there exist real numbers $a<b$ such that $a= \sup_{x\in M-f^{-1}(A) }\epsilon (x)$, and $b= \inf_{x\in f^{-1}(\overline{U})} \epsilon (x)$.

    For a regular value $c\in (a,b)$, $\epsilon^{c}= \epsilon^{-1}(-\infty, c])$ can be furnished with a smooth structure. The boundary component $\epsilon^{-1}(c)\cup \epsilon^{-1}(-\infty, c])\cap  \partial  M$ is mapped under $f$ to $X-U$.

    Then, the restriction of the cycle to $M_1$ gives a geometric 