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Let $A$ and $C$ be two unital simple C*-algebas with tracial rank zero.Suppose that $C$ is amenable and satisfies the Universal Coefficient Theorem.Denote by ${{KK}}_e(C,A)^{++}$ the set of those $\kappa$ for which$\kappa(K_0(C)_+\setminus\{0\})\subset K_0(A)_+\setminus\{0\}$ and$\kappa([1_C])=[1_A]$. Suppose that $\kappa\in {KK}_e(C,A)^{++}.$  We show that there is a unital monomorphism $\phi: C\to A$ such that$[\phi]=\kappa.$ Suppose that $C$ is a unital AH-algebra and $\lambda:\mathrm{T}(A)\to \mathrm{T}_{\mathtt{f}}(C)$ is a continuous affine map forwhich $\tau(\kappa([p]))=\lambda(\tau)(p)$ for all projections $p$ in allmatrix algebras of $C$ and any $\tau\in \mathrm{T}(A),$ where $\mathrm{T}(A)$is the simplex of tracial states of $A$ and $\mathrm{T}_{\mathtt{f}}(C)$ is theconvex set of faithful tracial states of $C.$ We prove that there is a unitalmonomorphism $\phi: C\to A$ such that $\phi$ induces both $\kappa$ and$\lambda.$  Suppose that $h: C\to A$ is a unital monomorphism and $\gamma \in\mathrm{Hom}(\Kone(C), \aff(A)).$ We show that there exists a unitalmonomorphism $\phi: C\to A$ such that $[\phi]=[h]$ in ${KK}(C,A),$ $\tau\circ\phi=\tau\circ h$ for all tracial states $\tau$ and the associated rotation mapcan be given by $\gamma.$ Applications to classification of simple C*-algebrasare also given.

Lifting KK-elements, asymptotical unitary equivalence and classification of simple C*-algebras