During last decade, researchers in speaker recognition have been working over the same acoustic space, regardless of whether the data lie on a linear space or not. Our proposal is to take into account the inner geometric structure of the speech in order to obtain a new space with a better representation of the speech data. A topological approach based on manifolds obtained thanks to Laplacian and Isomap algorithms is proposed. In this first work, the proposal is evaluated in terms of dimension reduction of the supervector space, known to have a high redundancy. The experiments are done in the NIST-SRE framework. It appears that the proposed approach allows to reduce by a factor four the dimension of the supervector space without losses in terms of EER. This first result highlights the potential of topological approaches for speaker recognition.