Cardiac T1 mapping can evaluate various clinical symptoms of myocardial tissue. However, there is currently a lack of effective, robust, and efficient methods for motion correction in cardiac T1 mapping. In this paper, we propose a deep learning-based and topology-preserving image registration framework for motion correction in cardiac T1 mapping. Notably, our proposed implicit consistency constraint dubbed BLOC, to some extent preserves the image topology in registration by bidirectional consistency constraint and local anti-folding constraint. To address the contrast variation issue, we introduce a weighted image similarity metric for multimodal registration of cardiac T1-weighted images. Besides, a semi-supervised myocardium segmentation network and a dual-domain attention module are integrated into the framework to further improve the performance of the registration. Numerous comparative experiments, as well as ablation studies, demonstrated the effectiveness and high robustness of our method. The results also indicate that the proposed weighted image similarity metric, specifically crafted for our network, contributes a lot to the enhancement of the motion correction efficacy, while the bidirectional consistency constraint combined with the local anti-folding constraint ensures a more desirable topology-preserving registration mapping.