Researchers propose using finite zero-dimensional persistent homology to represent the topology of ill-posed questions within large language models. The method models contextual hidden states as point clouds, summarizing each transformer layer with three descriptors: mean finite lifetime, normalized lifetime entropy, and largest-lifetime concentration. These descriptors are concatenated across layers to form a unified topological representation of the query's internal state. The study introduces topology-conditioned activation steering, which retrieves similar examples to construct interventions that encourage clarification or abstention. Evaluations on AmbigQA, SituatedQA, and CLAMBER show this approach outperforms prompt-based baselines, improving classification accuracy from 67.4% to 78.9% on AmbigQA. On SituatedQA, accuracy increased from 79.9% to 88.5%, while CLAMBER saw gains from 57.6% to 69.6%. Additionally, the steering mechanism raised the average total acceptable response rate from 61.4% to 70.6% across three open-weight LLMs.