I don’t actually know much about topology, apart from the edge-case of graph theory. Often the coarse topology of networks is of interest, or the more metaphorical knowledge topology, or actual continuous fields or knots or whatever. Sometimes I care about the induced topology of a metric convergence, but this is hardly fancy stuff.
Apparently grown-up topology has applications too? See for example
Abrams, A, and Robert Ghrist. 2002. “Finding Topology in a Factory: Configuration Spaces.” The American Mathematical Monthly 109 (2): 140–50.
Brüel-Gabrielsson, Rickard, Bradley J. Nelson, Anjan Dwaraknath, Primoz Skraba, Leonidas J. Guibas, and Gunnar Carlsson. 2019. “A Topology Layer for Machine Learning.” May 28, 2019. http://arxiv.org/abs/1905.12200.
Chen, Chao, Xiuyan Ni, Qinxun Bai, and Yusu Wang. 2018. “A Topological Regularizer for Classifiers via Persistent Homology.” June 27, 2018. http://arxiv.org/abs/1806.10714.
Cohen-Steiner, David, Herbert Edelsbrunner, and John Harer. 2007. “Stability of Persistence Diagrams.” Discrete & Computational Geometry 37 (1): 103–20. https://doi.org/10.1007/s00454-006-1276-5.
Gebhart, Thomas, Paul Schrater, and Alan Hylton. 2019. “Characterizing the Shape of Activation Space in Deep Neural Networks.” January 27, 2019. http://arxiv.org/abs/1901.09496.
Ghrist, Robert. 2008. “Barcodes: The Persistent Topology of Data.” Bulletin of the American Mathematical Society 45 (1): 61–75. https://doi.org/10.1090/S0273-0979-07-01191-3.
Ghrist, Robert W. 2014. Elementary Applied Topology. http://www.math.upenn.edu/~ghrist/notes.html.
Liu, Jen-Yu, Shyh-Kang Jeng, and Yi-Hsuan Yang. 2016. “Applying Topological Persistence in Convolutional Neural Network for Music Audio Signals.” August 26, 2016. http://arxiv.org/abs/1608.07373.
Petri, G., P. Expert, F. Turkheimer, R. Carhart-Harris, D. Nutt, P. J. Hellyer, and F. Vaccarino. 2014. “Homological Scaffolds of Brain Functional Networks.” Journal of the Royal Society Interface 11 (101): 20140873. https://doi.org/10.1098/rsif.2014.0873.
Petri, Giovanni, Martina Scolamiero, Irene Donato, and Francesco Vaccarino. 2013a. “Networks and Cycles: A Persistent Homology Approach to Complex Networks.” In Proceedings of the European Conference on Complex Systems 2012, edited by Thomas Gilbert, Markus Kirkilionis, and Gregoire Nicolis, 93–99. Springer Proceedings in Complexity. Springer International Publishing. http://link.springer.com/chapter/10.1007/978-3-319-00395-5_15.
———. 2013b. “Topological Strata of Weighted Complex Networks.” PLoS ONE 8 (6): e66506. https://doi.org/10.1371/journal.pone.0066506.
Toiviainen, Petri. 1997. “Optimizing Self-Organizing Timbre Maps: Two Approaches.” In Music, Gestalt, and Computing, edited by Marc Leman, 335–50. Lecture Notes in Computer Science 1317. Springer Berlin Heidelberg. http://link.springer.com/chapter/10.1007/BFb0034124.
Zomorodian, Afra, and Gunnar Carlsson. 2005. “Computing Persistent Homology.” Discrete & Computational Geometry 33 (2): 249–74. https://doi.org/10.1007/s00454-004-1146-y.