One of the fundamental theorems with many potential physical applications in Quantum Algebraic Topology (QAT) is the Generalized van Kampen theorem (GvKT). GvkT potential applications include the construction of global quantum state space (QST) representations in the physical context of relativistic quantum measurements or locally-covariant quantum gravity, and the precise embedding of smaller structures into larger ones in order to obtain exact solutions for the former. See also a downloadable publication in PDF file format:
http://www.bangor.ac.uk/~mas010/pdffiles/BBG-158-NAC0STQG.pdf
http://www.bangor.ac.uk/~mas010/publicfull.htm
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Open access:
http://www.bangor.ac.uk/~mas010/publicfull.htm
http://www.bangor.ac.uk/~mas010/pdffiles/BBG-158-NAC0STQG.pdf