Abstract
In this chapter, we study the rack space from homotopy theory, and see some results on the quandle homotopy invariant. More precisely, in Sect. 6.1, we first introduce a monoid structure of the rack space. In Sect. 6.2, we describe the classifying map of the rack space, and discuss a relation to second group homology. After that, in Sect. 6.3, we discuss the homotopy type of the rack space of the link quandle; In Sect. 6.4, we give a topological meaning of the quandle homotopy link-invariant. Finally, we provide a method of computing the third quandle homology; see Sect. 6.5.
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Every based loop space is equivalent to a monoid by the composite of loops. Conversely, every topological monoid \(\mathscr {M}\) with connected CW-structure is homotopy equivalent to a loop space. Indeed, the principle \(\mathscr {M} \)-bundle \(\mathscr {M} \rightarrow E\mathscr {M} \rightarrow B\mathscr {M} \) implies \(\mathscr {M} \simeq \Omega B \mathscr {M} \) because of the contractile space \(E \mathscr {M}\).
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Nosaka, T. (2017). Topology on the Quandle Homotopy Invariant. In: Quandles and Topological Pairs. SpringerBriefs in Mathematics. Springer, Singapore. https://doi.org/10.1007/978-981-10-6793-8_6
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DOI: https://doi.org/10.1007/978-981-10-6793-8_6
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Publisher Name: Springer, Singapore
Print ISBN: 978-981-10-6792-1
Online ISBN: 978-981-10-6793-8
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