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loop space
There are substantial connections between the homology and cohomology of the free loop space \(L X\) of a space \(X\) and the Hochschild homology and cohomology of the DGAs \(C_{*} \Omega X\) and \(C^{*} X\). We state the key results that we employ below and survey the other.
Circle action
# Loop homology
- The loop product arises from a combination of the degree \((-d)\) intersection product on \(H_{*}(M)\) and of the Pontryagin product on \(H_{*}(\Omega M)\) induced by concatenation of based loops. Consequently, the loop product also exhibits a degree shift of \(-d\) : \begin{align*} \circ: H_{p}(L M) \otimes H_{q}(L M) \rightarrow H_{p+q-d}(L M) . \end{align*}
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In order that \(\circ\) define a graded algebra structure, we shift \(H_{*}(L M)\) accordingly:
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Denote \(\Sigma^{-d} H_{*}(L M)\) as \(\mathbb{H}_{*}(L M)\), called the loop homology of \(M\), so that \(\mathbb{H}_{q}(L M)=H_{q+d}(L M) .\)
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Under this degree shift, \(\Delta\) gives a degree-1 operator on \(\mathbb{H}_{*}(L M) .\) The key result of Chas and Sullivan is that \(\circ\) and \(\Delta\) interact to give a BV algebra structure on \(\mathbb{H}_{*}(L M) .\) As discussed in Section A.1.6, this BV algebra structure gives a canonical Gerstenhaber algebra structure, and the resulting Lie bracket, denoted \(\{-,-\}\), is called the loop bracket. The loop bracket can also be defined more directly using operations on Thom spectra \([6,42]\).
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- \(H_*({\Omega}M)[d]\) Carries a product: Chas-Sullivan product
Infinite loop spaces
