---
date: 2022-02-07 22:35
modification date: Monday 7th February 2022 22:35:46
title: loop space
aliases: [free loop space, based loop space, loop homology]

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Last modified date: <%+ tp.file.last_modified_date() %>


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- Tags 
	- #homotopy/stable-homotopy 
- Refs:
	- #todo/add-references
- Links:
	- [homotopy](Unsorted/homotopy.md)
	- [smash product](Unsorted/smash%20product.md)

---

# 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.

![](attachments/Pasted%20image%2020220401122633.png)

# Circle action

![](attachments/Pasted%20image%2020220207223603.png)
# Loop homology

- The **loop product** arises from a combination of the degree $(-d)$ intersection product on $H_{*}(M)$ and of the [Pontryagin product](Pontryagin%20product) on $H_{*}(\Omega M)$ induced by concatenation of based loops. Consequently, the loop product also exhibits a degree shift of $-d$ :
$$
\circ: H_{p}(L M) \otimes H_{q}(L M) \rightarrow H_{p+q-d}(L M) .
$$
- In order that $\circ$ define a graded algebra structure, we shift $H_{*}(L M)$ accordingly:
	- Denote $\Sigma^{-d} H_{*}(L M)$ as $\mathbb{H}_{*}(L M)$, called the **loop homology** of $M$, so that $\mathbb{H}_{q}(L M)=H_{q+d}(L M) .$

	- 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 $[6,42]$.
- $H_*(\Loop M)[d]$ Carries a product: Chas-Sullivan product

# Infinite loop spaces

![](attachments/Pasted%20image%2020220209190549.png)