---
tag: 33GY6P5A4T
kind: concept
name: Casson invariant
msc: [57K31]
sources:
  - url: https://en.wikipedia.org/wiki/Casson_invariant
same_as:
  wikidata: Q5049507
  wikipedia: Casson invariant
---

## Summary

The Casson invariant is an integer-valued invariant of oriented integral [[homology 3-spheres|HK8NS02NNE]], introduced by Andrew Casson.
Informally it counts, with signs, the irreducible representations of the fundamental group into $SU(2)$, up to conjugation, divided by $2$.
It reduces modulo $2$ to the Rokhlin invariant, and it has a simple behaviour under [[Dehn surgery|ZA405B0Z1S]] on knots.
It was one of the first gauge-theoretic style invariants of $3$-manifolds and led to instanton Floer homology.

## Explanations

### L1 human

A homology $3$-sphere looks like the $3$-sphere to homology, but may be very different.
The Casson invariant is a whole number attached to each one, and it is $0$ for the $3$-sphere.
When it is not $0$, the fundamental group has a nontrivial representation into the group of unit quaternions, so the manifold cannot be the $3$-sphere.
For the Poincaré homology sphere it equals $\pm 1$, depending on the orientation.