notation used in algebraic topology

Question

i have some confusion in notations used in my algebraic topology class.

$\approx$ homeomorphic

$\simeq$ homotopy

$\cong$ isomorphic

Please correct me for the above.

Answer 1

There is some variation but the following is quite standard. In a general category, isomorphism is denoted by $\cong $, so that $x\cong y$ means $x$ and $y$ are isomorphic objects. In the category of topological spaces, isomorphism goes by the name homeomorphism, and so homeomorphic spaces are typically denoted by $X\cong Y$. Similarly, in the category of groups, isomorphism goes by the name isomorphism, and isomorphic groups are typically denoted by $G\cong H$. Since homotopy is a weaker notion than isomorphism, it is typically denoted by something line $\approx$ or $\simeq$.

Having said that, there are sufficient variations. Some author like to use equality also to mean "canonically isomorphic", i.e., when there exists a preferred, uniformly coherent, choice of an isomorphism between two objects. Others may prefer to have, in the context of algebraic topology, different notation for isomorphism of groups versus homeomorphism of spaces. In other words: proceed with caution.