Homology with a star symbol $H_*$

Question

Just to ask what is the meaning of the symbol $H_*$ in the context of homology.

For instance $H_*(X)$, where $X$ is a simplicial complex.

Does it mean $H_k(X)$, where $k$ can be any natural number, i.e. the $n$th homology group?

Thanks.

Answer 1

Your guess is partly right, $*$ is more or less a variable standing in for a natural number. But, also, $H_*(X)$ sometimes means the graded module $H_0(X) \oplus H_1(X) \oplus \cdots$.

This usage is even more common with cohomology, where $H^*(X)$ will often mean the graded ring $H^0(X) \oplus H^1(X) \oplus H^2(X) \oplus \cdots$ where the product operation is cup product.