Let me suppose that $M$ is a closed manifold of dimension $d$, and let $\varphi : M \to M$ be a diffeomorphism / homeomorphism which is homotopic to the identity, and choose such a homotopy $h_t$. The mapping torus $X_\varphi$ of $\varphi$ is a smooth / topological manifold fibering over $S^1$, and our choice of homotopy $h_t$ yields a preferred homotopy equivalence
$$ H : X_\varphi \longrightarrow S^1 \times M.$$

For each cohomology class $v \in H^{d-4i+1}(M;\mathbb{Q})$ we may form
$$\int_{X_\varphi} p_i(TX_\varphi - H^*TM) \smile H^*(1 \otimes v) \in \mathbb{Q}$$
where $p_i$ is the (rational) Pontrjagin class. This defines a linear functional $H^{d-4i+1}(M;\mathbb{Q}) \to \mathbb{Q}$ which is Poincare dual to a class $\xi_i \in H^{4i-1}(M;\mathbb{Q})$.

Morally this class measures the following: on the one hand $\varphi$, by virtue of being a homeomorphism, preserves the rational Pontrjagin class $p_i(M)$, so preserves a given cocycle representative $c$ up to a "preferred" coboundary. On the other hand, the homotopy $h_t$ gives another coboundary for $c-\varphi^*c$; the difference between these cobounding cochains is thus a cycle, and it represents the cohomology class $\xi_i$.

In principle $\xi_i$ is an invariant of the pair $(\varphi, h_t)$. However, a different choice of homotopy $h_t'$ yields a homotopy equivalence $H'$ which differs from $H$ by postcomposition by a homotopy automorphism of $S^1 \times M$ which i) is over $S^1$ and, ii) fixes a fibre $M \times \{*\}$. This means that for each $v$ we have
$$(H')^*(1 \otimes v) = H^*(1 \otimes v) + H^*(u \otimes \bar{v})$$
for some $\bar{v}$, where $u \in H^1(S^1;\mathbb{Q})$ is the canonical class. But
$$\int_{X_\varphi} p_i(TX_\varphi - H^*TM) \smile H^*(u \otimes \bar{v})$$
can be written as
$$\int_{S^1 \times M} (H^{-1})^*(p_i(TX_\varphi-H^*TM)) \smile (u \otimes \bar{v}) = \int_M p_i(TM-TM) \cup \bar{v}=0$$
so in fact $\xi_i$ only depends on $\varphi$.

As $X_{\varphi \circ \psi}$ is cobordant to $X_{\varphi} \sqcup X_{\psi}$, one sees that $\xi_i(\varphi \circ \psi) = \xi_i(\varphi) + \xi_i(\psi)$. In particular if $\xi_i(\varphi) \neq 0$ for some $i$ then $\varphi$ has infinite order.

There is a theorem due to Sullivan that a simply-connected high-dimensional manifold is determined "up to finite ambiguity" by its homotopy type and rational Pontrjagin classes. The analogous statement for automorphisms of simply-connected (or perhaps 2-connected?) manifolds says that elements in
$$\mathrm{Ker}(\pi_0(Top(M)) \to \pi_0(G(M)))$$
(or the smooth analogue) are determined up to finite ambiguity by their associated $\xi_i$'s. This can (surely?) be proved using the surgery exact sequence.