# Progress in Topological Mechanics
## 6 Conclusions and Future Directions  
In the previous sections, we have shown the breakthroughs in this research field made in recent years. Now, we can regulate the elastic waves in different topological protection methods. However, the development of topological physics in mechanical systems still lag behind the electronic, photonic and acoustic systems. As a result, we will outline those future developments of topological mechanics as follows:  
(1) 3D higher-order topological insulators  
Strikingly, 3D HOTIs have gapped surface and hinge states and corner states in the gap. Even though implementing 3D HOTIs is a big challenge, it has been observed in acoustics [100; 162; 163; 164; 165] and electric circuits [166; 167] via the 3D tight-binding model of octupole TIs or the generalization SSH models [15]. We can refer to these advances to design 3D HOTIs in mechanics.  
(2) Higher-order topological pump  
More recently, the topological pump, the second central, has been observed in photonics [23; 129] and acoustics [133] in higher-order topological band theory. Phenomenologically, a higher-order topological pump could produce edge-to-edge and/or corner-to-corner transport in certain metamaterial platforms. Physically, the higher-order topological pump could be characterized by the second Chern numbers or the position of the Wannier centers versus the centers of their assigned unit cells. Consequently, we can investigate how to realize a higher-order topological pump in mechanics based on this discovery.  
(3) Fractional charges  
Fractional charges have been an important feature to recognize the topological crystalline insulators (TCIs), which offer another experimental identification when TCIs lack robust spectral features [144]. Recently, fractional charges have been observed at corners [168] and disclinations [144; 145] of the TCIs. Generally, fractional charges could be found exploiting the real-space localized Wannier representation of TCIs [19]. In addition, we can attempt to design elastic structures to trap fractional charges.  
(4) Non-Hermitian topological states  
The abovementioned achievements are investigated based on the hypothesis of the Hermiticity of a Hamiltonian. Recently, topological phases of non-Hermitian systems have attracted extensive research, whose exceptional points and complex-valued spectra can produce many notable properties, including the non-Hermitian skin effects [169; 170], bulk Fermi arcs [171] and Weyl exceptional rings [172], to name a few. At present, HOTIs of non-Hermitian systems have been realized in sonic crystals [173; 174]. As such, we can study how to connect the higher-order topological physics with non-Hermitian systems in mechanics by inducing friction.  
(5) Application implementation  
Even though there are plenty of achievements about undulation control using TIs in mechanics, the actual devices still have not appeared. We could explore how to apply these advances in practice, such as shrinking the mechanical structures to a micro-scale, designing low loss devices, manufacturing vibration sensors, signal processing, microparticle manipulations and so on.  
Writing--original draft preparation, S.Z. and G.D.; writing--outline of the paper, review and editing, supervision, funding acquisition, B.X. All authors have read and agreed to the published version of the manuscript.  
This research was funded by the National Natural Science Foundation of China, grant number 12072108, and the postgraduate Scientific Research Innovation Project of Hunan Province, grant number CX20210392.  
**Institutional Review Board Statement:** Not applicable.