Establishing "H&S culture" in an organization is the ultimate responsibility of HSE Manager or it applies on the whole management of that company?

The rate of success of any systems and culture depends on the level of involvement and commitment of the Management.  The HSE Manager is the Facilitator in establishing the culture and leading the organization through providing the necessary inputs , training, monitoring and control through audits of the processes to ensure compliance to needed safe practices, regulatory requirements and seek out proactive measures to run a risk free operations.  Health and Safety of self and of others is  every individuals prime responsibility and in the sense its the Effort and Participation of the whole management of the company to establish a successful H&S culture in an organization.

Whole Management is responsible for promoting H&S culture, but specifically HSE manager is more responsible because it's his duty to make sure that (a problem) does not occur.

We often say that safety is the line responsibility. Hence, it cannot be the ultimate responsibility of HSE manager only. Every line supervisors, every departmental managers are expected to lead in H&S culture. Add more, the top management commitment is must because we are talking about the culture and behavioral changes. 

I think HSE managers are facilitator, vigilant/policemen who will coordinate with all in an organization to build a safety culture. 

It is not his ultimate responsibility, but MUST have his and all UPPER MANAGEMENT support, up and down the information support team members.

It applies on the whole management

The Whole Management of The Company.

It applies TOP-DOWN in an organisation.

It might be considered as a sole responsibility of HSE manager to implement H&S culture. If it is so, then workers might feel pressured by HSE manager to comply with safety rules without positive example of their company leaders. H&S culture can be established easier if whole management team follow safety rules and procedures. It will state that complying with safety rules is important to that company as well as safety and health of employees. Workers will follow their leaders knowing that they care for them. So, implementing for good H&S culture is responsibility of everyone. 

To promote a safety culture of an organization that,s are not following properly HSE procedures are the prime responsibility and commitment of the top managements.

That depends on commitment of Higher Management towards HSE issues ,its responsibility of all staffs not only HSE Manager