You need to understand what syscalls are. On Linux, they are the lowest possible user land API (in contrast Win32 API probably mixes real kernel syscalls with some libraries functions. libc
also does such mix on Linux). fork(2), execve(2), open(2), pipe(2), mmap(2), read(2), poll(2), close(2), dup2(2), sigaction(2) are important syscalls (but there are about 300 of them, see syscalls(2) for a list, which depends upon your precise Linux kernel).
Don't expect each Windows functionality to be available on Linux (and vice versa).
Don't even think of such an equivalent.
Get a different mindset on Linux.
(In particular, processes are very different on Linux and on Windows).
Don't forget that Linux is free software, and you can dive into the source code of every function you are using on Linux. Read it, search it, improve it.....
Read the intro(2) man page first, and several other man pages (notably syscalls(2), intro(3) etc...). Read also e.g. Advanced Linux Programming and Advanced Unix Programming.
Some libraries try to factor out and provide a common abstraction for both Posix (e.g. Linux) and Windows. In particular Qt (and also Gtk or FLTK or POCO, and Wt for web applications, and sqlite for databases).
Some open source server software (e.g. lighttpd, exim, postgresql, etc...) can run on both Linux and Windows (of course, after recompilation)
If you are interested about graphical interface, understand the important role of X11 (notice that the X11 server is nearest to screen & keyboard; most graphical applications are X11 clients). In 2016 or 2020, X11 tend to be superseded by Wayland (but you won't notice that implementation "detail" - a really major one - if you code against Qt or GTK)
If you write an application using only Qt and/or POCO calls (those not documented as being specific to Linux or Windows) in addition of standard C++ functions, it should be source portable from Linux to Windows and vice versa.